robust pricing of options & optimal transportation · relationship between different sets of...
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Robust Pricing of Options & Optimal Transportation
James Thomas
St Anne’s College
Oxford University
A thesis submitted in partial fulfilment of the MSc in
Mathematical Finance
September 30th 2013
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0 Abstract
A robust methodology for pricing and hedging options looks to make as few as possible assumptions about the
behaviour of financial instruments and thus reduces the overall level of model risk inherent in classical modelling
methods.
A recent approach, suggested in the paper ‘Model Independent Bounds for Option Prices: A Mass Transport
Approach’, by Mathias Beiglbock, Pierre Henry-Labordere, and Friedrich Penkner, introduces a systematic
method for deriving model independent bounds on exotic options based on techniques used in solving classical
Monge-Kantorovich optimal transportation problems.
For an exotic path dependent option in a multi period discrete model, the primal formulation of pricing an option
as the expectation of a minimal martingale measure with given marginals is equated with a dual formulation
under a Kantorovich-style duality theorem, which has a financial interpretation as a semi-static subhedging
strategy.
This dissertation will review the proposed approach suggested in this paper, and examine the main result, that
there is no duality gap between the primal and dual formulation under certain conditions. We shall compare this
form of the duality to alternative Optimal Transportation approaches recently published in the literature, most
notably by Galichon, Henry-Labordere and Touzi in the article ‘A stochastic control approach to no-arbitrage
bounds given marginals, with an application to Lookback options’. In addition, we will evaluate bounds implied by
the optimal transportation approach using numerical methods in a two period trinomial model, as well as a more
general -period trinomial setting.
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Contents
0 Abstract................................................................................................................................ 2
1 Model Risk and Robustness ................................................................................................... 6
1.1 What is Model Risk? ....................................................................................................... 6
1.2 Model Risk, the Financial Crisis and Regulatory Response ................................................. 8
1.3 Model Risk and Exotics ................................................................................................... 9
1.4 ‘Classical’ Financial Mathematics .................................................................................. 10
1.4.1 Classical Market Framework.................................................................................. 10
1.4.2 Complete Markets and Replication ........................................................................ 11
1.5 Robust Pricing & Hedging ............................................................................................. 12
1.5.1 European digital option – robust pricing / hedging.................................................. 12
1.5.2 One touch digital option – robust pricing bounds ................................................... 13
1.5.3 Model Independent Pricing and Hedging – the methodology .................................. 14
1.5.4 Approach Summary – Classical v Robust................................................................. 17
1.5.5 Marginal Distribution of Final Stock Price – Illustration ........................................... 18
2 Monge – Kantorovich problems ........................................................................................... 20
2.1 Introduction & Context................................................................................................. 20
2.2 Basic Framework and Monge-Kantorovich Problem ....................................................... 20
2.2.1 Kantorovich’s Optimal Transportation Problem ...................................................... 20
2.2.2 Monge’s Optimal Transportation Problem ............................................................. 21
2.3 Simple Examples of Monge – Kantorovich type problems ............................................... 22
2.3.1 Example 1 – Mass at a single point......................................................................... 22
2.3.2 Example 2 – Multiple points of mass ...................................................................... 22
2.3.3 Example 3 - n discrete locations with equal mass.................................................... 23
2.4 Kantorovich Duality Overview ....................................................................................... 24
2.4.1 Kantorovich Duality Theorem ................................................................................ 24
2.4.2 Outline Proof of Duality Theorem .......................................................................... 25
2.5 Kantorovich Duality – Examples .................................................................................... 26
2.5.1 Example 1 - Dirac Measure at source.................................................................... 26
2.5.2 Example 2 - Duality with multiple points of mass .................................................... 27
2.6 Numerical Techniques for Optimal Transportation ......................................................... 27
2.6.1 Overview & Framework of Constrained Linear Optimisation ................................... 27
2.6.2 Simple Numerical Example – Optimal Transportation ............................................. 29
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3 Relevance of Monge – Kantorovich problems to Financial Markets ........................................ 31
3.1 Monge-Kantorovich and Financial Markets .................................................................... 31
3.1.1 Overview of Discrete Market Framework ............................................................... 31
3.2 Dual Formulation ......................................................................................................... 32
3.3 Duality Theorem & Optimal Transportation ................................................................... 34
3.4 Outline Proof of Duality Theorem for Discrete Time Markets .......................................... 36
3.4.1 Preliminary Results ............................................................................................... 36
3.4.2 Details of Proof of Duality Theorem ....................................................................... 36
3.5 Some comments on Martingale Optimal Transport Theory ............................................. 38
3.6 Alternative Frameworks for Monge-Kantorovich Problems ............................................. 39
3.6.1 Continuous Time Market Framework - Quasi-sure Hedging ..................................... 39
3.6.2 Duality in Continuous Time Framework.................................................................. 41
3.6.3 Continuous Time Market – Pathwise robust hedging .............................................. 43
3.6.4 Duality in Continuous Time Market – Pathwise set up ............................................. 44
3.6.5 Quasi-sure Hedging and Pathwise Hedging............................................................. 45
3.6.6 Summary of different Approaches ......................................................................... 47
3.7 Skorokhod Embedding Problem and connection to Optimal Transportation .................... 47
3.7.1 Overview of SEP ................................................................................................... 47
3.7.2 Connection to Optimal Transportation Problem ..................................................... 49
4 Simple Discrete Market Models ........................................................................................... 50
4.1 One Period Trinomial Model ......................................................................................... 50
4.1.1 The Standard Binomial Model ............................................................................... 50
4.1.2 Trinomial Model – an incomplete market............................................................... 51
4.1.3 Simple Numerical Example – Call and Put options in one period trinomial model ..... 53
4.1.4 Classical Financial Mathematics Approach – the Trinomial Method ......................... 55
4.1.5 Trinomial model with static trading in options and stocks ....................................... 57
4.2 Two Period Trinomial Model – Robust Hedging and Pricing ............................................ 58
4.2.1 Robust Hedging - Linear Programming Set Up ....................................................... 58
4.2.2 Robust Pricing – Linear Programming Set up .......................................................... 60
4.2.3 Path Dependent Options – Robust hedging and Pricing ........................................... 63
4.2.4 Additional Market Information – Market prices for Call Options Example ................ 66
4.2.5 Additional Market Information – Robust Hedging (i.e. minimum superhedging cost) 66
4.2.6 Additional Market Information – Robust Pricing ..................................................... 69
4.2.7 Two period Model – Multiple Constraints and market completeness ....................... 75
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4.2.8 Summary of two period Trinomial Model ............................................................... 81
5 N-period Trinomial Model & Conclusions.............................................................................. 82
5.1 Mathematical extension to N-period Model .................................................................. 82
5.2 Matlab Implementation................................................................................................ 87
5.2.1 Implementing the Model: Varying Parameters ....................................................... 87
5.2.2 Implementation Output: Results............................................................................ 92
5.3 Model Results .............................................................................................................. 93
5.3.1 Two period Trinomial Model – Exotic Options ........................................................ 94
5.3.2 Three period Trinomial Model – Exotic Options ...................................................... 95
5.3.3 Four period Trinomial Model – Exotic Options ........................................................ 98
5.3.4 Five period Trinomial Model – Exotic Options....................................................... 101
5.3.5 Some Remarks on the Limits of the algorithm ...................................................... 102
5.3.6 Illustration of Performance of Robust Pricing Algorithm for higher values of n ....... 102
5.3.7 Varying Parameters – Interest Rate...................................................................... 104
6 Conclusions and Further Research...................................................................................... 105
6.1 Summary of Dissertation & Conclusions ...................................................................... 105
6.2 Further Areas of Exploration ....................................................................................... 107
7 Bibliography...................................................................................................................... 108
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1 Model Risk and Robustness
1.1 What is Model Risk?
The classic Merton-Black-Scholes formula for option pricing is regarded as a seminal achievement in classical
Mathematical Finance, earning its surviving authors the 1997 Nobel Prize for Economics. Based on a set of
assumptions about the behaviour of the underlying risky asset and a set of market parameters i.e. ‘a market
model’, it provides a simple method to price and hedge vanilla call and put options. However in making the step
of applying this formula to the financial markets to price or hedge real financial instruments, we expose ourselves
to what is described as Model Risk.
Model Risk is the risk that a financial model used for pricing or risk management is an inappropriate, inaccurate
model for the reality of the financial markets it aims to describe. Any financial position based on that particular
financial model is consequently mispriced i.e. a mark to market value derived from the financial model is
substantially different from the value at which that position might trade.
Model Risk is related to the concept of uncertainty, as distinguished by Frank Knight in his classic 1921 text ‘Risk,
Uncertainty and Profit’ [21]:
‘Uncertainty must be taken in a sense radically distinct from the familiar notion of Risk , from which it has never been prope rly
separated.... The essential fact is that 'risk ' means in some cases a quantity susceptible of measurement, while at other times it
is something distinctly not of this character…..’
Within a particular model, we may use a particular risk measure to quantify the probability of an adverse outcome.
However, we face Knightian uncertainty when considering the question of whether we can be confident in our
model at all, or the true value of the parameters in our model. As such, we might question whether quantifiable
measures of risk (or pricing) based on a particular assumed model are in fact appropriate when facing the real life
complexities of the behaviour of financial markets.
There are numerous different ways in which a model can be an inappropriate or inaccurate model for financial
reality. In particular, two key ways are described below:
A) The model may be an inaccurate oversimplification of a complex external reality, misrepresent
relationship between different sets of variables or be inappropriately applied for a certain purpose.
One of the key assumptions in the Black-Scholes model referenced above is that the risky asset has a constant
volatility, whereas simple time series of standard deviations of log stock returns reveal volatility clustering / auto-
regressive features, and wide variation between volatility at different points of lifetime of a stock. Stock prices do
not follow Geometric Brownian Motion, as the basic Black-Scholes model assumes.
The existence of volatility surfaces in the market i.e. the different levels of implied volatility for varying strikes and
maturities, can’t be explained or captured in a simple Black-Scholes model framework, and requires the
development of local volatility or stochastic volatility models to explain these features.
Similarly, a particular financial model may inaccurately describe or predict the interaction between different
variables in a model. An example might be the use of Gaussian copulas to model correlation across underlyings
such as mortgage payments in different US States, which underpinned CDO pricing and hedging.
Another example might be the use of local volatility models, used to address some of the deficiencies of the
Black-Scholes by allowing calibration to volatility surfaces. Local volatility models permit such calibrations, and
thus reproduce vanilla option prices for a given maturity in a self-consistent arbitrage fashion. However as Hagan,
Kumar, Lesniewski and Woodward describe in [11]:
‘the dynamic behaviour of smiles and skews predicted by local vol models is exactly opposite the behaviour observed in the
marketplace: when the price of the underlying asset decreases, local vol models predict that the smile shifts to higher prices;
when the price increases, these models predict that the smile shifts to lower prices’ [12]
The result is that hedges based on local volatility models may perform worse than hedges based on the native
Black Scholes model, despite local volatility models being able to calibrate better to market data.
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The model suitable for one purpose may not necessarily be suitable for a different one. As Kienitz and Wetterau
in [20] highlight, stochastic volatility or jump diffusions models may capture features of stock price movement that
are important for pricing of exotic options; however they may not translate into actionable hedging strategies due
to the incompleteness of the market (e.g. the inability to hedge jumps through tradable market instruments).
B) There may be significant calibration issues, as the model may be overly complex; or may allow over-
fitting to a data series.
In attempts to address and more accurately model the dynamics of the underlying in financial markets, more and
more complex models with free parameters can be introduced to capture these features. However, the risk then
becomes that in an attempt to calibrate the model by fitting to a data series, the model fits to random noise rather
than data that accurately reflects the underlying process.
For example, in volatility modelling in discrete time, an ARCH model is built up by recursively defining volatility in
terms of previously established values to capture the observed auto-regressive nature of volatility in the financial
markets. An ARCH(q) model uses the q previous values of volatility to set the subsequent value, i.e.
where are parameters to be calibrated, are random normal increments, and
are the values of the
volatility. The danger with this structure is that by introducing such a large number of free parameters, we overfit
the model when trying to calibrate resulting in poor out-of-sample predictions.
Alternatively, some of the more exotic stochastic volatility models that have been proposed have additional
parameters and features that don’t necessarily improve modelling accuracy. Gatheral in [9] compares a
stochastic volatility model with jumps in the underlying (SVJ model) as well as a stochastic volatility model with
jumps in both the stock price and volatility (SVJJ). Modelling the volatility of volatility as a stochastic process with
jumps might seem intuitively sensible - if we allow the stock price to jump then surely the volatility would see a
step change at the same time. However, Gatheral concludes that:
‘the SVJ fits the observed implied volatility surface reasonably well……. [we] might wonder whether mak ing dynamics more
reasonable by including jumps in volatility as in the SVJJ model might generate surfaces that fit even better………. [however]
not only does the SVJJ model have more parameters than the SVJ model, but it’s harder to fit to observed option prices ’ [9]
Kienitz and Wetterau offer further examples of how calibration issues may cause model risk. Firstly, calibration
aims to minimise some distance measure between market and model prices, and so leads to a ‘best guess’ of
model parameters. Some models may be poorer than others in minimising the distance measure and therefore
have a higher degree of ‘model risk’.
Additionally, the time period chosen over which to perform the calibration may dramatically affect the value of the
model parameters. For instance, Kienitz and Wetterau highlight in particular the dangers of parameter instability
when calibrating a model to daily data, where despite the fact that observable market parameters are relatively
stable, we observe frequent jumps in the parameters through calibration.
A related example of being overly dependent on the time period used to calibrate the models comes from Value
at Risk models used in financial institutions to quantify market risk. These models are often based on what is
referred to as the ‘historical method’, where a historical data set is used to approximate a future distribution for
the various risk factors which drive the value of the overall portfolio. This requires financial institutions to decide
whether to use e.g. one year, three year, 5 year, or any other time period for the length of the historical data set.
However, with no theory to guide us, the choice of a timeframe or weightings seems arbitrary and yet can have
significant implications on the value calculated. Morgan Stanley’s 2012 Q3 results [23] provided a recent example
of the dangers of this approach. Following a change in the timeframes for their VAR methodology, the daily VaR
for the Credit Portfolio changed from £104m at a 95% confidence level in Q3 2011 to £69m on the new model at
the same confidence level – a significant difference for an arbitrary change in model assumptions.
Additional complexity in a model then, even if it’s introduction is driven by efforts to capture features that are
observed in the real world such as volatility clustering, can result in poor predictive properties of a model from
over-fitting or being harder to calibrate to observed prices.
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Finally, it is worth noting that issues of Knightian uncertainty and model risk are not limited to pricing and hedging
in financial markets. Robust control in economics relates to understanding decision rules for economic agents
that perform well across a variety of different models, where models in the case generally relate to a specification
of a probability distribution over outcomes that are of interest to the modeller. The selection of a particular model
introduces model risk, so we instead might consider a range of different alternative models and aim to evaluate a
decision rule across this wider set. Instead, we might look to minimise a worst case scenario over a set of models.
Approaches like that developed by Gilboa and Schmeidler in [10] aim to describe how uncertainty aversion might
affect how economic agents make decisions in such situations.
1.2 Model Risk, the Financial Crisis and Regulatory Response
Many commentators have highlighted a poor understanding of model risk as a key factor in the problems
underpinning the financial crisis from 2007 onwards, as well as other well publicised financial losses. For
example, the use of the Gaussian copula in models used to price CDO’s was highlighted in articles such as
‘‘Recipe for Disaster: The Formula That Killed Wall Street’ [28], where the author Salmon writes:
‘Li's Gaussian copula formula will go down in history as instrumental in causing the unfathomable losses that brought the world
financial system to its knees’ [28]
The role of Value at Risk in over simplifying risk management (and potentially increasing the risk a financial
institution is exposed to) has been highlighted by commentators such as Taleb in [31]:
‘My first encounter with the VaR was as a derivatives trader in the early 1990s when it was first introduced. I saw its
underestimation of the risks of a portfolio by a factor of 100 …. Worse, there was no way to get a handle on how much its
underestimation could be…’ [31]
Taleb continues by highlighting deficiencies of VaR including having an ‘anchoring’ behavioural impact on risk
taking, as well as encouraging trading strategies that can be characterised as delivering a series of frequent
small gains (which deliver traders substantial bonuses) but occasional huge blowouts (despite which bonuses are
not clawbacked).
Other well respected commentators have themselves highlighted the failure of quantitative financial models as a
contributing factor to the financial crisis. Lord Turner in the ‘Turner Review – Regulatory Response to the
financial crisis’ [33], writes of the ‘misplaced reliance on sophisticated maths’ and references Knight’s distinction
between risk and uncertainty, writing:
‘More fundamentally, however, it is important to realize that the assumption that past distribution patterns carry robust
inferences for the probability of future patterns is methodologically insecure….. instead, we need to recognise that we are
dealing not with mathematically modellable risk , but with inherent ‘Knightian’ uncertainty’ [33]
However, despite the identification of model risk as a contributing factor to the financial crisis by both regulators
and commentators, the current regulatory framework is structured in a way that arguably encourages a greater
level of model risk in the financial markets.
In particular, introduced as part of the Basel II regulatory framework and retained as part of Basel III, banks have
a choice on how to calculate the overall level of market risk or credit risk they are exposed to, and thus how much
capital they must hold. For market risk, the first alternative for banks to calculate the overall size of this risk is
through use of a standardised method that uses published percentages applied on particular defined financial
instruments to define an ‘overall market risk’ level e.g. an 8% risk charge is applied to value of bonds rated BB+
to B-. The second alternative is to use an ‘Internal Models’ approach where banks are able to use their own
internal models (based on 99th
percentile VaR) to calculate the risk banks are exposed to. A similar framework is
used for Credit Risk, with banks offered a choice to use either a standardised model or an Internal Ratings based
approach.
The framework offers a perverse set of incentives. Aggressive banks may aim to reduce overall capital
requirements to boost shareholder returns; this can be achieved by avoiding a standardised approach and
building more sophisticated internal models that reduce capital requirements. The system is fundamentally non–
robust; the incentives are to move from a set of objective measurements of risk to a set of model-dependent risk
measures. As per the Morgan Stanley example described above where we saw the impact of a change of
assumptions in the VaR calculation, this introduces a substantial level of model risk into the system.
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In commenting on the introduction of the concept of the regulatory trading book and permitting banks to use
internal models to calculate regulatory capital against market risk, Andy Haldane puts it thus in [12]:
‘With hindsight, a regulatory rubicon had been crossed. Internal risk models were allowed as a means of calibrating credit ri sk.
….. For a large, complex bank, this has meant a rise in the number of calculations required from single figures a generation ago
to several million today. That increases opacity….. This degree of complexity also raises serious questions about the
robustness of the regulatory framework given its degree of over-parameterisation. ’
Both Haldane and Taleb agree on a potential way forward – to introduce greater ‘robustness’ into the system.
Taleb writes explicitly in terms of using simple robust measures to regulate capital requirement in [31]:
‘Regulators should understand that financial markets are a complex system and work on increasing the robustness in it…. This
implies reliance on "hard", non-probabilistic measures rather than more error-prone ones. For instance "leverage" is a robust
measures (like the temperature, it does not change with your model), while VaR is not ’.
In this context, a measure is ‘robust’ if it is not exposed to model risk i.e. if the value of the measure is
independent of particular assumptions we might make in a model. It may be a directly observable parameter, or
immediately derivable from one, or simply be derived from a reduced set of assumptions.
In summary then, there is a growing recognition that model risk is hugely important in ensuring the stability of the
financial system, and that the current regulatory framework is not optimised in a way to reduce it, and may even
encourage it. Ensuring robustness in the models and metrics used to manage their businesses is a key challenge
for financial institutions.
1.3 Model Risk and Exotics
The issues of model risk and robustness are particularly acute when considering exotic options. Beiglbock,
Henry-Labordere and Penkner state the problem succinctly in [1]. Noting that numerous alternative models exist
for pricing exotic path dependent options, they state:
‘These models depend on various parameters which can be calibrated more or less accurately to market prices of liquid options
(such as vanilla options). This calibration procedure does not uniquely set the dynamics of forward prices which are only
required to be (local) martingales according to the no-arbitrage framework. This could lead to a wide range of prices of a given
exotic option when evaluated using different models calibrated to the same market data. ’ [1]
In other words, the constraint of having to calibrate a model to the market price of options is not sufficient to
choose between various different stochastic models, as a wide number of these models can successfully
reproduce the terminal distribution for risky assets that is derivable from a given maturity of vanilla call option
prices (we shall see later through the Breedan and Litzenberger Lemma how this is done). However, since all
these calibrated models assume different underlying stochastic processes that impact on the price of a path
dependent option, it is not the case that the marginal distributions of the path dependent (e.g. associated
maximum or minimum) processes will be the same for each model. This leaves us exposed to a high level of
model risk as the various models (calibrated to the same benchmark data) calculate different prices for the path
dependent exotic options.
This issue is illustrated by Schoutens, Simons and Tistaert in [29]. They take seven different models standardly
employed in the market to price exotic options, calibrate these models to market prices of vanilla call options, and
calculate the resulting values for a set of exotics. The resulting values have wide variation depending on the
model employed, highlighting that the choice of model is hugely important in pricing / hedging options. The
models tested include three standard stochastic volatility models (the Heston model; the Heston model with
Jumps; Barndor-Nielsen-Shephard Model); and four Levy process models with stochastic time change
(considering two Levy processes - Variance Gamma and Normal Inverse Gamma – and two stochastic clocks –
Cox-Ingersoll-Ross (CIR) and Ornstein-Ulenbeck (OU)).
Schoutens, Simons and Tistaert successfully calibrate all these different models to the same set of vanilla options
using the characteristic function method. They note that all of the models can be adequately fitted to the same set
of market data, including any smile-conforming properties; and conclude there is limited basis for choosing the
‘correct’ model based purely on ability to fit to the market data alone.
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Having calibrated these diverse models, they then calculate a set of prices from each model for certain path-
dependent exotic options and compare results. The table below shows the calculated prices for a Lookback
option under each model.
Stochastic Volatility Models Levy
Model used to price
exotic: Heston
Heston with
Jumps
Barndor-
Nielsen-
Shephard
Variance
Gamma –
CIR
Variance
Gamma –
OU
Normal
Inverse
Gamma –
CIR
Normal
Inverse
Gamma –
OU
Price of Lookback Option 838.48 845.19 771.28 724.80 713.49 730.84 722.34
Table 1.3.1: Lookback options, priced under various models cf. [29]
Looking at this table we see that the Lookback option prices calculated are not robust – i.e. there is substantial
variation between prices depending on the model chosen. The results are worse for the other exotic options the
authors consider:
‘Lookback prices vary over about 15 percent, the one-touch barriers over 200 percent, whereas for the digital barriers we found
price differences of over 10 percent. Finally for the cliquet premiums a variation of over 40 percent was noted’ [29]
These results have been replicated in similar studies by e.g. Kienitz and Wetterau in [19, Chapter 10]. The
conclusion therefore is that even if we calibrate models to the same set of market data, we will be left with a wide
range of exotic option prices, demonstrating the extreme lack of robustness that is present in the pricing of
options even when using a supposedly ‘advanced’ pricing model.
1.4 ‘Classical’ Financial Mathematics
In this section, we start to examine in a little more detail the standard approach to financial mathematics, and
begin to examine the reasons why this approach does not necessarily lead to a robust output in terms of prices
and hedges.
The classical approach typically proceeds as follows. A stochastic model is postulated for an underlying risky
asset, with dynamics often prescribed in order to capture some observed feature of that underlying (e.g. mean
reverting stochastic volatility). Expressions for prices of financial instruments are derived based on discounted
expectations under a risk neutral measure. We then reverse engineer and fix free parameters in the stochastic
model through calibration to market prices of vanilla options. Calibrated models are then used to price or hedge
more exotic instruments, and recalibrated to market prices at appropriate frequent intervals.
1.4.1 Classical Market Framework
To describe this more precisely, we shall follow Obloj [25] and describe the inputs and assumptions in the
classical approach. In particular, in terms of ‘inputs’ we have:
Beliefs – a set of assumed dynamics for risky assets in the market, semi martingales on a probability space with
a filtration and a particular empirical probability measure .
Information – market quotes on financial instruments (typically, vanilla options) which are used to fix the free
parameters in the stochastic model
Rules – we can adopt a self-financing trading strategy with no transaction costs, between the risky asset and a
risk free asset, the gains from which are described by a stochastic integral of the form:
∫
where ( ) are the discounted portfolio (stock) value, and the amount of stock held at time t.
Underpinning these inputs, under the classical approach we then have a set of what Obloj [25] calls ‘reasoning
principles’, primarily:
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Efficient Markets – The classical approach assumes markets are efficient i.e. there are no arbitrage
opportunities in the market. We then make use of the Fundamental Theorem of Asset Pricing that asserts that no
arbitrage in the market is equivalent to the existence of a risk neutral measure equivalent to (see for
example, Bjork [2] Chapter 10). Under this measure discounted stock prices for risky assets are martingales
Under the classical approach, given these ‘inputs’ and ‘reasoning principles’ we can use self-replicating portfolio
arguments and risk neutral expectations to derive prices and hedges for financial instruments; as well as apply
these classical principles to other areas such as portfolio optimisation and risk management.
For an illustration of this, we have the below assumed dynamics of a risky asset under the risk neutral measure
(i.e. it is a martingale):
for some (discounted, or assuming zero interest rates) stock price process , volatility process , and a
Brownian motion.
We proceed by articulating our beliefs about the behaviour of this volatility. Our beliefs about what dynamics to
choose might be influenced by the need to capture some behaviour of the underlying itself (e.g. mean reversion,
a smile or skew effort across different strikes); but also may be influenced by factors not related to the underlying
itself – for example, the existence in our model of closed form solutions or the ease of calibration to market data.
In particular, in the case above we might choose:
a) , for some constant; the standard Black Scholes model
b) i.e. the volatility process is a deterministic function of the random asset price process and time; a
local volatility model
c) ; for some random process ; i.e. a stochastic volatility model.
We can take these models further in order to capture features of volatility in the markets that aren’t currently
included; for example by the inclusion of jumps in the underlying dynamics or jumps in the volatility process (as
we referenced in Section 1.1 with Gatheral in [9]). Finally, then we use our rules to determine prices for assets
through calculating risk neutral expectations, or calculating cost of self-financing replicating portfolios. Under the
classical model, market frictions such as transaction costs, liquidity problems are typically not included.
1.4.2 Complete Markets and Replication
We briefly recap the theory of self-financing replication and risk neutral markets in the standard Black Scholes
market framework, under which the risky asset moves under Geometric Brownian. The Brownian motion is the
only source of randomness in the model, and the market can be shown to be complete i.e. the payoff of every
contingent claim can be replicated through a self-financing portfolio containing the stock and risk free bond. This
is shown in outline below.
We saw above that the self-financing condition can be written as (where represents value of a portfolio under a
particular trading strategy, and represents the discounted portfolio value):
∫
We use the risk neutral formula to define the value of the contingent claim with payoff (which is
measurable, where is the filtration generated by the Brownian motion ). In other words, we set:
[ |
Then this process is a martingale (by iterated conditioning) and we can apply the Martingale Representation
theorem to get the expression:
∫
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Since we require = to ensure replication of the payoff, then no arbitrage arguments impose that:
and in particular, = = . We can therefore write the hedging strategy in the below form, using
to derive :
∫
∫
∫
With this representation, we can hedge any contingent claim in this market, and thus the risk neutral method of
pricing, defined as , is equivalent to saying that the price of an option is the initial capital required
to set up a self-financing portfolio that replicates the option payoff.
1.5 Robust Pricing & Hedging
As discussed in Section 1.1, the upfront postulation of a model in the Classical Framework can lead to situations
where we are exposed to a high level of model risk. Having reviewed the classical approach to financial
mathematics, and articulated some of the problems facing it, the objective is to propose a framework that
addresses some of these challenges – particularly around reducing the level of model risk that we are exposed to.
As referenced in Section 1.2, a robust pricing or hedging methodology is one that is independent of the model
selected to price or hedge a financial instrument. We start with a quote by Hobson in [17] who describes the
approach a ‘robust’ methodology’ might take. After describing the classical approach and pointing out the risk that
models that are all consistent with market prices may produce different prices, he says:
‘Instead, one might attempt to characterise the class of models which are consistent with the market prices of options. This is a
very challenging problem, and a less ambitious target is to characterise the extremal elements of this set, and especially the
models for which the price of the exotic is maximised or minimised’
In other words, we accept that there are a number of stochastic models and probability measures that are
consistent with the market prices of options. But instead of trying to pick one of these (like the classical approach
does), we look at the extreme elements of these sets in order to say what constraints (in terms of upper and
lower bounds) these elements put on prices and hedges, without claiming reliance on a particular stochastic
model that underpins these prices.
Since a robust pricing methodology is independent of a specific model, we have made fewer assumptions about
the behaviour of the financial instrument in the market than compared to the classical financial mathematics
approach. As such, the model risk is dramatically reduced. The trade-off however is likely to be a lack of
exactness i.e. we are not looking to calculate actual values for prices as before, but upper and lower bounds on
these prices. So the broader question for the robust methodology is to whether the benefits of reduced model risk
outweigh the drawback of not being able to be as specific on price as the classical approach, and whether in fact
these bounds will be practically useful in the market or of purely theoretical interest.
We will reconsider the overall robust methodology in more detail in section 1.5.3. First, we will look to illustrate
the concept of robustness through two simple examples, the first a European digital option and the second a
digital one touch option.
1.5.1 European digital option – robust pricing / hedging
In this first example, we will see how it is possible to price exactly some more exotic option payoffs without
postulating a stochastic model, and instead using the principle of replicating portfolios and the assumption of no
arbitrage in the market.
13
Following Crosby [5], a digital European option (or binary cash or nothing option) has a payoff of one dollar if the
end stock price is greater than the strike, and pays out zero otherwise. Equivalently we can write the payoff as
, where is the indicator function with value 1 if .
We can consider the following replicating portfolio: for , take a long position in
of vanilla call options
with strike , and a short position in
vanilla call options with strike . Consider now the
payoffs of this replicating portfolio:
( )
In other words, as , we see that the portfolio described above exactly replicates the payoff for a digital
European option. If we write ( ) for a vanilla call option with strike K and implied volatility , we get:
As a point of note,
is the vega of an option multiplied by the volatility skew. We have therefore derived a
price for a European digital option (in the presence of volatility skew) without making any assumptions on the
dynamics on the underlying, only requiring the no arbitrage principle ensures that a portfolio with the same payoff
as an option will have the same value as the option prior to maturity.
1.5.2 One touch digital option – robust pricing bounds
The second example follows Hobson in [17] in developing robust price bounds for a particular exotic option,
rather than an exact price as in the previous example. The method is to use a semi-static replicating portfolio of
vanilla call options, along with the use of a self-financing trading strategy (in this case, using forward
transactions) to derive a pathwise inequality for the payoff, and then apply the principle of no arbitrage to claim
that the cost of setting up this portfolio must be an upper bound for the price of the option.
Hobson considers a digital one touch option, where the option pays out 1 if at any point prior to maturity the stock
price exceeds a certain level B. In other words, the payoff is written as:
{ } { }
Hobson notes the following pathwise inequality ((i.e. inequality which holds) along all possible stochastic paths),
valid for :
( )
We can easily verify this, by considering the possible combinations of values at time T;
We can also give a financial interpretation of the above inequality: the first term on the RHS of the inequality
is equivalent to having
call options at strike K; and the second term on the RHS
( )
is
14
equivalent to entering
units of a forward transaction in X the first time that the barrier B is crossed (if ever)
(assuming zero interest rates).
Hobson notes that we therefore have a super-replicating strategy for the one touch digital option, where super-
replicating means the portfolio will also have value greater than or equal to the option payoff at maturity. Note that
the strike was arbitrary, and given that the forward transaction is costless, we have that (where denotes the
price of the one touch option, and denotes the price of call option strike K):
As in the previous example in Section 1.5.1, we have not relied upon the dynamics of an assumed stochastic
model to derive a price (or in this case, an upper bound on price from a super-replication strategy); instead we
have set up a static hedging strategy in both instances where the payoff is equal to (or bounded by) our portfolio.
We have then applied the principle of no arbitrage to derive the fact if the payoffs are equivalent (or related by an
inequality), the value of the option prior to maturity must be equivalent (or less than) the value of the portfolio.
The resulting value then is robust – as long as no arbitrage holds in the market, regardless of the actual
stochastic dynamics of the underlying, we have an accurate price / bound for the exotic option.
1.5.3 Model Independent Pricing and Hedging – the methodology
Having then illustrated the concept of robust upper and lower bounds of prices in the above examples, we can
start to build a more formal framework within which we can develop a robust pricing and hedging methodology
following the outline we gave in the introduction to Section 1.5. There are several different ways we might look to
do this.
Robust Pricing – Weak Constraints on the probability measure
In Section 1.4, we outlined the classical financial model for the markets, and used the framework of a process
with dynamics under the risk neutral measure of:
i.e. the stock price is a Martingale under the risk neutral measure. In this framework, the critical step was in
deciding how to make the choice of appropriate dynamics for the volatility process , and this was what led us
into the issues around model risk on the classical approach.
Following Touzi [32]. we can aim to proceed without making any assumption about the volatility process i.e. we
allow that it is unknown. Say we have an exotic option, and let be price process for an exotic defined as
. Following Hobson’s suggestion in [17] which was quoted earlier in Section 1.5, it is natural
to consider the notion of a robust price as the ‘extremal points’ of the set of models that are consistent with
market prices. We are led then to introduce ‘robust’ bounds for the price of this option as:
where ranges over all volatility process such that martingale constraint is satisfied.
This price bound is ‘robust’ in the sense that whatever particular volatility function we choose, the upper price for
an option with payoff will be greater than the value of the price derived from any particular model (as the
supremum will range over a set that includes the value of for the particular chosen under the model.
As per Hobson’s suggestion, our ‘robust price bounds’ are the extremal elements of a particular set i.e. the set of
models where the martingale constraint for the price process is satisfied.
However, without further constraints on the measure that is chosen, the bounds from this ‘weak constraint’ may
be trivial or too wide to be of use. Furthermore, we may be able to provide tighter bounds by considering
additional market information that may act as a stronger constraint on this initial robust price bound. We shall
discuss this in theory in the next section (and shall see a numerical demonstration of it in Chapter 4 and 5).
Robust Pricing – Stronger Constraints on the probability measure
15
We therefore are looking for additional constraints which will allow us to produce tighter upper and lower price
bounds for the exotic options. In this approach, we now look to use the observed market option prices in order to
further constrain the eligible set of martingale measures.
Without postulating a model we can derive from the market some information about the process and therefore
the set of eligible martingale measures. Informally, we can differentiate the risk neutral valuation formula with
respect to the strike to determine an expression for the marginal distribution at maturity of the underlying stock
price implied by the call option. This is stated formally as below (from Hobson [17]):
Lemma 1.5.3 - Breeden and Litzenberger Lemma (Hobson [17])
For fixed maturity , assume that European call option prices are known for all strikes , then assuming
call option prices are calculated as the discounted expected payoff under a model :
Then we have (assuming is twice differentiable with respect to K):
i.e. the marginal distribution of under is known
So we can derive a formula for the risk neutral density of the stock price at time T if we know option prices for all
strikes at that maturity T. This Lemma 1.5.3 – the Breeden and Litzenberger Lemma – is an absolute key result
which we shall reference frequently as we develop the robust information. Essentially, it is a means of translating
extraneously given market information into constraints on a probability measure and model.
When combined with the Kolmogorov backward equations and assuming we have a complete set of European
option prices for all strikes as well as all expirations, this equation can be used to derive an expression for the
local volatility in terms of call prices for different strikes (Dupire’s Equation see e.g. Gatheral [8, Chap 1]). So
while we still don’t know the dynamics of the process , we do know the marginal distribution of under .
So we have some additional constraints on the risky asset price process i.e. the marginal distribution of the stock
price after time T must satisfy the above equations relating to the market price of call options for all strikes. We
denote the law of under as (i.e. ) and define as the set of all martingale measures
such that the marginal distribution of is equivalent to the law :
{ }
Then we can redefine robust price bounds on the value of the exotic option with an additional constraint, still
based on Hobson’s idea in [17] of the extremal elements of the set:
The preceding argument can be extended to more than one maturity. If we know call prices for all strikes for all
maturities, then we can redefine the set of allowable measures as { } and the
above bounds will be redefined similarly. These bounds are sometimes referred to as upper, or lower, martingale
prices.
We see then that one potential route to finding robust price bounds is through finding infimums and supremums
of expectations over probability distributions that are constrained by Lemma 1.5.3 applied to the market prices of
call options. This observation provides the basis for introducing the Monge-Kantorovich optimal transportation
framework, which we shall outline in Chapter 2.
Robust Hedging – Minimal Super replication costs
In Section 1.3 we briefly reviewed the perfect replication that is achievable in a complete market in the classical
approach, and noted that the price of an option is equal to the initial capital required to set up a portfolio that
16
replicates exactly the payoff of the option. In the classical framework, under the assumption of Geometric
Brownian motion in the typical Black Scholes market framework and an arbitrage free market, not only does a
risk neutral measure exist (the first Fundamental Theorem of Asset Pricing), but also that this risk neutral
measure is unique and therefore the market is complete (the second Fundamental Theorem of Asset Pricing).
We saw in Section 1.3 that we could provide an explicit representation, through a stochastic integral representing
a self-financing trading strategy that gave a perfect hedging strategy for a contingent claim.
From the perspective of a robust framework, without assuming the Fundamental Theorem of Asset Pricing, we
can appeal to principles of no-arbitrage to set limits of prices of exotic options through construction of super (or
sub) replicating portfolios.
In other words, an alternative approach to deriving robust bounds for price is to look for super and sub replicating
portfolios for an exotic options payoff, and to appeal to no-arbitrage principles to justify that the costs associated
with setting up these super / sub hedging portfolios must be bounds on the price of the option. For if this is not
the case we have a simple arbitrage. For example, if the price of an exotic option is strictly greater than the
cost of setting up a super hedging portfolio i.e. , then we sell the exotic option at the initial time for and
use the proceeds to set up a super hedging portfolio costing , with then remaining. Then at maturity
the long position in the super hedging portfolio covers the exotic option payoff by assumption, and we have
locked in a profit of at least . The principle of no-arbitrage then implies that superhedging costs function as
robust bounds on price of an exotic option.
This approach of hedging portfolios was used in the examples in sections 1.5.1 and 1.5.2. We set up a market
framework, and within that framework we look to define admissible trading strategies and super replication. We
can then set an upper bound to the price in terms of it being the minimal cost of setting up a super replicating
portfolio.
Depending on our market framework, we might set different restrictions on admissible trading strategies in the
market. In particular, we may allow the following:
1) Dynamic trading, under a self-financing constraint, of a set of risky assets, or costless forward transactions
2) Static trading of a set of vanilla call options
Galichon, Henry-Labordere and Touzi in [8] describe super replication in a probabilistic framework, where they
assume interest rates are zero. We briefly outline this framework below, and we will consider in more detailed in
Section 3.6.
Briefly, firstly consider a market where only dynamic trading of an underlying risky asset is allowed, and for a
portfolio process we have the same self financing portfolio value process as in the classical framework:
∫
where the integral is defined with respect to some measure , defined earlier. The model free super-replicating
bound is then described as:
{ }
They then introduce the possibility that we are able to statically trade vanilla call options, with a single maturity
for all strikes . By Lemma 1.5.3 (the Breedan and Litzenberger lemma), we can determine the - marginal
distribution for the stock price, denoted by (where denotes the set of all probability measures
on ). Then for a derivative with payoff (where ) we have the no-arbitrage price of that derivative
as:
∫
This result is not immediately obvious so we briefly provide some justification. Galichon, Henry-Labordere and
Touzi in [8] reference the existence of a replicating portfolio of down and in Arrow securities at all strikes for any
17
payoff , as outlined in Carr and Chou [3]. In this paper, the authors note that butterfly spreads with vanilla
call options form payoffs which approximate Dirac functions in the limit i.e. a butterfly spread of the form below:
has payoff that tends to the Dirac function as . In this way (or using Down and In barrier options which can
be statically replicated from vanilla calls), we can form what are described as Arrow-Debrau securities, which
have value (from above equation in the limit) of
. By then buying and holding a portfolio of Down and In
Arrows at all strikes, we can replicate the arbitrary payoff exactly (by holding of arrows at strike ).
Since we have already seen that
through Lemma 1.5.3, we can use this along with
principles of no arbitrage to argue that the no-arbitrage price of that derivative is ∫ as
referenced above.
With this above equation established, we can then define the enhanced model free super replicating bound as:
{ }
In other words, we define the model free super replicating bound in this market as the minimum initial capital
required to set up a portfolio that super replicates the option payoff, where that portfolio includes both a set of
self-financing transactions on risky asset ( ∫
) but also a portfolio of call options that cost
to set up and has payoff .
We have seen then two different approaches which we might consider in determining a ‘robust price’. Firstly,
following Hobson’s suggestion in [17], we can consider the ‘extremal elements’ of a set and consider the robust
price as the supremum over eligible martingale measures of the expectation of the payoff under that measure.
Secondly, the principle of no arbitrage ensures that the ‘robust hedging cost’ (defined as the infimum of the set up
cost of a super replicating portfolio) is an upper bound on the price.
The obvious question then is whether two values derived from these two approaches are equal i.e. does some
form of duality relation hold between them. We shall return to this question in Chapter 3 (answering it in the
affirmative) after first reviewing in Chapter 2 Monge-Kantorovich problems and some of the associated machinery.
1.5.4 Approach Summary – Classical v Robust
Referring back to Obloj’s framework in [25], we can summarise the Inputs and Reasoning Principles involved in
the model independent approach and compare this to the classical framework.
Classical Financial Mathematics Model Independent Approach
Inputs Beliefs – a set of assumed dynamics for risky
assets in the market, semi martingales on a
probability space with a filtration and a
particular empirical probability measure .
Information – market quotes on financial
instruments (typically, vanilla options) which
are used to fix the free parameters in the
stochastic model
Rules – we can adopt a self-financing trading
strategy with no transaction costs, between the
risky asset and a risk free asset
Beliefs – no assumed dynamics for risky asset, set of
possible measures
Information – market quotes on financial instruments (call
options) used to determine marginal distributions of stock
prices at maturity of options
Rules – we can adopt a self-financing trading strategy with
no transaction costs, between the risky asset and a risk free
asset. Generally assume working with forward price
processes or that interest rates are zero.
Reasoning
Principles
Efficient Markets – assuming that there are
no arbitrage opportunities in the market, we
use the Fundamental Theorem of Asset
Pricing that asserts that no arbitrage in the
market is equivalent to the existence of a risk
neutral measure Q equivalent to , and under
this measure discounted stock prices for risky
Efficient Markets – as per Classical model, we assume that
there are no arbitrage opportunities in the market. No
arbitrage principles can be used to demonstrate that super /
sub hedging strategies are robust bounds on price.
Price bounds can be viewed as extremal elements of sets of
models that are consistent with extraneously given market
18
assets are martingales prices of vanilla call options
Table 1.5.1 – Comparison of Classical Financial Mathematics and Model Independent Approach
1.5.5 Marginal Distribution of Final Stock Price – Illustration
In summary then, the key difference between the classical approach and the robust approach is how the robust
approach uses market prices of vanilla options to determine constraints on the risk neutral measure used to price
more exotic options, whereas in the classical approach the stochastic dynamics of the risky underlying are
assumed and then parameters calibrated using market call prices.
In particular, the Breeden and Litzenberger Lemma plays a critical role in converting market information (call
option prices at a continuum of different strikes) to the marginal distribution of the stock price at maturity of those
options. In this section we will give a very simple illustration of the concept behind this lemma by demonstrating
its validity in an idealized market.
We shall consider the Black-Scholes market, in which the risky asset is assumed to move under Geometric
Brownian motion. Under the risk neutral measure, this has assumed dynamics:
where
is Brownian motion under the unique risk neutral measure . Using Ito’s Lemma applied to , we
derive an alternative representation as:
(
)
Since is normally distributed (with mean and variance ) we can then immediately identify that is also
normally distributed, with:
(
)
In other words, the stock price is log-normally distributed, with the mean and variance of given by the
above expressions.
One of the key assumptions behind the Breeden and Litzenberger lemma is that European call option prices are
known for all strikes . In the idealized market in the Black Scholes framework, we have the standard
Black Scholes formula for the price of a call option for a given maturity T. This allows us to generate a series of
option prices for a continuum of strikes, and then apply the Breeden and Litzenberger lemma to derive the
distribution of the stock price at maturity T. In this instance, we have a method to validate that the derived stock
price distribution is correct, as we can compare the distribution derived from the Breeden and Litzenberger
lemma with the lognormal distribution which we saw was implied directly by consideration of the stock price
dynamics under the risk neutral measure.
This is done in the below Figure 1.5.1. Figure 1 on the top left hand side shows the option prices for a series of
strikes calculated using the Black Scholes formula. Figure 2 shows a graph of the first derivative (with respect to
the strike K) of , which by Breeden and Litzenberger is the same as the cumulative distribution function for
the stock price at time T. Figure 3 then shows a graph of the second derivative (with respect to the strike K)
of , which by the Breeden and Litzenberger Lemma is the probability density function of the stock price at
time T. Figure 4 then shows the probability density function derived directly from the dynamics of Geometric
Brownian Motion i.e. log-normally distributed with the parameters described above.
The key point is that the distributions in Figure 3 and Figure 4 are identical, demonstrating that the marginal
distribution of the stock price derived through the Breeden and Litzenberger Lemma is indeed the distribution we
expect via direct consideration of the assumed dynamics.
19
Figure 1.5.1 – Illustration of Breeden and Litzenberger lemma in the Black Scholes market. Figures calculated in Matlab.
Parameters: r =0.05; sigma = 0.2; T = 2; S = 100. Black Scholes market, Geometric Brownian Motion
.
As opposed to the constant volatility assumptions in the Black-Scholes market model, real financial markets
exhibit implied volatilities that vary across strikes, typically in either a negative or a positive skew. The
subsequent diagrams, in Figure 1.5.2, demonstrate the implied risk neutral probability distribution for the stock
price at maturity where volatility exhibits these characteristics. From examining the charts, we see that the
negative skew (green line in Figure 1.5.2) has a fatter left tail than the constant volatility lognormal distribution –
i.e. there is a greater probability of extreme events than lognormal distribution implies. This translates to a higher
price for a deep out of the money put compared to the constant volatility price.
The key observation underpinning the robust approach is that the Breeden and Litzenberger Lemma holds even
in cases such as these where implied volatility varies; the only assumption that is made is that there prices can
be expressed as discounted expectations under a risk neutral measure.
Figure 1.5.2 – Lemma 1.5.3 (Breeden and Litzenberger Lemma) applied to volatility at different levels.. Example Negative and
Positive implied volatility skews (of the form
where for negative skew, and where for positive skew).
Call option prices calculated using Black-Scholes formula.
0 50 100 150 2000
20
40
60
80
100Fig 1 - Strike against Call Price (from Black-Scholes equation)
Strike
Option P
rice
0 50 100 150 2000
0.2
0.4
0.6
0.8
1Fig 2 - Stock Value against d/dK Call Price
Stock Value
Cum
ula
tive D
istr
ibution F
unction
0 50 100 150 200
0
0.01
0.02
0.03Fig 3 - Stock Value against d2/dK2 Call Price
Stock Value
Pro
babili
ty D
istr
ibution F
unction
0 50 100 150 200
0
0.01
0.02
0.03Fig 4 - Stock Distribution from Geometric Brownian Motion
Stock Value
Pro
babili
ty D
istr
ibution F
unction
0 50 100 150 2000
0.1
0.2
0.3
0.4Volatility against Strike
Strike
Implie
d V
ola
tilit
y
Constant Vol
Negative Skew
Positive Skew
0 50 100 150 2000
20
40
60
80
100Call Option prices
Stock Value
Option P
rice
0 50 100 150 2000
0.2
0.4
0.6
0.8
1Stock Value against d/dK Call Price
Stock Value
Cum
ula
tive D
istr
ibution F
unction
0 50 100 150 200
0
0.01
0.02
0.03Stock value against d2/dK2 Call Price
Stock Value
Pro
babili
ty D
istr
ibution F
unction
20
2 Monge – Kantorovich problems
2.1 Introduction & Context
In Chapter 1, we considered some issues associated with the classical approach to financial mathematics,
primarily related to model risk. At the end of the chapter, we outlined the beginning of a different approach that
would be more robust to model misspecification. In particular, we noted that we could use market prices of call
options to give information about marginal distributions of stock prices, and set upper and lower bounds of exotic
option prices. In this chapter, we will outline the mathematics of Optimal Transportation which is related to how
we might look to minimise a given cost function as we ‘move’ between two known probability distributions. In
Chapter 3 we shall this Optimal Transportation framework to the Financial Markets to develop further the robust
framework outlined in Chapter 1.
Mathematical Optimal transportation problems are historically derived from real world transportation problems
such as how to transport a given pile of sand to completely fill a hole in the most cost effective manner. The
French mathematician Monge introduced the problem in his 1781 treatise ‘Memoire sur la theorie des deblais et
des remblais’, and more recently in the 1940’s the Russian mathematician Leonid Kantorovich developed a new
approach for a particular relaxation of Monge’s original problem. These optimal transportation problems are now
described as Monge-Kantorovich problems.
2.2 Basic Framework and Monge-Kantorovich Problem
In this section, we briefly introduce the mathematical framework and statement of the Monge-Kantorovich
problem. Following Villani [34], given two measure spaces X and Y, we consider two probability measures
defined on these spaces – for subsets , represent the ‘amount of sand’ in A and B
respectively (where the total mass of sand has been normalised to 1). In addition, we introduce a non-negative
measureable cost function defined on that models the effort or cost involved in transporting a unit
of mass from location to .
Transportation plans are then defined as probability measures on the product space ; where
can be interpreted as the amount of mass moved from location to . Based on the consideration that all the
mass from a particular point in X must be moved to a new location in Y somewhere; and that all the mass in Y
must have come from somewhere in X, we require:
∫
∫
This is more properly stated from a measure-theoretic standpoint as per below, where are measureable
subsets of :
More formally, these two statements are equivalent to the condition, for ) :
∫
∫
∫
We define a set to be the set of all probability measures such that this condition is satisfied; these
measures are called the transferences plans and have marginals and .
2.2.1 Kantorovich’s Optimal Transportation Problem
We can now state the Kantorovich Problem:
21
Problem 2.2.1 - Kantorovich Problem
For probability measures (as defined above), minimise the functional:
∫
We write to represent the minimal cost of transportation
In other words, over the set of transportation plans with marginals equivalent to the measures on spaces and Y,
we look for a transportation plan that minimises the cost to ‘transport’ mass from location to .
Kantorovich’s problem can be given a probabilistic interpretation, as we can consider the total transportation cost
as equivalent to the expectation of the cost function under a probability measure which is the transference plan.
Given probability measures and , we look to minimise over all pairs of random variables in , and in
, such that (i.e. , and ), the below expression:
Transference plans are all possible laws of the couple and the expectation under these transference plans
of the cost function gives the transportation cost.
2.2.2 Monge’s Optimal Transportation Problem
Continuing to follow Villani [34], we briefly outline the difference between Monge optimal transportation problems
and Kantorovich’s relaxation of this problem.
Informally, Kantorovich’s problem is different only to Monge’s original question in that Kantorovich allows for
mass from each location in X to be split as it travels to its destination in Y. Monge’s problem is a restriction of
Kantorovich’s, with the additional constraint that the mass in X must have a unique destination in Y, rather than
being transported to several different locations i.e. the mass cannot be split in Monge type problems.
We can characterise the Monge Problem as looking at a restricted set of the transference plans already
introduced above. We can characterise those transportation plans that are suitable for the Monge
problem as those that satisfy:
where is the dirac measure on x. In other words, for a location , the only mass that contributes in the
transference plan is the mass at point , where is a measurable map.
In terms of measurable sets, we can characterise the Monge problem as requiring that for a map , the
mass mapped to is such that:
( )
i.e. for all subsets of Y, the mass moved to the subset by the function is equal to the original mass of that
subset. In terms of notation, for maps that satisfy this requirement we say that is the push forward or image
measure of by and we write .
Following this, the transportation cost for those satisfying the above conditions and for a measurable
map is as follows:
∫ ∫
i.e. the relevant transportation plans are those such that the destination of x is the unique destination T(x), and
the integral can be taken over X using the measure on this space.
We can now characterise the Monge Problem.
22
Problem 2.2.2 – Monge Problem Minimise the below functional over the set of all measureable maps such that :
∫
2.3 Simple Examples of Monge – Kantorovich type problems
2.3.1 Example 1 – Mass at a single point
A simple example of a Monge – Kantorovich problem then is when we have all the mass located at a single
location, either at the source or the destination.
If all the mass is located at a single point at the destination , then we have a probability measure in the form of a
Dirac measure, say . In this case, the only available transference plan is to transport all mass to point ,
and so we have the transportation cost:
∫
Alternatively, say we have a single location of mass at the source. This example will be relevant when we return
to considering financial markets as, when we apply the Monge-Kantorovich framework to these markets, one of
the pieces of market data we will reference will be the initial stock price is known i.e. the probability measure that
describes the initial distribution of a stock price is the Dirac measure with = initial stock price.
If we assume that for some i.e. all the mass is located in a particular location, then the only
solution to the Monge problem is if there is a single location for the mass i.e. the measure on Y is also a dirac
measure.
For the Kantorovich problem, if we say that on Y is such that , then the unique transference
plan is to distribute the mass located at X onto Y exactly according to the measure – called the independent
coupling. So we have:
The transportation cost therefore is:
∫ ∫
2.3.2 Example 2 – Multiple points of mass
Let us again consider a typical discrete example, this time with mass in two different places in each of the
measure spaces X and Y.
Let the following measures be defined on X and Y, with the below cost function:
i.e. a quarter of the mass is located at one point and three-quarters in another discrete point in X, and similarly for
the measure space Y.
In terms of Monge-transport plans, there is an obvious unique mapping defined by , { }
where the transportation cost is
.
In terms of broader Kantorovich-transference plans, we can easily derive the following equations that must be
satisfied in any transference plan:
23
For { } let be the mass transferred from to for a cost The following equations and inequalities
must hold:
∑∑
Note that these equations imply that we can rewrite these all as functions of :
We can therefore express as a linear function of just , as per below:
In order to minimise this, depending on whether is positive or negative we can easily find
the maximum: if is positive, is minimised by choosing ; and if
is negative then choose as the maximum possible i.e.
following the set of inequalities (2)
described above.
Note that when we have , the solution of the Kantorovich problem is
, which is
equivalent to the solution to the Monge Transport problem as we saw above. However, when
, the solution of the Kantorovich problem is is equivalent to sending all the mass from point to
(i.e.
) , and then splitting the remaining mass from between (through
) and (through
). The transportation cost in this instance is then
.
We see in this instance therefore that firstly, the solutions to the Monge and Kantorovich solutions do not
coincide; and secondly, whether or not the solutions coincide can be dependent on the choice of cost function.
2.3.3 Example 3 - n discrete locations with equal mass
For the next example, consider X and Y as discrete spaces where all points have the same mass, in other words:
(∑
)
(∑
)
For a Monge-Transport plan, where mass is not split between points, this is about finding a permutation
which rearranges the points in X onto the points in Y for a minimal cost. The cost of transportation for the Monge
problem is then:
∑
There are a wider range of potential solutions for the Kantorovich problem, where we allow the splitting of the
mass cantered at each point. If the mass from each point is split then sent to various locations, then the total
mass transported from that point must be equal to the original mass at that point.
In this case, any measure that is a potential transference plan must satisfy the below equations (if we scale the
mass at each point to be 1):
24
∑
∑
We can therefore represent the transference plans as a bistochastic matrix, where bistochastic means that
all the are non negative and the above condition holds i.e. the set defined by:
{ ( ) ∑
∑
}
The Kantorovich problem is then written as:
{
∑
}
It can in fact be shown that in this instance, the solutions of the Monge problem and the Kantorovich problem
coincide. Following Villani [34], we note that the set is a bounded convex set and by Choquet’s theorem the
problem admits solutions which are extremal points of ; where extremal points are defined as those which
can’t be written as a nontrivial convex combination of two points in . Birkhoff’s theorem then tells us that the
extremal points of are the permutation matrices. Note that is a permutation matrix if = {1,0} and the
above conditions hold.
Thus we can see that in this particular example, the solutions to the Kantorovich problem are in fact the same as
the solutions to the Monge problem.
2.4 Kantorovich Duality Overview
As well as developing a relaxation of the Monge problem that was articulated above, Kantorovich also developed
a duality theorem for optimal transportation problems. The content of this theorem is that finding the minimum of
the linear functional that computes the transportation cost under an allowable transportation plan is actually
equivalent to finding the maximum of the sum of two measurable functions whose sum is always less than the
cost function. More accurately, Villiani [34] states the theorem as:
2.4.1 Kantorovich Duality Theorem
Theorem 2.4.1 - Kantorovich Duality Theorem (Villani [34])
Let X, Y be two measure spaces with probability measures , and let be a non-negative measureable
cost function defined on ; define a set containing all the measurable functions such that:
Then:
∫
∫
∫
We say ∫ ∫ is the dual of the primal problem ∫
Informally, Villani describes the content of this theorem as follows:
if we suppose an industrialist is managing a series of mines and factories, and needs to transport coal from the mines to the
factories, at a cost of c(x,y) depending on location x of the mine and location y of the mine. Imagine an independent trader
makes the industrialist an offer that he will manage all the shipping of the coal and we just will pay for the initial loading of the
coal from a mine, and unloading of the coal into a factory. The trader says the loading of the coal will cost , and the
unloading of the coal will cost ; and moreover, the sum will always be less than the cost of transporting the
coal from x to y.’ [34]
25
Such a deal would be immediately accepted as the cost for transporting from x to y will always be greater than or
equal the cost that would be paid to the trader. However the content of Kantorovich’s duality theorem says that if
the trader sets prices appropriately he will be able to charge a sum that is equivalent to the best price the
industrialist could have achieved if he had optimised the transportation of goods using his original cost function.
From a mathematical perspective, note the Duality Theorem has potentially simplified the search for a solution for
the Kantorovich problem. The primal Kantorovich problem is to minimise a functional over a set of transference
plans. The duality Theorem converts this optimisation problem about maximising a functional over a set of
functions satisfying the constraint . Moreover, whereas the original Monge problems are
based on non-linear constraints i.e. , the duality formulation is in a linear form; making it potentially
easier to handle and find a solution.
Before detailing the various outline steps in the proof of the duality in the next Section 2.4.1, we shall note that
one part of the duality theorem is relatively immediate. This is the inequality:
By definition of , as a transference plan has marginals on X and Y, we have:
∫
∫
∫
Further note that the condition holds for almost all almost all . So we
have that the condition holds -almost everywhere, so we have that
∫
∫
The core content of Kantorovich’s duality theorem is then to say that the inequality holds in the other direction
and therefore that we have equality.
2.4.2 Outline Proof of Duality Theorem
Following Villani [34], we shall give a brief overview of a proof of the duality theorem, as some of the methods
used here will be relevant when we examine the application of the optimal transportation framework to the
financial markets in Chapter 3. With this in mind we shall outline the key steps in the formal proof of the equality;
which consist in rewriting the original infimum problem as an inf – sup problem, and then applying a ‘Mini-max’
principle to convert this to a ‘sup-inf’.
Step 1 - Expanding the infimum expression
Starting with the expression as defined in previous section, we can rewrite as per the below:
( {
)
where is the set of nonnegative Borel measures on X x Y, and, ∫ as described
earlier. This step is valid, as if then the LHS and RHS are clearly equal; and if ,
then the ‘penalises’ the RHS to ensure that a lower infimum is not obtained.
Step 2 – Rewriting the infimum as a supremum
We can then use the following identity on the 2nd
part of the final expression in Step 1:
{
∫
∫
∫
This allows us to rewrite the expression in Step 1 as:
26
( ∫
∫
∫
)
Note that we can take the ∫ inside the expression as is not dependent
on .
Step 3 – Applying the Mini-Max Principle
Proceeding informally, we assume that appropriate conditions are satisfied and we can apply a min-max theorem
to the expression from Step 2. We get as a result the expression:
(∫
∫
∫
∫
)
Step 4 – Deriving the Duality Theorem
Finally, then we note that we have a supremum / infimum problem with functionals that are linear, separating out
components that are independent of the variables, and using we get:
(∫
∫
∫
)
Taking the second term i.e. the final expression, we argue that if at some
point , then by choosing the Dirac measure , and letting we get that
. If however, for all , then supremum is obtained for . So then
we can see we have arrived at the final Kantorovich Duality Theorem:
∫
(∫
∫
)
We shall see in Chapter 3 how an adaption of this proof can be used to prove a key duality result in financial
markets that we hinted at in Section 1.5.3 – the relationship between an upper ‘robust price’ (defined as the
maximum of expectations of the (discounted) payoff) and a ‘robust hedge’ defined as the minimum of the initial
capital required to hedge a portfolio.
2.5 Kantorovich Duality – Examples
We can briefly return to the examples from the previous section to illustrate the concept of Kantorovich duality,
and how it can be used to solve optimal transportation problems.
2.5.1 Example 1 - Dirac Measure at source
Example 2.3.1 included a Dirac measure at the source i.e. we had for some i.e. all the mass is
located in a particular location. Given a measure on Y, then the unique transference plan was to distribute the
mass located at X onto Y exactly according to the measure i.e.
The transportation cost is therefore ∫ ∫ . From a duality perspective, we need to
find functions such that , so set and so we have an equality
throughout and we have:
∫
Since we have equality between and , this proves that the are optimal.
27
2.5.2 Example 2 - Duality with multiple points of mass
Recall the example from Section 2.3.2, where we considered multiple points of unevenly distributed mass i.e.
we had the following definition of two measures and cost function:
For { } let be the mass transferred from to for a cost We saw in the previous section that the
solution to the Monge problem was the mapping defined by , where the transportation cost is
. The solution to the Kantorovich problem depended on the sign of ; if
, then the solution to the Kantorovich problem was the same as that to the Monge
problem; if , then the optimal solutions are different, and the Kantorovich problem is
solved by setting ,
. We can demonstrate that use of the duality theorem will
return us the same result.
By the Duality Theorem, we are looking for functions such that:
{ }
We then look to maximise the below functional over the functions that satisfy this constraint:
∫ ∫
In this case, in order to maximise , we need to set , to be as large as possible. We have from
the above inequality that so choose , such that . Similarly,
set , so that all the inequalities related to are satisfied as equalities. Finally, note that
the inequality can be rearranged as ; and then we can
define { } .
Finally then, we can evaluate and verify that it is equal to the minimal transportation cost calculated in
Section 2.3.3:
We see then that the duality theorem holds and that the maximum value of is equal to the minimum
transportation cost calculated in Section 2.3.2.
2.6 Numerical Techniques for Optimal Transportation
In this section we shall briefly introduce linear programming as a numerical technique and demonstrate its
application to solving optimal transportation problems. We shall make heavy use of this technique in Chapter 4
when we study the application of the Kantorovich duality to the financial markets in a simple trinomial model.
2.6.1 Overview & Framework of Constrained Linear Optimisation
Linear programming is a set of techniques used to determine solutions to constrained optimisation problems
where the function being optimised is a linear function. Following outline in Nocedal and Wright [26], we can
describe constrained linear optimisation problems in the below way:
28
where represents an index set for the equality constraints and an index set for the inequality constraints.
The function is called the objective function which is minimised (or maximised). A vector is called a
feasible solution if satisfies all the constraints; the set of all feasible points is the called the feasible set i.e.
{ }
Then we define a vector to be a local solution of the optimisation problem if and there is a
neighbourhood such that . A strict local solution is defined similarly but with strict
inequalities.
We define the Active Set at any feasible to be:
{ }
In other words, this set tells us for any particular feasible point which of the inequality constraints are in fact
equalities i.e. at that point. An inequality constraint is said to be active (at a point ) if and
inactive if .
Duality theory is also used heavily in constrained linear optimisation problems. In this context, to define the dual,
we firstly introduce the Lagrangian function, defined as:
where is the vector containing the inequality constraints, and . The dual
objective function is then defined as:
The dual problem then is to find:
We consider a duality in this form in Chapter 3, Section 3.6, from Galichon, Henry-Labordere and Touzi in [8].
Finally, constrained linear optimisation problems are typically expressed using vectors in the following standard
form:
where . The dual problem is then written as:
There is a strong duality result fundamental to the theory of linear programming that says that the solutions of the
primal formulation and the dual are the same (and if either problem is unbounded, so is the other).
Again following Nocedal and Wright [26], we note that Active Set methods are one class of algorithms for
constrained linear (and nonlinear) optimisation, and the most common of these is the Simplex method. Active Set
algorithms at each step maintain estimates of Active and Inactive index sets; at each iteration of the algorithm the
basis is the current estimate of the inactive set. The overall strategy of the algorithms is to estimate the active
set and move towards the solution of a reduced problem where the constraints in the Active Set are satisfied as
equalities.
In the simplex method, at each step a single index is swapped out of the basis , and on most steps, the value of
the primal objective function (i.e. ) is decreased. The simplex method covers a number of different variants:
the revised simplex method, as well as the dual simple method where the above duality result is used.
29
An alternative class of methods to Active Set are the Interior Point methods, named as they enforce inequality
constraints in the problem to be satisfied strictly. As opposed to active set methods, where generally there are
large number of inexpensive iterations, a single iteration on the Interior Point method is expensive to compute,
however makes significant progress towards the solution. The Interior Point methods approach the boundary of a
feasible set only in the limit, whereas the simplex method works its way around the boundary of feasibility testing
each point until it finds an optimal solution.
Within this set of methods, the Primal Dual Interior point method is commonly used for linear optimisation. This
method uses the dual formulation described above, and the Karush-Kahn-Tucker conditions to modify search
directions and step lengths so a set of inequalities are satisfied strictly at every step. The ‘desirability’ of each
potential point in the search space for each step is calculated using a duality measure, and the search direction
itself calculated using a variant of Newton’s method for non-linear equations.
In Chapter 4 and Chapter 5 we will use these techniques extensively via use of the Matlab function ‘linprog’.
For large scale problems, Matlab uses primal dual interior point methods to solve linear programming problems;
and for medium scale it uses simplex or other Active Set methods. Primal Dual Interior point methods have also
been used by Henry-Labordere in [13] to provide numerical solutions for his application of Monge-Kantorovich
concepts to the financial markets, which will be the content of Chapter 3.
2.6.2 Simple Numerical Example – Optimal Transportation
We finish this chapter by giving a simple example of solving an optimal transportation problem using linear
programming methods in a Matlab implementation. We describe below a simple optimal transportation problem
and its solution.
Example 2.6.2 - Optimal Transportation: Numerical Solution
A. Set up We introduce two probability measures on a discrete probability space, and a cost function representing the cost associated with moving from point to point .
The probability measure may represent for example the supply of a certain commodity from a set of mines, and the measure the demand for that commodity among a set of factories. We assume we have 3 mines
and 3 factories, with the below distribution of supply and demand.
Supply
Demand Mine 1 - 0.4 Factory 1 - 0.1 Mine 2 - 0.3 Factory 2 - 0.8 Mine 3 - 0.3 Factory 3 - 0.1
Table 2.6.1 – Indicative Supply & Demand for example Optimal Transportation problem
We shall assume a cost function , described by the matrix below:
[
]
We know that any transference plan must satisfy , and i.e. the mass moved
from / to any one location must equal the total original mass in that location. In this simple discrete time example, this drives a set of equality constraints of the form:
∑
{ } ∑
{ }
where represents the amount of mass moved from to i.e. the transference plan .
We can encode all these constraints into the standard form linear programming problem as per below:
30
where , and , and are
described as:
[ ]
[ ]
B. Results Given this standard form, we can use the Matlab function linprog to solve this problem – the solution is shown
below (along with algorithm used and iterations)
Algorithm Used Large Scale
Interior Primal Dual
Medium Scale
Simplex
Medium Scale
Active Set
Iterations used 7 2 4
Minimum of cost function 0.8200 0.8200 0.8200
Table 2.6.2 – Output of Linear Programming routine for example Optimal Transportation problem
The optimal transference plan that minimises the objective function is described as:
[
]
where represents the amount of mass moved from to
31
3 Relevance of Monge – Kantorovich problems to Financial Markets
3.1 Monge-Kantorovich and Financial Markets
In Chapter 1, we discussed model risk and introduced the concept of model independent financial mathematics,
considering how we might make financial prices and hedges more robust to model misspecification. In Chapter 2
we gave an outline of Monge-Kantorovich style problems and introduced the notion of duality that allowed
solutions to these problems to be more easily found. In this Chapter 3 we turn our attention back to the financial
markets and examine how some aspects of Monge-Kantorovich theory find a natural interpretation in terms of
robust pricing and hedging.
Beiglbock, Henry-Labordere and Penkner make the connection between Monge-Kantorovich problems and the
financial market in their article ‘Model-Independent Bounds for Option Prices: A Mass Transport Approach’ [1]. In
this paper, they demonstrate how the techniques of optimal transportation developed by Kantorovich can be
applied to financial markets in the context of a discrete time model.
In this chapter then, we will start by giving an overview of the discrete time model that Beiglbock, Henry-
Labordere and Penkner develop. In particular, they develop a duality in this market that equates the lower
martingale price of a path dependent option (i.e. the infimum of expectations of the payoff over martingale
measures) with the supremum of the initial capital required to set up a subhedging portfolio for this option. They
describe how we can relate this duality to the Kantorovich framework we introduced in Chapter 2, and describe
how the cost function of the Kantorovich problem is related to the hedging cost.
Several attempts have also been made in continuous time to use the optimal transportation framework to
establish a similar type of duality between robust prices and super / sub hedging strategies. In particular, we shall
also review the continuous time framework in Dolinsky and Soner [7], as well as the previously referenced
Galichon, Henry-Labordere and Touzi in [8]. We shall finish by discussing the link of optimal transportation back
to the Skorokhod Embedding Problem which was the traditional way to establishing robust bounds in financial
markets, thorough for example Hobson [18].
3.1.1 Overview of Discrete Market Framework
We begin with describing the Discrete Market Framework introduced by Beiglbock, Henry-Labordere and
Penkner in [1]. They describe a simple model for the financial market, with an exotic option which depends on the
value of a single asset S at discrete times , with payoff denoted by . For simplicity,
zero interest rates and a zero dividend yield are assumed.
Under a no arbitrage assumption, in the classical approach, we postulate a probability measure on under
which the stock price process defined by:
is a martingale in its own filtration. Note in this discrete time context, the martingale property is equivalent to
= (note we don’t assume the price process is Markov).
Under the classical approach, the fair value of the exotic option is given by the expectation of the payoff:
∫
As in the classical approach to financial mathematics, we require that our model is calibrated to the market. We
assume there is a continuum of call options with payoffs ( ) for each maturity , with
price:
[ ] ∫
32
This allows us to apply the Breeden and Litzenberger Lemma (Lemma 1.5.3) to determine the marginal
distributions of the stock price at the different maturities. We have therefore different probability measures,
corresponding to the different maturities of options. In other words, the one dimensional marginals of satisfy:
where the are probability measures on the real line determined by the above equation [ ]
and the application of the Breeden and Litzenberger Lemma (Lemma 1.5.3).
Define then a set containing all the martingale measures on the pathspace with marginals
for as described above. We can characterise the set with the below two properties:
[ ]
We are now then in a position to define what in Section 1.5.3 we called the lower Martingale price. Note that for
the moment we continue to look at the lower price bound of the option as this will subsequently provide the most
direct parallel with optimal transportation problems, where typically the lowest cost of transportation is considered.
We will see subsequently that the results will apply equal to upper martingale prices.
With this notation established then we can formulate what Beiglbock, Henry-Labordere and Penkner describe as
the ‘primal problem’, which is an expression for the lower martingale price:
{ }
To summarise then, we have simply revisited in a discrete time setting the approach to setting up robust price
bounds for exotic options that we outlined in Chapter 1, Section 1.5.3. We have used the Breeden and
Litzenberger Lemma (for the different maturities referenced in this model) to derive information about the
marginal distributions of the stock price process, and then used this, along with the martingale property, to
introduce constraints on the price of the exotic option. We note that , which we can describe as the robust lower
price bound, or lower martingale price, is defined following Hobson’s idea of looking at the ‘extremal elements’ of
a set of permissible models.
With the benefit of our discussion in Chapter 2 of Monge-Kantorovich problems, we can now immediately see the
parallels to the optimal transportation framework. We have a functional (in this case { })
which we are trying to minimise over a set of probability measures (the set ). This is analogous to the
optimal transportation framework where we looked to minimise the expectation of a cost function under a
permissible transference plan. We shall examine more closely the relationship between these two frameworks
after having reviewed the dual formulation that Beiglbock, Henry-Labordere and Penkner next develop.
3.2 Dual Formulation
Having defined the lower martingale price bound in terms of the minimum of the expectation of the payoff of an
exotic option over a constrained set of martingale measures, we can now consider the natural duality between
prices through martingale measures and through the cost of replicating portfolios that we saw used in the
discussion of complete markets in Section 1.4.2 and considered in the case of a robust market framework in
Section 1.5.3. In other words, we examine the cost of constructing replicating portfolios to either super or sub
replicate the payoff of a contingent claim, and try to establish a relationship (ideally, equality) between the cost of
this and the value of the claim as determined through the minimum of the expectation of a payoff under some set
of martingale measures. In this instance, since we are considering the lower price bound, we shall examine
trading strategies that sub replicate the option payoff i.e. the final portfolio value under our trading strategy will
always be less than the option payoff.
Following Beiglbock, Henry-Labordere and Penkner in [1] closely, we therefore now consider construction of a
replicating portfolio. In this paper, the authors hypothesise a market consisting of vanilla call options and a risky
asset that can be dynamically traded at finitely many times . Allowing then the market
participant to initially set up a portfolio of call options at which is then kept static, as well as allowing dynamic
trading strategy in the risky stock at specified times, we generate a ‘semi-static subhedging strategy’ which has a
payoff at time of the form:
33
( ) ∑ ∑
where , are measurable, and functions are integrable. The functions represent
the payoffs from initial static positions including vanilla call options with maturity (that are purchased at time ).
For our purposes, we restrict the set of functions to be contained in the set defined by :
{ ∑
}
Clearly, the represents an initial position in the money market account, is an initial position in the stock and
represents the amount of the option with maturity with payoff that is initially purchased. In other
words, we consider functions that represent a linear combination of call options with maturity for different
strikes , as well as an initial position in bond and the stock. For example, is the initial position,
and ∑
is the payoff at from a linear combination of call options with maturity .
The expression ∑ is then the discrete time equivalent of the continuous time self-
financing strategies we described in Section 1.4.2 i.e. ∫
(although it doesn’t include the initial
capital required to set up the portfolio). The random variable in this expression indicates that each of
the times the portfolio is potentially rebalanced, with the rebalancing dependent on the
previous stock prices . The ‘discrete time stochastic integral’ ∑ thus represents
the final gain or loss associated with this trading strategy.
We assume that this portfolio sub-hedges the payoff i.e. for particular we have the following inequality,
which holds for all possible stock price paths:
( )
Then the principle of no arbitrage gives us, for all the pricing measures :
( )
We have already referenced the fact that is the fair value of the exotic option payoff. In addition, the
expression ( ) can be considered as the cost of setting up the sub-replicating portfolio. Further, since
the asset price process is a martingale under if , then we have that
[ ] , and so:
[ ( )] ∑
] = ∑
The expression ∑ therefore is the initial cost of setting up the semi-static replicating portfolio, and in
particular represents the costs of setting up the static vanilla call option portfolio as well as initial capital invested
in stock and money market account. Note, we have already seen that the price of a call option with maturity is
∫
; thus the cost at the initial time of the set of call options, stocks and bonds with
payoff is (for each i=1,..,n).
Finally, then we can consider a dual formulation to the above ‘primal problem’, in terms of initial cost of
construction of the sub-replicating portfolio, where we define D, the subhedging cost, as per below:
{ ∑
( ) }
From the inequality ( ) referenced above, we have:
34
{ } { ∑
( ) }
We have a clear financial interpretation of this inequality from no-arbitrage principles: If we say we can purchase
the exotic option for a price p < D, that then there is a potential arbitrage opportunity i.e. if then if we sell short the
portfolio D described by ( ) , then because we have ( )
we can cover the payoff on the
portfolio with the payoff from the exotic option, thus locking in the arbitrage profit.
Before moving on, we again briefly mention the similarity of this problem to the Monge-Kantorovich framework
outlined in Chapter 2. In that context, we saw that the problem of finding a minimal Kantorovich transference plan
for a cost function was equivalent to finding the maximum expectation of two functions over
under marginal distributions that matched the original transference plan, where . In the
context of the financial markets, we again have a dual formulation to the initial problem of funding a minimal
expectation of a ‘cost function / payoff’ over a set of martingale measures, and this duality is around finding the
maximum over a set of functions , such that ( ) holds. Therefore, purely through analogy with the
Kantorovich problem, it seems natural to ask whether this dual formulation is in fact an equality relation, i.e. not
only but .
3.3 Duality Theorem & Optimal Transportation
From a financial markets perspective, we investigate whether i.e. whether the minimal martingale price is
equal to the maximum subhedging cost. The main result of [1] is to as demonstrate that there is no duality gap
under certain mild conditions; in other words we do have the equality . While this result holds in classical
financial as a corollary from the Fundamental Theorem of Asset Pricing in classical financial mathematics, the
this paper demonstrates this result in a model free environment.
In optimal transference terms, we have a result that closely follows the Duality Theorem which was detailed in
Section 2.4.2.
Under the market framework described in Section 3.1 and 3.2, the full result is [1] is stated as follows:
Theorem 3.3 - Duality Theorem (Beiglbock, Henry-Labordere, Penkner in [1])
Assume are probability measures on so that is non-empty. Let be an upper
semi continuous function so that the following holds, for some :
Then there is no duality gap i.e. , or in full form:
{ } { ∑
( ) }
Moreover the primal value P is attained i.e. there exists a martingale measure such that
Note that not only does Theorem 1 state that there is no duality gap i.e. ; but also that there actually exists
a martingale measure such that . So in this market, even though it is not
complete, there is a unique martingale measure that can be used to generate robust bounds for prices which are
equal to maximum sub hedging costs. Note that Theorem 1 has a natural extension to upper price bounds; by
considering instead of then we can derive a similar relation to the one above for price upper bounds and
minimal super replication cost i.e.
35
{∑
{ ( ) }
In the next section, we shall give an overview of the proof of this duality result. However before this we will
describe in more detail how the framework of optimal transportation relates to this market setting and how a
dimensional version of the Kantorovich Duality theorem is needed in establishing Beiglbock, Henry-Labordere
and Penkner’s proof.
Firstly, in the financial market described above there are times at which the marginal
distribution of the stock price under the measure is known, corresponding to the continuum of
call options with payoffs ( ) we have assumed available for each . We therefore have a
measure with different known marginals. In typical Monge-Kantorovich frameworks however,
there are only two probability measures, corresponding to the ‘initial’ and ‘final’, or ‘source’ and ‘destination’
measures that define the transference plan . This then, is a key difference between the two approaches; really
we are considering an dimensional optimal transportation problem in our financial market.
Similarly, the cost function, denoted , required for establishing Beiglbock, Henry-Labordere and Penkner’s proof
is dimensional i.e. we assume a measurable function that is bounded from below by functions
such that:
This then is a direct analogue to the Monge-Kantorovich Duality set-up; however, instead of the inequality
for the cost function, we have extended the inequality to dimensions.
In this -dimensional setting, the Monge-Kantorovich primal problem then is to minimise the cost functional
over the set of admissible transportation plans i.e. :
∫
As a consequence of the cost inequality above, we have the below dual formulation as an immediate inequality
(assuming functions are -integrable):
∫ ∑∫
We can then define the functional and seek to maximise it over the set of admissible functions :
{∑∫
∑
}
We have therefore outlined an exact analogue to the original Monge-Kantorovich problem, except extended to
dimensions rather than the original two dimensions. In two dimensions, Theorem 2.4.1 - Duality Theorem stated
that . Beiglbock, Henry-Labordere and Penkner make use of an equivalent version of
the Kantorovich Duality Theorem that holds in dimensions, and is stated below without proof, along with
associated conditions.
Proposition 3.3.1 – dimensional Version of Kantorovich Duality Theorem (Beiglbock, Henry-Labordere,
Penkner in [1])
Let be a lower semi continuous function satisfying:
Then:
36
{ } {∑ ∫
}
Where is the set of admissible functions for the dual maximisers:
{ ∑
}
Having outlined this -dimensional version of a Kantorovich Duality Theorem, we shall see how it is applied to the
financial market structure outlined in Section 3.1 and 3.2.
3.4 Outline Proof of Duality Theorem for Discrete Time Markets
In this section we provide a brief sketch of the proof of the main result Theorem 3.3, directly following [1]. We
shall describe in particular how the Proposition 3.3.1 - Multi-dimensional Kantorovich Duality theorem described
in the previous section is applied to the financial market structure.
3.4.1 Preliminary Results
There are two key preliminary results that Beiglbock, Henry-Labordere and Penkner use of in [1] in the proof of
their duality theorem which we shall state below with proof.
The first result is to make use of the following Min-Max decision theorem from linear programming, similar to that
assumed in Section 2.4.2 when reviewing the proof of Kantorovich’s duality:
Theorem 3.4.1.A – Min-Max Decision Theorem (Beiglbock, Henry-Labordere, Penkner in [1])
Let , be convex subset of vector spaces where is locally convex and let . If:
a) is compact
b) is continuous and convex on K for every , and
c) is concave on for every .
Then:
The second result is to establish the following:
Theorem 3.4.1.B (Beiglbock, Henry-Labordere, Penkner in [1])
The set is compact in the weak topology
3.4.2 Details of Proof of Duality Theorem
Following [1], we provide a brief outline of the proof of Theorem 3.3.
Step 1 – Setting up the basic inequality
Recall that by definition of in Section 3.2 we have :
{ ∑
( ) }
37
We have the immediate inequality from definition of D and using ∫ :
( ) ∑
( )
∑ ∫
Note that we are here assuming that and i.e. are continuous bounded functions on and
.
Step 2 – Separating out the supremums
By definition of ( ) and the self-financing condition, we have the below expression:
( ) ∑ ∑
So, we rearrange to get:
∑ ∑
Using this, we can therefore rewrite the above inequality from Step 1 as the below:
∑ ∑
∑∫
Step 3 – Applying the Kantorovich Duality Theorem (Proposition 3.3.1)
We are now able to apply the Multi-Dimensional Kantorovich duality theorem, above as Proposition 3.3.1. In this
instance, are defined as in Proposition 3.3.1 and the ‘cost function’ is defined as:
∑
In this context then, Proposition 3.3.1 can be restated as:
∑ ∑
∑ ∫
( ∑
)
Putting this expression back into the final inequality from Step 2, means we derive the below inequality:
∫ ∑ ( )( )
Step 4 – Applying the Mini-max principle
The next step in proving the duality is to invoke Theorem 3.4.1.A – the Minimax principle previously stated, with
the function as:
( ( )) ∫ ∑ ( )( )
38
Applying this principle to the final inequality in Step 3 we have:
∫ ∑ ( )( )
Step 5 – Applying the martingale constraint onto the functional
The final step in the proof is then as follows. The expression ∫ ∑ ( )(
) is maximised when we have:
∫∑ ( )( )
This is the case when the transference plan is also a martingale measure. However if
and also is a martingale measure, then we actually have ; so we may rewrite the
final inequality in Step 4 as:
∫
In addition, we have from above that ; meaning that we have shown as claimed in the theorem.
Step 6 – Relaxing continuity assumption on payoff
We shall omit a detailed review of the next step in the theorem where the assumption that the payoff function is
continuous is relaxed, and instead assume that it is merely lower semi-continuous. The proof proceeds by
choosing a subsequence of bounded continuous functions such that (so that
), and showing that which tends to .
Step 7 – Obtaining the primal bound
The final step in the theorem is to demonstrate that the primal bound is actually obtained, i.e. that there actually
exists a measure such that . We firstly assume the following lemma:
Lemma 3.4.1.C (Beiglbock, Henry-Labordere, Penkner in [1])
If ∫ is lower semi-continuous, then if a sequence of measures in converges
weakly to a measure , then ∫ ∫ .
We note that then if , the infimum is trivially attained, so assume , and pick a subsequence in
such that:
∫
Now, by Theorem 3.4.1.B, is compact, so converges to some measure along a
subsequence of , and therefore by above lemma:
∫ ∫
Therefore is a primal minimizer i.e. for this measure .
3.5 Some comments on Martingale Optimal Transport Theory
In this section, we provide a brief comparison on some of the Martingale Optimal Transportation Theory back to
the original Monge -Kantorovich concepts introduced in Chapter 2.
39
In the market framework introduced by Beiglbock, Henry-Labordere and Penkner in [1], we have a measure
with different known marginals corresponding to the different maturities of vanilla call options.
As described in the Section 3.3, this meant that we needed an dimensional Kantorovich duality theorem,
though this was a simple extension from the two-dimensional framework.
In addition, there are clear parallels between the techniques used in proving Theorem 2.4.1 - Kantorovich’s
Duality Theorem and Theorem 3.3. Both proofs rely on ‘expanding out’ single supremum or infimum conditions to
become or ; then rewriting the second expression as a converse i.e. to get or ;
then finally applying the minimax principle to derive an inequality in the desired form. Finally, both proofs use a
wider set such as (the set of all Borel measures on ) in the initial supremums / infimums, then
restrict the or to range over a narrower set by using the specific conditions required to obtain supremums
e.g. in the proof of Theorem 2.4.1, is used to ensure the final supremum ranges over
as required. A similar technique is used at the end of the proof of Theorem 3.3 to restrict the supremum from
ranging over to range over .
The key difference between the standard Monge-Kantorovich framework and the financial market framework in
[1] however is that the financial market framework requires the measure to be a martingale
measure – no such constraint exists in standard Monge-Kantorovich theory. Henry-Labordere in [15] therefore
describes this Martingale adaption of Monge-Kantorovich problems as ‘Martingale Optimal Transport’, and
attempts to update several key results of Monge-Kantorovich theory to include this additional restriction. In
particular, he outlines a revised version of Brenier’s theorem for Martingale Optimal Transport (see also [14]),
which relates to establishing conditions for when a transference plan is optimal (iff it is concentrated on the
subdifferential of a convex function for probability measures and with finite moments of order 2).
3.6 Alternative Frameworks for Monge-Kantorovich Problems
In this section we outline some alternative frameworks for Monge-Kantorovich approaches to robust mathematics,
briefly detailing some results broadly parallel to those of Beiglbock, Henry-Labordere and Penkner already
reviewed. In particular, there are two results in continuous time that we shall briefly review and compare back to
the earlier Theorem 3.3 - Duality Theorem that was outlined in section 3.4.
Firstly, Galichon, Henry-Labordere and Touzi establish a duality result for a continuous time market in the paper
‘A stochastic control approach to no-arbitrage bounds given marginal, with an application to Lookback options’ [8].
We have already provided a brief overview in Section 1.5.3 of the market framework that Galichon, Henry-
Labordere and Touzi set up in [8]. In this section, we expand on this description and highlight several key
differences with the discrete time market framework introduced in [1] that was discussed in Section 3.4, as well
as discuss the duality result that they establish.
The first difference is that the framework in [8] focuses on a continuous price process for the stock price and
allows trading at any point, as opposed to the Beiglbock, Henry-Labordere and Penkner discrete framework in [1]
where the asset is traded only discretely many times. Secondly, Galichon, Henry-Labordere and Touzi’s
framework involves a single maturity for tradeable call options, with a continuum of available strikes at that
maturity; in the alternative framework outlined in [1] in Section 3.4 we have different maturities of calls options
that can be traded, with a continuum of strikes at each maturity. Finally, given the move to continuous time,
Galichon, Henry-Labordere and Touzi makes use of the notion of quasi-sure inequalities, described more fully in
the section below.
3.6.1 Continuous Time Market Framework - Quasi-sure Hedging
The authors in [8] describe a market where an investor can take static positions on European call (or put) options
with a single maturity T, as well as continuously trade the risky asset. This market was outlined in Section 1.5.3
and is briefly recapped below.
We let { } be the canonical space and the canonical process on this space; the
Weiner measure and { } the filtration generated by . Interest rates in this market are set to zero.
We define, for some value :
40
For an - progressively measureable process , we define the probability measure on as:
∫
Then we have that X is a - local martingale. We then shall consider the set which is the set of all such
probability measures in ; and in fact take the subset defined by:
{ }
We can then define for any portfolio process , the portfolio value process by:
∫
In this case, denotes the initial capital required to set up the portfolio, and ∫
the gains or losses from
continuous trading in the risky asset. Let be an measurable random variable, and we define a subset of
as follows:
{ }
We then define as the set of admissible continuous trading strategies for the risky asset, specifically:
{ }
We can define the model-free super-hedging problem (without vanilla call options) as:
{ }
In other words, the model free superhedging bound is the minimum initial capital required such that a portfolio
process (consisting of continuous trading of the risky asset) exists and the final value of the portfolio exceeds the
payoff of the random variable almost surely, for all probability measures under which the risky asset is a uniformly
integrable martingale.
This type of bound is described as ‘quasi-sure’ – meaning that the inequality is required to hold -
almost surely for all probability measures in . Dolinsky and Soner in [7] refer to this as a quasi-sure super
hedge, and we shall briefly clarify later how this differs from the pathwise approach. The above expression
therefore gives the quasi-sure robust super-hedge when only dynamic trading in the risky asset is permitted for
hedging.
We next look to define a more accurate robust quasi-sure superhedge by allowing static trading in call options.
So in addition to the above, we assume that the investor is also able to take static positions in call options with
maturity T, for all possible strikes . Given this, and applying the Breedan and Litzenberger Lemma outlined
in Section 1.5.3, we derive the marginal distribution of the asset price at time T, represented as , where
denotes the set of all probability measures on .
Then for any scalar function (so may represent a linear combination of options held in different
amounts with differing maturities) the T-maturity derivative defined by payoff has no-arbitrage price:
∫
In other words, the price of a set of derivatives with payoffs is the expectation of that payoff under the
marginal distribution of the asset price at maturity; we refer to Section 1.5.3 where we outlined the derivation of
this expression.
We can then define the final value of a self-financing portfolio process which consists of continuously trading the
risky asset, as well as taking an initial static position in call options. This is represented by:
41
The represents the final value of the continuously traded asset under the strategy , as defined above; the
is the initial cost of the static call positions; and the is the payoff from the position in the options. Note
that the market has zero interest rates, so the value at time T of the initial cost of the call option portfolio is equal
to its value at time 0.
We next define the set of eligible static call positions:
{ }
Finally then we can define the improved no-arbitrage superhedging bound (improved as the bound now includes
the statically traded call options so will be tighter) as:
{ }
Again, we see here that the superhedging bound is the initial capital required such that there exists a portfolio
process and a position in the static call options such that quasi-surely. Note that in
this instance we assume that we are borrowing at time (at zero interest rates) to fund the static call
position, which we must repay at time T, hence the value of the final portfolio process is reduced by .
Therefore the initial capital required to set up the position is just , and the minimum value of this is defined
to be the minimum superhedging cost for the exotic option payoff.
3.6.2 Duality in Continuous Time Framework
Within this framework, Galichon, Henry-Labordere and Touzi in [8] develop a duality result relating the upper
bound on price with the minimal capital required to super hedge (quasi-surely) the portfolio. In this section we
shall briefly review this duality.
The proof of the duality is taken directly from an earlier result in a separate paper by Soner, Touzi and Zhang [30]
that shows the duality holding in a market which only allows continuous trading of the underlying asset. This
result is stated below [30]:
Theorem 3.6.1 - Duality for continuous time market, trading of risky asset only (Soner, Touzi and Zhang
[30])
Let be an measurable random variable, such that . Then:
{ }
Interpreting this, we have a version of the duality similar to that stated in Theorem 3.3 that states that the
minimum capital required to set up a super-hedging portfolio is equal to the maximum value of the expectation of
the payoff under eligible martingale measures.
As an extension of this theorem, Galichon, Henry-Labordere and Touzi apply this to the market which includes
the statically traded call options.
Recalling the definition { }, we consider
, as defined above; and follow through the various below equalities based on the definitions
introduced:
∫
∫
42
In effect then, we have shown that:
Note that we need to include the additional term at the front of the expression as the bound is only
defined over continuous trading of the risky asset, whereas the bound includes the possibility of static
trading of the call options also. We must therefore minimise over the set of possible positions of these call options,
which is what the terms represents.
Applying then the above Theorem 3.6.1 to and then taking the infimum of this value over
to get an expression equal to , we derive the below proposition, which is formally stated as:
Theorem 3.6.2 – Duality Theorem Including Call Options (Galichon, Henry-Labordere and Touzi [8])
Let , (where on is a collection of all uniformly continuous maps on ), and be such that
for all . Then for all , we have:
{ }
{
In words then, the result Theorem 3.6.2 above states that the infimum of the minimal capital required to
superhedge the portfolio over all probability measures is equal to the infimum over of the supremum over
probability measures of the value .
Having described the duality developed in the market framework in [8], we can see that it is in a different form to
the earlier result we saw in the discrete time market framework from Beiglbock, Henry-Labordere and Penkner in
[1]. We recall the duality proved in [1] and covered in Section 3.3 was a result about the lower martingale price
bound and subhedging; whereas here in [8] the result relates to upper martingale price bound and super hedging.
However, the result from Beiglbock, Henry-Labordere and Penkner can be easily converted to be in the form of
an upper martingale price bound and superhedging strategy. This is restated below for comparison purposes:
Theorem 3.6.3 – Restatement of Duality Theorem 3.3 for Upper Price Bound (Beiglbock, Henry-Labordere
and Penkner [1])
Assume are probability measures on so that is non-empty. Let be an upper
semi continuous function so that the following holds, for some :
Then there is no duality gap i.e. , or in full form:
{ } { ∑
( ) }
The supremum is obtained, i.e. there exists a maximising measure.
To summarise – we now have two slightly different versions of the Duality Theorems in a financial markets setting.
Theorem 3.6.3 from Beiglbock, Henry-Labordere and Penkner in [1], establishes that the minimum superhedging
cost is equal to what was described as the upper martingale price i.e. the supremum over set of martingale
measures of the expectation of the payoff. Theorem 3.6.2 from Galichon, Henry-Labordere and Touzi in [8]
relates the minimum superhedging cost to a different expression i.e. equating it to {
. We can see then that there is a difference in the form of these duality expressions, which is worth of
closer examination, which we now do.
The Duality Theorem result in Beiglbock, Henry-Labordere and Penkner in [1] is derived from a straight
application of the optimal transportation formulation as shown in the proof described in Section 3.4.2.
43
In this instance, we have n different probability measures that are the risk neutral marginal distributions
of the risky asset at times , and the objective is to maximise an expectation over these
constrained probability measures. The duality formulation then states that this is equivalent to the minimum cost
of the sub replicating portfolio. The key point is that the constraint that the risk neutral distribution is equal to the
implied marginal distribution from actual market call prices is directly used in determining the set over which the
supremum ranges. When discussing this Optimal Transportation type method, Galichon, Henry-Labordere and
Touzi describe it as:
‘directly imbedding in the no arbitrage bounds the calibration constraint that the risk neutral marginal distribution of is given
by ’. [8]
Notice that this contrasts with the duality that is developed by Galichon, Henry-Labordere and Touzi themselves
in [8]. In this case, we firstly find the supremum of an expression over the unconstrained set
. In other words, we do not embed the constraint derived from the market call prices directly in the
probability measures to be maximised over. Subsequently, we then find the infimum over a set of functions .
So, the function which represents the static position in call options in the above equation ‘encodes’ the T-
maturity call option price constraint in the same way that a Lagrange multiplier does.
Dolinsky and Soner in [7] describe the Galichon, Henry-Labordere and Touzi Duality Theorem 3.6.2 as follows:
‘the minimal super-replication cost is given as the infimum over Lagrange multipliers and supremum over martingale measures
without the final time constraint, and the Lagrange multipliers are related to the constra int’. [7]
We summarise this key difference in the dualities in the table below:
Reference Duality Theorem Description
Beiglbock, Henry-Labordere and Penkner in [1]
Theorem 3.6.3
{ }
‘directly imbedding in the no arbitrage bounds the calibration constraint that the risk neutral marginal distribution of is given by ’ [8]
Galichon, Henry-
Labordere and Touzi in [8]
Theorem 3.6.2
{
‘infimum over Lagrange multipliers and supremum
over martingale measures without the final time constraint, and the Lagrange multipliers are related to the constraint’ [7]
Table 3.6.1 – Comparison of duality expressions for the minimum superhedging cost
Finally, we note that while Beiglbock, Henry-Labordere and Penkner develop a market framework in which there
are different maturities of call options Galichon, Henry-Labordere and Touzi’s framework only admits one.
However, we shall not focus on this difference, as in a later paper ‘Maximum maximum of martingales’ [16] the
authors demonstrate that duality result proved above is in fact extendable to a case where we have a finite set of
intermediate maturities with corresponding tradeable call options and marginal distributions
, where the exotic option to be considered is a Lookback option.
3.6.3 Continuous Time Market – Pathwise robust hedging
The final market framework that we will examine is a continuous time market similar to [8] with a single risky
asset that is continuously tradeable, as well as calls options for all strikes for a given maturity T that can be
statically traded. However, in this market framework described by Dolinsky and Soner in [7], the notion of super
hedging is defined pathwise, rather than through the quasi-sure definition described above in section 3.6.1.
Dolinsky and Soner also derive a result that tidies up and equates the various forms of upper martingale prices,
superhedging bounds and duality formulations that we have considered above.
The financial market consists of a risky asset with initial assumed price of without loss of
generality. We denote the set of all strictly positive functions such that by , and so
note that an element of can be interpreted as a potential stock price process.
For the payoff, we consider a path dependent European payoff. If we denote by the set of measurable
functions ; then we let the payoff function be where is some measurable map G:
44
(i.e. the value of is dependent on the path the asset takes up to the maturity T). In addition,
Dolinsky and Soner make some Lipschitz like assumptions on the regularity of the payoff functional G.
Again, we let denote a probability measure on such that for , the price of an option with payoff
is given by:
∫
In order to define admissible portfolios in our continuous time trading strategy, Dolinsky and Soner adopt a
pathwise approach by defining for any function of finite variation and continuous through
integration by parts:
∫
∫
With this definition Dolinsky and Soner now define the notion of pathwise super replication. A semi-static portfolio
is firstly defined as a pair where and is a (progressively measureable) map
; i.e. maps a potential path for the stock price to an amount of the asset held at any one time. The
discounted portfolio value is then given as:
{ ∫
We restrict the set of semi-static portfolios to those that are admissible by requiring the below condition to hold:
A super replicating admissible semi-static portfolio then is defined as:
Note that as opposed to the quasi-sure definition of super replicating portfolios described in Section 3.6.1, the
definition here does not refer to probability measures and require that the inequality holds with probability one for
all these measures. Instead, the inequality is required to hold for all paths of the asset price (i.e. pathwise), where
the set of available paths is the set of functions ; however no notion of probability has been used to define
the value of the portfolio.
Finally, then we can define the minimal super hedging cost for a given payoff as:
{ ∫ }
Dolinsky and Soner define the minimal super hedging cost as the cost required to set up the static call position,
which is ∫ ; this includes the cost of setting up an initial position in a bond and the stock. This aligns
with the form used in [1] who define the initial capital required to set up the super hedging portfolio ∑ ∫
(where the functions were described above).
3.6.4 Duality in Continuous Time Market – Pathwise set up
In this market framework, Dolinsky and Soner prove a version of the duality result, and also relate it to the quasi-
sure inequalities that are developed in [8] and discussed in Section 3.6.3. In this section, we briefly review the
duality they develop, and compare to those previously established in discrete time setting in [1] and in continuous
time by Galichon, Henry-Labordere and Touzi in [8].
In order to describe this duality, we first briefly describe the probabilistic framework required to describe and
relate quasi-sure hedging.
We set { } the set of positive continuous functions on ; and let be canonical
process defined by ; and be the canonical filtration. A probability measure
is a martingale measure if is a martingale under the measure , and almost surely. For a
45
given probability measure , we define the set to be the set of all martingale measures such that the
probability distribution of under is .
With this probabilistic framework, Dolinsky and Soner prove a duality result, which is formally stated below:
Theorem 3.6.4 – Duality in Continuous Time Market Pathwise hedging (Dolinsky and Soner [7])
Assume that European claim has payoff defined by (where satisfies relevant Lipschitz conditions), and
the probability measure is such that:
∫ (or equivalently ∫ )
Then the minimal super hedging cost is given by:
{ ∫ }
Note that the form of this duality is similar to the duality in [1], described in Theorem 3.3 for lower martingale
prices and Theorem 3.6.3 for upper martingale prices. In particular, we have that the infimum of the cost required
to set up the superhedging portfolio, including the static call options and initial position in the bond and stock
(which is ∫ in the case of Dolinsky and Soner, and ∑
in the case of [1]) is the same as the
supremum of the expectation of the option payoff over the probability measures, where these measures are
constrained by the marginal distribution of the measure at time T being .
In addition to this, the supremum on the RHS of the expression ranges over probability measures that includes
the constraint distribution derived from the T-maturity call options. This aligns with the Beiglbock, Henry-
Labordere and Penkner duality - the only difference is that in the case of the discrete time framework of [1] we
have n different constraints on the probability measure , corresponding to the n different maturities of call
options, whereas with Dolinsky and Soner we only have the one constraint, corresponding to call options with the
maturity T. As discussed in the preceding Section 3.6.3, this is different to the duality in [8], where the supremum
is taken over the unconstrained set of probability measures, then Lagrange multipliers are used to encode the
constraint that .
3.6.5 Quasi-sure Hedging and Pathwise Hedging
How does the notion of pathwise superhedging relate to the notion of quasi-sure superhedging? A second result
in the Dolinsky and Soner paper [7] addresses this relationship. Note that the definition of the robust bound
(repeated below) doesn’t reference the dependence of the super hedge on a probability measure.
{ ∫ }
Instead, the result says that for each path which represents a potential stock price process, a portfolio
is super-replicating if we have:
Since we have this holding i.e. for all possible stock price paths, then we would expect the robust
bound defined by this notion of pathwise super replication to dominate a robust bound that was defined
quasi-surely.
We recall the previously defined notion of probabilistic super replication i.e. in Dolinsky and Soner’s notation, that
the portfolio is super-replicating if:
∫
As before, if we say that a property holds quasi-surely for a set of probability measures if its holds almost surely
for all probability measures in that set, then we can define a robust upper bound on price by:
46
{ ∫ }
Then we arrive at the simple inequality that:
In other words, the superhedging bound defined pathwise is greater than the superhedging bound defined quasi-
surely.
Finally, Dolinsky and Soner show that the inequality holds the other way also. Recalling from [8] and Theorem
3.6.2 that we have the duality as follows:
{∫ }
Since we have the general inequality
, we can derive:
{∫ }
Now if we have , then is not a martingale measure and hence the {∫
} is minus infinity. Hence we can restrict to in above expression, and apply Theorem 3.6.4
described above to arrive at:
{∫ }
Hence the quasi-sure super hedging and pathwise super hedging result in the same upper bound, and all the
inequalities in the above derivation are in fact equalities. In fact, Dolinsky and Soner have also shown that the
two different forms of the superhedging dualities that we highlighted in Table 3.6.1 are in fact equivalent
We summarise these results in the below Theorem 3.6.5 to allow for ease of reference in subsequent Chapters 4
and 5:
Theorem 3.6.5 – Equivalence of Quasi-sure and Pathwise Duality formulations in Continuous Time Market
Pathwise hedging (Dolinsky and Soner [7], Proposition 3.1)
Based on the below definitions, as per above, with the payoff of an exotic option
{ ∫ } – Pathwise superhedging
{ ∫ } – Quasi-sure superhedging
We have the below expressions as equivalent:
{∫ }
As referenced above, we see that this Theorem 3.6.5 ties together the various different duality formulations that
we have been examining. Firstly, it says that the minimum superhedging cost defined pathwise is equivalent to
the minimum superhedging cost defined quasi-surely. Secondly, it says that this is equal to what we described as
the upper martingale price i.e. the supremum over martingale measures of the expectation of the payoff of the
exotic option. Finally, it states that the alternative duality formulation of the form {∫
} is equal to both the minimum superhedging cost and the upper martingale price. In Chapter 4
and 5, we shall make extensive use of this final duality expression to derive a variety of robust bounds on exotic
options in a discrete time model.
47
3.6.6 Summary of different Approaches
We have now reviewed three different approaches to the use of concepts from optimal transportation in setting
up dualities relating hedging costs and price bounds in both a discrete and continuous time market setting.
Having reviewed three slightly varying duality results in various market frameworks, we summarise below the key
features of the market frameworks that are developed:
Market Feature Beiglbock, Henry-Labordere
and Penkner [1]
Galichon, Henry-Labordere
and Touzi [8]
Dolinsky and Soner [7]
Trading in risky asset Dynamic in Discrete Time, n
maturities
Dynamic in Continuous Time Dynamic in Continuous Time
Robust Hedging
defined ?
Pathwise Quasi-sure Pathwise
Availability of options
to statically trade
n different maturities,
continuum of strikes
Single Maturity Single Maturity
Interest Rates Assumed at zero Assumed at zero Assumed at zero
Duality Result Supremum of expectation of
payoff over constrained
probability measures equals
infimum of capital required to
superhedge
Infimum over functions
representing static call option
hedges, and supremum of
expectation of payoff over
unconstrained probability
measures equals infimum of
capital required to superhedge
Supremum of expectation of
payoff over constrained
probability measures equals
infimum of capital required to
superhedge
Other comments Supremum ranges over
probability measures that
include the constraint that the
marginal distributions at
are derived
from the call option price for
each maturity (i.e.
)
Supremum ranges over
martingale measures without
the final time constraint, and
then infimum acts as Lagrange
multipliers that are related to
the constraint
Supremum ranges over
probability measures that include
the constraint that the time T
marginal distribution is derived
from the call option price for each
maturity (i.e. )
Table 3.6.2 – Comparison of different market Financial frameworks and dualities in [1], [8] and [7]
3.7 Skorokhod Embedding Problem and connection to Optimal
Transportation
3.7.1 Overview of SEP
In this section, we will briefly examine Optimal transportation in relation to other approaches for developing a
robust framework for pricing and hedging. In particular, we shall outline and review an approach that relies on
use of solutions to the Skorokhod Embedding Problem (SEP) to derive upper and lower bounds on prices for
exotic options, and following Galichon, Henry-Labordere and Touzi in [8], highlight some results that shows
equivalence of the results derived from the Optimal Transportation approach to the results from the SEP
approach.
The SEP approach to pricing / hedging financial exotic options, introduced by Hobson in the paper ‘Robust
hedging of the Lookback option’ [18], starts with the observation that market trading in vanilla call options is
relatively liquid for a large range of strikes, and that we can treat these instruments as primary assets who prices
are given exogenously to any financial model we construct. As before, this allows us to apply the now familiar
Lemma 1.5.3 (Breedan and Litzenberger Lemma) i.e. that knowledge of vanilla call prices of all strikes for a given
maturity T determines the marginal distribution of the stock price under a risk neutral pricing measure. Hobson
summarises it thus in [17]:
48
If we know call prices of all strikes for a single maturity, then we know the marginal distribution of the asset price, but t here may
be many martingales with the same marginal at a single fixed time. Any martingale with the given marginal is a candidate price
process.
So far, this exactly follows the methodology outlined in Section 1.5.3 on the Robust approach, and developed
when expanding the Optimal Transportation approach in Chapter 3. However, at this stage, the SEP approach
and the Optimal Transportation approach diverge. Rather than use techniques from optimal transportation
techniques to develop robust pricing and hedging dualities, we use results from the Skorokhod Embedding
Problem to establish robust price bounds. We outline this technique below.
Firstly, we introduce the Skorokhod Embedding Problem itself. Following Hobson [17], the Skorokhod Embedding
Problem can be stated as follows.
Problem 3.7.1 - Skorokhod Embedding Problem (Hobson [17])
For a given stochastic process , and a measure on the state space of , find a particular stopping
time such that the stopped process has law (we write to denote this)
Most often the process is taken to be Brownian Motion (denoted ).
Secondly, we reference a key result regarding the properties of Martingales (see for example Revuz and Yor [27],
Chapter V). This is stated in full below; informally as per Obloj [24], it states that any continuous local martingale
is a time-changed Brownian motion (and moreover, that the quadratic variation process explicitly gives the time
change)
Theorem 3.7.2 – Dambis, Dubins-Schwarz (Revuz and Yor [27])
If is a continuous local martingale (with filtration on probability space ) vanishing at 0 and such that ⟨ ⟩ , and if we set:
{ ⟨ ⟩ }
Then,
is a -Brownian Motion and ⟨ ⟩
To recap then, we have that any martingale with a particular given marginal (inferred from the prices of market
vanilla call options for all strikes) is a candidate price process; and also any given martingale is a time change of
Brownian Motion. However, the problem of finding a stopping time for Brownian Motion so that the law of is
the given law is precisely the statement of the Skorokhod Embedding Problem.
In other words, following Hobson [17], say we have a continuous Martingale where . Then by Theorem
3.5.2, we have that ⟨ ⟩ , and so ⟨ ⟩ is a solution of the SEP for and .
Hobson in [17] therefore summarises:
‘There is a 1-1 correspondence between candidate price processes which are consistent with observed prices, and solutions of
the Skorokhod embedding problem. …. extremal solutions of the Skorokhod embedding problem lead to robust, model
independent prices and hedges for exotic options’
There are in fact a wide variety of known solutions to the Skorokhod Embedding Problem, surveyed by Obloj in
[24]; and this correspondence between solutions of the Skorokhod Embedding Problem and candidate price
processes can be exploited to develop robust price bounds and hedges for exotic options. This has been done in
papers such as Hobson [18], [17] in the case of Lookback Options, and in Cox and Obloj [4] for double touch
barrier options.
We give a brief overview of result for the robust bound for a Lookback option derived in Hobson [17]. We firstly
define the barycentre function as follows, where is a probability measure with unit mean and support
contained in :
49
∫
Note that for a stock price with law , this is equivalent to defining as .
If X is a random variable with continuous distribution then , the Hardy-Littlewood transform of , is the law of
. Hobson then determines the below result:
Theorem 3.7.3 – Robust bound for a Lookback option (Hobson [17])
Given stock price process , for fixed maturity , assume that European call option prices are known for all
strikes , such that by Lemma 1.5.3 (Breeden and Litzenberger) we have for some probability
measure i.e. is the marginal distribution of the stock price .
Then defining the payoff of a Lookback option as:
we have the upper price bound on the Lookback option as:
∫
where is the Hardy-Littlewood transform of
Hobson demonstrates that this is in fact the least upper bound for price by showing that firstly it is possible to
purchase a portfolio of call options for cost that super-replicates (so by no-arbitrage is an upper bound),
and secondly, that there is a market model for which the unique price of the Lookback option is (so it is a least
upper bound).
3.7.2 Connection to Optimal Transportation Problem
The natural question that arises is how any robust price bounds calculated through the Skorokhod Embedding
Problem approach compare to robust price bounds calculated through the Optimal Transportation approach. As
referenced above, Galichon, Henry-Labordere and Touzi in [8] demonstrate that for the particular robust price
bound for the Lookback option described above, the two approaches do in fact lead to the same outcome.
We state below this result without proof from Galichon, Henry-Labordere and Touzi in [8] as an example of the
equivalence of the two approaches for this particular exotic:
Theorem 3.7.4 – Equivalence of Robust Price bound for SEP & Optimal Transportation approach (Galichon, Henry-Labordere and Touzi in [8])
Let and payoff for some nondecreasing function satisfying
} , and
(where and are as defined above)
Then we have:
where ( ); and ∫ ∫ ( )
and is the lower support of
In other words, the robust superhedging bound for an exotic option with payoff is equal to the Hardy-
Littlewood transform of , which is the value calculated for the robust bound by the SEP approach; and as per
previous result this is equal to the duality formulation {∫ }.
50
4 Simple Discrete Market Models
4.1 One Period Trinomial Model
In this Chapter, we will introduce a simple financial market framework based on the one period trinomial model to
illustrate some of the concepts introduced in Chapter 3 and start to develop a numerical implementation to
examine the robust pricing and hedging dualities we reviewed. The trinomial model gives a simple example of a
market which can be incomplete, and thus admit multiple potential risk neutral measures. We first examine the
binomial model in order to see that it is not sufficiently rich to examine some of the concepts explored in the
previous chapter.
4.1.1 The Standard Binomial Model
The standard one period binomial model is not sufficiently rich to demonstrate the concepts of robustness in
terms of price, as it is a complete and thus admits a unique risk neutral measure that unambiguously sets prices
in that market.
In particular, in a market where we have one risky asset , which takes values at time 0, we can draw the
binomial tree:
Now, assume further we have a risk free bond such that and . As we set an assumption of
no arbitrage in our model, we derive the following conditions on and :
For say this inequality didn’t hold, for example . Then there is a simple arbitrage – we can sell
short and invest the proceeds in the risk free bond, ensuring that at time 1, we have , which by
assumption is greater than , which is the cost to buy back the stock and cover the short position. Note if we
have then the risky asset never moves in price.
We can easily see that there exists a risk neutral measure in this market. Under the risk neutral pricing measure
in this market, the discounted stock price is martingale i.e. the following equality must hold:
We also have the constraint that as the represent risk neutral probabilities. Writing these in matrix
form we get:
[
][
] [
]
We have already established through the no-arbitrage condition, and so since the matrix
[
] has the determinant
, it is invertible and we can solve the above matrix equations to get the
unique risk neutral probabilities:
[
] [
]
[
]
[
]
Furthermore, we can easily show that the market is complete. For a contingent claim with payoff , we need
to describe a replicating portfolio with invested in the risky asset at time , and in the bond. We therefore
have the system of equations:
[
] [
] [
]
– probability
- probability
51
Again, since the determinant of the matrix [
] is non-zero by the no-arbitrage condition, we have a
unique solution for , which is:
[
]
[
] [
]
[
]
Note the expression for the amount of stock required to replicate is the standard delta hedge for the binomial
model
. With this portfolio, we can replicate the payoff of any contingent claim and the market is thus
complete.
4.1.2 Trinomial Model – an incomplete market
Incomplete Market
With this context, we now examine the one period trinomial model. Again, we assume we have a risky asset ,
which takes values at time 0, and a risk free bond with initial value 1 and terminal value . This time the
risky asset can take three values, which we describe via the below tree:
Again, we impose the no-arbitrage condition on the model, which sets the conditions that
In looking for the risk neutral measure, we use the requirement that the discounted stock price must be a
martingale (and so the stock price must grow at the risk free rate) to derive the constraint:
We also have the constraint as the represent risk neutral probabilities. Given these two
constraints in three unknowns, we can rearrange to get:
Using similar rearrangement for , we can derive the matrix equations:
[
]
[
]
Given the condition , we can derive from the first row of the matrix equation:
Similarly, we have from the second row of the matrix equation:
– probability
– probability
– probability
52
We have already argued that the assumption of no arbitrage is equivalent to condition ; and
also that we have assumed . Putting these expressions together, we see that there are an infinite
number of which satisfy the above inequality, and therefore that there are an infinite number of
satisfying the above equations.
In summary then, in this model, we can find infinitely many such that and the stock price is a
martingale under this probability measure. We therefore have an infinite number of equivalent martingale
measures in this model.
Duality in Trinomial Model
We now consider the use of superreplicating portfolios in this market, which along with the principles of no-
arbitrage, set upper and lower bounds for the price of an option.
We introduce a contingent claim with value at of , and initial value . We look for a super
replicating portfolio that where invested in the risky asset at time , and in the bond. Specifically, we
require the following three inequalities to hold, corresponding to the three potential states of the risky asset at
time :
If we have these inequalities hold, then we use apply the no-arbitrage principle to get:
In other words, the initial capital required to set up the super replicating portfolio sets an upper bound
on price (as we saw in Chapter 1 when discussing robust pricing and hedging).
Given this upper bound on , and the fact that we have an infinite number of martingale measures in this market,
it is natural to ask whether a form of the Duality Theorem we considered in Chapter 3 also holds in this market.
Following Kohn in [22], we can give a direct proof that this is in fact the case through a proof which closely follows
the method outlined by Villani in Chapter 2, in particular making use of the Min-Max principle to derive the duality.
Note that the market framework is not as rich as that described by Beiglbock, Henry-Labordere and Penkner [1]
in Chapter 3, in particular we have not permitted the use of vanilla call options to hedge exotic options payoffs,
and for now we are limited to considering one period trinomial models.
Step 1 – Introducing an additional maximum term
Firstly we define { }, i.e. the set of all such that the
payoff of the contingent claim is superhedged by a portfolio consisting of of the stock and of the bond. We
consider the expression for the minimum hedging cost given by:
Note that, similar to Villani’s proof of the duality theorem considered in Chapter 2 (Theorem 2.4.1) we can
introduce a maximum term by noting that for :
( ) { ( )
Here, ( ) are such that and satisfy the constraint (i.e. they represent the risk neutral
probabilities)
We can then rewrite the original expression as:
53
∑ ( )
i.e. we can sum across the risk neutral probabilities
Step 2 – Applying the Min-Max principle
Assuming informally that we can apply the min-max principle, we derive the following expression:
∑ ( )
Rearrange this to get:
∑
∑
∑ ( )
Step 3 – Removing the minimum
Finally, then we note that this is equivalent to:
∑ ∑ ∑ ( )
As ∑ is achieved when ∑ , and similarly for ∑ . If
we then make the substitution , then this expression can be rewritten and we have proved the full
duality, where satisfy conditions to be risk neutral probabilities:
∑ ∑ ∑
( )
So as per Chapter 3 and in particular the Duality Theorem 3.6.3 from Beiglbock, Henry-Labordere and Penkner in
[1], we consider the upper martingale price for the contingent claim as the supremum over martingale measures
of the expectation of the discounted payoff, which is equivalent by the above proof to finding the minimum of the
initial capital required to set up a replicating portfolio.
4.1.3 Simple Numerical Example – Call and Put options in one period trinomial model
In the trinomial model described above, we have shown how the possible martingale measures can be
characterised by choosing a value for , which determines the value of probabilities and We have also
shown that a duality result, similar to the Duality Theorem 3.3 described in Chapter 3 (or rather the restatement of
this theorem for superhedging cost and upper martingale prices, described in Theorem 3.6.3), holds in this
market. We shall now give a simple numerical example of these two concepts and investigate the validity of the
duality in our simple trinomial market.
Firstly, let’s choose a one period trinomial model that meets the constraints outlined in the previous section. The
model is illustrated below:
We set the interest rate at r = 0. As per the section above, we have the below expressions for the probabilities
:
– probability
– probability
– probability
54
[
]
[
]
This means we can easily describe the infinite set of martingale measures as per the below simple graph, where
the derived values of and are shown mapped against the input value of :
Figure 4.1.1 – Martingale measures on one period Trinomial Model
In a similar fashion, we can then very easily plot the discounted expectation of any given payoff under these
different measures against (which we are using to order the measures). In the examples, below we choose a
call option with strike , and a put option with strike . The maximum price under these various measures is
given by the below expression:
∑ ∑ ∑
( )
This is shown on the graphs below:
Figure 4.1.2 and 4.1.3 – Maximum call price under equivalent martingale measures, for call option (strike K = 0.9) and put
option (strike K = 1.1)
As per the above graphs, the maximum price for the call option (strike at 0.9) over all martingale measures is
, and the maximum price for the put option (strike at 1.1) over all martingale measures is .
The duality theorem that we demonstrated above shows that this maximum price should be equal to the minimum
initial capital required to fully hedge the option’s payoff. In particular, as per previous section, we want to
minimise (where is invested in the risky asset at time 0, and in the bond), but with the following
inequalities satisfied:
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10
0.1
0.2
0.3
0.4
0.5
0.6
0.7
Value of q2
Valu
e o
f q1 a
nd q
3
Martingale measures on one period Trinomial Model
q1
q3
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10.095
0.1
0.105
0.11
0.115
0.12
0.125
0.13
0.135
0.14
Value of q2
Price (
dis
counte
d e
xpecta
tion u
nder
mart
ingale
measure
)
Maximum Call Price under equivalent martingale measures, Strike =0.9
Max Price =0.13636
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10.1
0.11
0.12
0.13
0.14
0.15
0.16
Value of q2
Price (
dis
counte
d e
xpecta
tion u
nder
mart
ingale
measure
)
Maximum Put Price under equivalent martingale measures, Strike =1.1
Max Price =0.14545
55
We can rewrite this in matrix form as:
[
] [
] [
]
This is a linear programming problem close to standard linear form as described in Chapter 2, Section 2.6, and
can be solved through Matlab.
With this implementation completed, we obtain the following results for the initial capital, i.e. the expression
and also show the required hedge for our examples:
Call Option Strike Put Option Strike
Amount of stock required
Amount in bond
Total Initial Capital Required
Table 4.1.1 – Matlab results of linear programming to solve duality expression for one period trinomial model, hedging with
risky asset only
We can see that as expected and in line with the duality result we have proved directly in this market, the minimal
initial capital required to set up a super-replicating portfolio is the same as the maximum expectation of the payoff
over all martingale measures. In other words, this is a numerical example of the duality that holds in this market.
In addition, solving the linear programming problem gives us the hedge to set up the super-replicating portfolio; in
the case of the call option it means we require units of the stock and invested in the bond.
We finish the example by making comments on the above example:
- Firstly, in this one period trinomial model, although it is rich enough to admit an infinite number of
martingale measures (unlike for example the one period binomial model), we have a method to fully
describe all these measures as we derived expressions for and in terms of . This meant it was
relatively easy to calculate the ‘robust’ upper martingale price i.e. the maximum value of the discounted
expectation of the payoff under the martingale measures.
- Secondly, we also were able to articulate a list of simple constraints based on the three inequalities that
were required to be met as the portfolio needed to be super replicating. This meant we could construct a
minimisation problem that could be solved by the linear programming methods that were introduced in
Chapter 2
- Finally, the example gives a very simple illustration of the derivation of the upper martingale price for the
call option. The minimal initial capital required to set up a superhedging portfolio is shown to be equal to
the maximum expectation of the payoff over the various risk neutral measures, and therefore we have
one simple ‘robust’ price.
4.1.4 Classical Financial Mathematics Approach – the Trinomial Method
We will now illustrate how the assumption of an underlying model for the stock price adds additional constraints
to the above framework that will determine the actual choice of risk neutral measure from the infinite number we
saw possible above.
The classical approach to use of the trinomial model, as detailed in for example Hull [19], assumes specific
dynamics for the stock and sets the parameters of the state space to align with the implied distribution from
assuming these dynamics. In our framework, we have already assumed values for the state space , and ,
however we will now add in a constraint related to the variance of the stock.
56
In a discrete time market following Hull [19], with time between each discrete point, the dynamics of the risky
asset are assumed to be:
√
where is the standard normal distribution and is the volatility of the stock price return. In this instance, we
have the return
of the stock given by:
[
]
Similarly, the expectation is given by:
[
]
Turning to our one period trinomial model where we set , if we expect the random variable to have ;
then we expect that the random variable will also have variance (as adding a constant will not affect
the variance). The random variable is defined in our model as:
{
where
; and so we require:
Using the relation , and noting that (as this is the risk neutral measure) we
have:
Then we have:
Therefore, if we have already set the state space values , then we have introduced an initial constraint
in our model, which we write below in matrix form:
(
)(
) (
)
The first line of the matrix relates to the fact that the form a probability measure so sum to 1; the next line is the
constraint that form a risk neutral measure under which the stock price grows at the risk free rate. These were
the two constraints we had previously in section 4.1.2. The final line then is the new constraint introduced by
assuming that the stock price follows the assumed dynamics, with the constant representing the volatility of the
stock price return.
Now we have three equations in three unknowns (with the additional constraints that ), and therefore,
given we know the volatility of the stock, we are able to determine the specific risk neutral measure by solving
these equations. The below Figure 4.1.4 builds from Figure 1.4.2 and illustrates this for two theoretical choices of
volatility, 10% and 15%. The additional information in the form of the volatility allows us to ‘choose’ appropriate
risk neutral measure to use to price the option, and we now have a lower, more precise specified price for the
option. The trade-off is that we have made additional assumptions about the movement of the underlying through
the imposition of an extraneously given parameter which in reality would be subject to significant Knightian
uncertainty in calculating the real value.
57
Figure 4.1.4 – Sample prices for call option assuming various extraneously given volatilities of stock price, 10% and 15%
4.1.5 Trinomial model with static trading in options and stocks
In the previous section, the one period trinomial model admitted only trading in the one risky asset. We saw that
the market was incomplete in such a setting, admitting an infinite number of equivalent martingale measures and
that a version of the duality theorem held in this setting.
In this section, we shall briefly extend the model considered in Section 4.1 to understand the effect of allowing
static trading in call options, as is assumed in the market of Beiglbock, Henry-Labordere and Penkner in [1]
discussed in Chapter 3, Section 3.3.
We postulate a market as per section 4.1 and described below, but now with an additional call option with price
denoting a vanilla call option with strike , that expires at . The price of we
assume is given exogenously from the market.
We then introduce another call option with price , that is struck K such that . We can then use
the call option with price to create a replicating portfolio for . In order to hedge the call option ,
we must have the below equations holding reflecting the 3 potential states of the risky asset at time , where
we denote as the amount of the stock held; as the amount in the bond, and let denote the amount of the
call option :
This set of 3 linear equations in 3 unknowns can be solved uniquely as:
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10.095
0.1
0.105
0.11
0.115
0.12
0.125
0.13
0.135
0.14
Value of q2
Price (
dis
counte
d e
xpecta
tion u
nder
mart
ingale
measure
)
Maximum Call Price under equivalent martingale measures with additional vol constraint, Strike =0.9
Max Price =0.13636
Price =0.12455; Vol = 0.15
Price =0.11091; Vol = 0.1
– probability
– probability , Call option with strike
– probability
58
The market is therefore now complete, and there is a unique martingale measure which can be used to price all
call options in this market.
4.2 Two Period Trinomial Model – Robust Hedging and Pricing
We will now start to evaluate the properties of a richer market setting, a two period recombining trinomial model.
The model is of the form described below:
.
Figure 4.2.1 – Two Period Trinomial Model Framework, with probabilities and Stock prices
The notation represents the price of the risky asset at times ; and the notation represents the
probability at time 1 or 2 of a move up, down and remain the same.
In the following section, we will fix the tree with the following parameters: letting , and and
and similarly for the 2
nd period. Therefore, we will use the above stock prices for
illustrative purposes in the following sections.
4.2.1 Robust Hedging - Linear Programming Set Up
We begin with a model that only contains the risky asset and we aim to find a robust price for a vanilla call option.
We do this by developing a linear programming problem for a robust hedge for a particular vanilla call option.
As per the one period trinomial model, for a given strike and option defined by the standard payoff function for
a call , our hedging portfolio must be greater than or equal to the payoff of the option at maturity. This
means that our superhedging portfolio must satisfy the inequality:
In the above equation, represents the amount of unit of the stock purchased at time , and represents
the money invested in a risk free bond earning interest rate in one period.
However, in a key difference to the one period trinomial model, the two period model allows for rebalancing of the
self-financing portfolio at the intermediate time . In addition, the amounts of stock and bond purchased will
depend on the value of the stock at , denoted i.e. a different amount of stock / bond will be required to
super replicate the option price depending on the intermediate stock price.
Following this logic, we therefore denote by the number of units of stock in the hedging portfolio at
, if the stock price goes up, stays level or goes down respectively. Similarly, we use the notation
59
to represent the amount invested in the risk free bond. The values are decided at ,
then remain constant until where the final value of the portfolio is determined.
We have therefore a set of nine inequalities that must be satisfied if the portfolio super hedges the payoff of the
option (sample shown below):
Finally, then we denote by the initial amount of stock in the hedging portfoilio, and the initial amount of the
bond chosen at . Since we require the portfolio to be self-financing, then the rebalancing of the portfolio at
time must satisfy the standard self-financing conditions, described below:
The robust hedging problem for this two period trinomial model therefore can be expressed as a standard linear
programming problem, with the set of inequality and equality constraints described below in matrix form.
Linear Programming Set-up 4.2.1
Robust Hedging in Two Period Trinomial Model with dynamic trading of risky asset
where represent the below matrices:
[ ]
[
]
[
]
And also:
[
]
[
]
[
]
60
With these matrices defined, we can execute a standard linear programming routine through Matlab, as outlined
in Chapter 2, to determine the minimum value of the initial capital required to set up a hedging portfolio. We show
the results below for a series of strikes.
Figure 4.2.2 – Minimum super hedging cost for 2 period trinomial model, plotted against strike for vanilla call option. The
minimum super hedging cost for strike K = 0.9 is shown explicitly
4.2.2 Robust Pricing – Linear Programming Set up
We can now demonstrate numerically that a robust pricing and hedging duality holds in this market, i.e. we have
that:
where in this instance is the payoff of a vanilla call option, and is a martingale measure in the market.
We can set up a simple linear programming routine to solve the equation:
Following the labelling convention introduced above in Figure 4.2.1, we have the following set of constraints on
the problem:
Constraint 1 - is a probability measure meaning we have:
We also have similar equations holding for the probability triples described above i.e. ; ;
. We therefore have 4 equality constraints and 9 inequality constraints.
Constraint 2 - is a martingale measure for the stock price process . In other words we require the
below equations to hold:
(and similar equations for and ).
Constraint 3 – The price of the option is the (discounted) expectation under the martingale measure of the option
payoff. We therefore have the equations:
0.6 0.7 0.8 0.9 1 1.1 1.2 1.30
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
Strike
Min
imum
Hedgin
g C
ost
Minimum Hedging Cost for Call Option, 2 period model
Strike K = 0.9, Hedging Cost=0.16116
61
(where denotes the intermediate value of the option at time , if the stock value is ; similar equations for
the other intermediate times). Note that although the payoff of the call option is given, we do not know the
intermediate value of the option at . The final price of the option at is then given by:
We can include all these constraints in a linear programming setup in order to determine the value of
, which can be implemented in Matlab as described below.
Linear Programming Problem Set up 2 – 4.2.2
Maximum of martingale measures in Two Period Trinomial Model with dynamic trading of risky asset
Note, in order to solve the problem we will first solve a linear programming problem to
determine the maximum intermediate value of the call option at for each of the states , , and then
we subsequently solve a second linear programming problem to determine the maximum initial value of the call
option (i.e. at ). The principle we are therefore using is that finding the maximum of the expectation over the
two periods is the same as finding the maximum price at each intermediate stage.
Step 1: Determine maximum intermediate value of option for each of , , ; using the below set of matrices
and solving:
Note we are looking to minimising , equivalent to maximising
Equality constraints:
[
] [
] [
]
Inequality constraints:
[
] [
] [
]
Step 2: Determine maximum initial value of option ; using the below matrices:
Equality constraints:
[
] [
] [
]
Inequality constraints:
[
] [
] [
]
Note that the above assumes that . For non-zero interest rates (briefly considered in Chapter 5) a factor of
must be included in the first two rows of equality constraint matrix to represent that we are aiming to find the
62
maximum expectation of the discounted payoff and that the discounted stock price is a martingale.
With this linear programming routine defined, we can implement via Matlab to determine the supremum of the
(discounted) expectation of the payoff over the set of martingale measures. This is shown in the graph below,
calculated for a series of strikes of the call option:
Figure 4.2.3 – Maximum price under martingale measures in 2 period trinomial model.
The value at strike K = 0.9 is shown.
Through comparison of the above Figure 4.2.3 to Figure 4.2.2 in Section 4.2.1 shows that for a given strike the
minimum capital required to robustly superhedge the call option is the same as the upper martingale price (i.e.
the maximum expectation of the payoff over martingale measures). Both graphs show the value of the robust
hedge / price for strike , where the maximum price is . We have demonstrated numerically that a
form of the ‘robust pricing / hedging duality’ holds in this market.
Finally, we can verify that the above linear programming problem has indeed found us the maximum price of the
call option and that we have determined the upper martingale price (and have a robust price bound). As we saw
in the previous section, the one period trinomial model is incomplete, and there are in fact an infinite number of
equivalent martingale measures which could be used in pricing an option. We have a similar situation in the two
period trinomial model i.e. we have an infinite set of measures under which the risky asset is a martingale.
We can use a simple Monte Carlo type concept to illustrate the potential range of prices under these different
martingale measures. Using the equation developed in the previous section to fully describe the martingale
measures;
[
]
[
]
We can use a Uniform distribution to generate a random selection of martingale measures (this technique is
described in full in Chapter 5, Section 5.2.1), and then evaluating the price of the option with strike as
the expectation of the payoff under this measure, we get below distribution of prices for the call option shown in
Figure 4.2.4 below.
0.6 0.7 0.8 0.9 1 1.1 1.2 1.30
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
Strike
Maxim
um
Price o
f m
art
ingale
measure
sSupremum of Price under martingale measures, 2 period model
Strike K = 0.9, Max Price=0.16116
63
Figure 4.2.4 – Sample distribution of option prices for call option with strike K = 0.9 under sample of equivalent martingale
measures, calculated through Monte Carlo with 100,000 iterations. Measures are randomly selected through use of uniform
distribution; methodology described more fully in Chapter 5 in Algorithm 5.2.1
Figure 4.2.4 (and supporting data) shows that note that the price of the option under randomly generated
martingale measures does not exceed the value of the robust bound already calculated of shown in
Figure 4.2.3. In addition, this is the minimum capital required to robustly hedge the portfolio and is indicated in the
previous Figure 4.2.2. The linear programming technique has successfully identified the robust upper bound on
price and demonstrated it is equal to the minimum superhedging cost i.e. the maximum over martingale
measures of the discounted expectation of the payoff is equal to the minimum capital required to set up a super
hedging portfolio for that option’s payoff.
4.2.3 Path Dependent Options – Robust hedging and Pricing
We can use the linear programming technique explained in the previous section to derive the minimum capital
required to robustly super hedge a path dependent option, not just the vanilla call option considered in Section
4.2.1.
As an example, we consider a path dependent option with payoff defined by:
This is similar to a standard Lookback option, except the initial value of the stock price is not considered in the
final payoff, only the maximum intermediate value at . The below demonstrates the payoff of this specific
option for the market parameters defined and strike ; note a fuller range of path dependent options are
considered in Chapter 5.
1.2 1.4 1.4 0.5
1.2 1.2 1.2 0.3
1.2 1 1.2 0.3
1 1.2 1.2 0.3
1 1 1 0.1
0.08 0.1 0.12 0.14 0.16 0.18 0.2 0.220
500
1000
1500
2000
2500Illustrative Distribution of Call Prices in 2 Period Trinomial Model, K = 0.9
Option Price
Max Price=0.16104
64
1 0.83 1 0.1
0.83 1 1 0.1
0.83 0.83 0.83 0
0.83 0.69 0.83 0
Table 4.2.1 - Illustration of payoff for the path dependent option
The below Figure 4.2.5 shows the minimum superhedging cost for this path dependent option:
Figure 4.2.5 – Robust hedging costs for path dependent option, strike .
Note that by comparing to Figure 4.2.2, we can see that the path dependent option has a higher value of initial
capital required to super-hedge its payoff for a same strike (for example, with strike , the path dependent
option requires to hedge, whereas the vanilla call option requires ). This is as expected, given that
the value of (used to determine the payoff of the path-dependent option) will always be greater than
or equal to , which is used in the payoff of the vanilla call option; and therefore the value of the payoff of the
path dependent option dominates the payoff of the vanilla purchase for all stock paths.
As per previous section, having derived the minimal hedging cost, we can verify that the robust price-hedging
duality holds in this market, as well as demonstrate through simple Monte Carlo that we have derived a robust
bound.
For each strike, a linear programming problem is solved to give the maximum value of the discounted expectation
under martingale measures. Figure 4.2.6 below shows that under this method, as expected, the value of the
robust price (i.e. maximum value of discounted expectation of payoff of path dependent option) is the same as for
the robust hedge (for strike , both methods give ).
0.6 0.7 0.8 0.9 1 1.1 1.2 1.30
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
0.5
Strike
Min
imum
Hedgin
g C
ost
Robust price for Path Dependent Call Option
Strike K = 0.9, Hedging Cost=0.21074
65
Figure 4.2.6 – Graph showing supremum over martingale measures of the discounted expectation of the payoff of the path
dependent option. The results demonstrate numerically that duality holds in this market
The final Figure 4.2.7 below verifies that we have in fact determined the upper martingale price through a simple
Monte Carlo routine as per the method described in section 4.2.2. We generate a set of measures through
randomly generating uniform distribution and using this to describe a martingale measure on the state space and
risky asset; and then use this measure to evaluate the price of the path dependent option based on .
The Figure 4.2.7 shows that over this set of randomly generated martingale measures that the price for the path
dependent option does not exceed the robust price (or robust hedge) that we have already calculated. The robust
price bound calculated above was ; the maximum price from the below Monte Carlo simulation for the path
dependent option is .
Figure 4.2.7 – Monte Carlo simulation (with 100,000 iterations) generating sample martingale measures on the state space via
a uniform distribution, and evaluating the value of the path dependent option using discounted expectation of payoff under that
martingale measure. For 100,000 samples, the maximum price calculated for the path dependent option is 0.2102; which
compares to the previously calculated robust price bound of 0.2107
0.6 0.7 0.8 0.9 1 1.1 1.2 1.30
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
0.5
Strike
Maxim
um
Price o
f m
art
ingale
measure
s
Supremum of Price under EMM, 2 period model exotic path dependent
Strike K = 0.9, Max Price=0.21074
0.08 0.1 0.12 0.14 0.16 0.18 0.2 0.22 0.24 0.260
200
400
600
800
1000
1200Simulation of price for path dependent call option, strike K = 0.9
Max Price=0.2102
66
4.2.4 Additional Market Information – Market prices for Call Options Example
In the one period model trinomial model, we saw that the introduction of a market price for a call option made the
market complete, and therefore that there was a unique equivalent martingale measure that could be used to
price the option.
In the two period model, with the extra degrees of freedom afforded by determining the probability measures over
two periods, we can introduce an extraneously given market price for a call option without determining a unique
martingale measure i.e. with the market being complete. Before formally showing this, we demonstrate this with a
simple example from the results already developed.
We arbitrarily set an extraneously given price for the vanilla call option with strike as
. From the previously developed examples, we can demonstrate that there are multiple martingale
measures which would produce this price for the call option. Four such example measures are shown in the
below table.
Value of Value of Value of Value of
Value of Call Option,
under measure
Value of Path Dependent Option,
under measure
Measure 1 0.9486
0.2921
0.0123 0.5965
0.1365
0.1894
Measure 2
0.2555
0.3049
0.5955 0.6956
0.1365
0.1678
Measure 3
0.7334
0.3609
0.1613 0.3360
0.1365
0.1785
Measure 4
0.4948
0.3754
0.1455 0.7838
0.1365
0.1752
Table 4.2.2 – Example equivalent Martingale measures for the two period trinomial model, each distinct measure valuing the
call option with strike at the value of 0.1365. The right hand column shows the value of the path dependent Lookback
option under each of the measures.
In the table above, the columns measures ‘Value of ’ indicate the values of the probability for the stock to take
that particular path, with the labelling described as per Figure 4.2.1. Each measure is fully characterised by
describing the values of the probabilities . The remaining probabilities in the measure (i.e.
) can then be derived using the formula referenced earlier for each node of the
stock price tree:
[
]
[
]
In the column ‘Value of Call option under measure’, we see these measures all give the same call option price i.e.
the expectation under that risk neutral measure of the discounted payoff of the call option is equal to the
extraneously given call option price of . We therefore see that we have a set of distinct equivalent
martingale measures that match this call option price, and we would be exposed to significant model risk if we
arbitrarily choose one over any of the others.
The final column entitled ‘Value of Path dependent Option under measure’ shows the value of the path
dependent Lookback option under each of the various probability measures. As can be seen from the table, the
value of this path dependent option varies according to the measure i.e. we have not established a unique price
for the option by including the additional market information of the call price with strike .
4.2.5 Additional Market Information – Robust Hedging (i.e. minimum superhedging
cost)
We saw in Chapter 3, Section 3.6 that Galichon, Henry-Labordere and Touzi in [8] firstly reviewed a duality
expression based on purely trading the risky asset, and subsequently introduced the ability for semi-static trading
of call options. Similarly, we now consider the robust hedging problem in the enhanced market described in the
previous Section 4.2.4, where we have the ability to not only dynamically trade the risky asset, but also to set up
a static position in vanilla call options at a single strike whose price is determined extraneously. We arbitrarily set
the strike as , and keep interest rates at .
67
The state space of the 2 period trinomial tree and overall market features are outlined below:
Figure 4.2.8 - The enhanced market framework, which allows dynamic trading in the risky asset in the described state space,
along with an initial static position in the vanilla call option with ‘market price’ = 0.1365
In this section, we shall examine the problem of determining the minimum superhedging cost for a path-
dependent option in this market framework.
In this enhanced market, we have the option of setting up at a hedging portfolio consisting of not only the
risky asset and bond, but also a position in the vanilla call option which has maturity at . At , we have
the opportunity to readjust our position in the risky asset and bond, but we keep static the position in the call
option. We denote the amount of call option that we choose to hold at as . Our portfolio therefore can be
described as below:
Here, represents the amount of stock held at , the amount invested in the bond, the final stock
price and the market price of the option.
In order for our portfolio to robustly hedge a path dependent exotic option, we require the below inequality to
hold:
Here, represents the payoff of the path dependent exotic option (note strictly speaking, we should write
{ } as the exotic option is path dependent, and so depends on the value of not only , but also and
). The stock price belongs to the set { } . and both are
dependent on the value of the stock at , so in fact we have { } and { }.
In addition to these linear inequality constraints, we have a set of equality constraints. In the previous section we
saw that the equality constraints were driven by the need to ensure that the portfolio was self-financing i.e. we
had the constraint:
Note that the introduction of the ability to take a static position in the vanilla call option does not necessitate any
additional equality constraints in our linear programming problem. This is because in the case of static trading,
the amount of call options is set at as , and this remains unchanged throughout the evolution of the
market.
In this particular problem, we consider the same path dependent option introduced in Section 4.2.2, i.e. which
has payoff defined by:
= 1
= 0.83
1.44
= 1.2
Vanilla Call Option, strike K = 0.9C(K) = 0.1365
68
Finally then, in setting up our linear programming problem, we look to minimise the initial capital required to set
up the super replicating portfolio i.e. we look to find:
In this particular instance, we set that as the extraneously given market price for the
vanilla call option (in reality, this has been generated from choosing a particular equivalent martingale measure
for this market, and finding the expectation of the discounted payoff of the vanilla call option under this measure).
We now can describe fully the form of the linear programming problem required to solve the robust hedging
problem.
Linear Programming Problem Set up – 4.2.5
Robust Hedging in Two Period Trinomial Model with dynamic trading of risky asset and static position in
call option (single strike only)
where represent the below matrices:
[
]
[
]
[
]
And also:
[
]
[
]
[
]
We solve this via Matlab with the below Table 4.2.3 summarising three sets of results.
Firstly, we repeat the example martingale measures used in the previous Section 4.2.3 to obtain the lowest and
therefore most accurate prices for the path dependent option, but ones which are ‘model dependent’ in that we
have needed to choose a particular model for how the risky asset moves. Secondly, we use the results generated
as part of Figure 4.2.2 that give the minimal capital required for setting up a super-replicating portfolio when only
dynamic trading of the risky asset is allowed. Finally, we describe the results of the above Linear Programming
Problem Set Up 4.2.5 i.e. the minimal capital required to set up a super-replicating portfolio when able to both
dynamically trade the risky asset and statically trade with one call option.
69
Pricing Method
Method 1
Selecting several Equivalent Martingale Measures
Measure 1 Measure 2 Measure 3 Measure 4
Price of path dependent option under measure:
0.1894 0.1678 0.1785 0.1752
Method 2 Robust hedging under dynamic trading of
asset, no static trading of call option 0.21074
Method 3 Robust hedging under dynamic trading of
asset, static trading of call option with
strike
0.19097
Table 4.2.3 - Price of a path dependent exotic option under three different pricing methodologies: selecting equivalent
martingale measures; robust hedging with just the risky asset; and robust hedging with a risky asset and a call option .
As we might expect, the addition of the ability to statically trade a call option in addition to dynamically trade the
risky asset reduces the minimum capital required to super hedge the payoff of the option (in this example, from
to ).
We repeat the above linear programming problem for a series of strikes for the path dependent option. We keep
fixed the market price that is known, which is ; but vary the strike of the path dependent
option. The below Figure 4.2.9 compares the robust hedging cost when we are able to trade the risky asset only,
and when we can trade both the risky asset and the vanilla call option.
Figure 4.2.9 - Robust hedging cost for path dependent option, in market framework with and without ability to statically trade a
call option. The graph shows that being able to statically trade the call option reduces the capital required to set up a robust
hedge.
4.2.6 Additional Market Information – Robust Pricing
Having calculated the initial capital required to set up a robust hedge for the path dependent option, we can
consider an alternative method for deriving a robust bound on the price of an exotic option.
In the market with no additional vanilla call option market price to act as a constraint, we saw that the following
duality held:
0.6 0.7 0.8 0.9 1 1.1 1.2 1.30
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
0.5
Strike
Min
imum
Hedgin
g C
ost
Robust price for Lookback Option - Risky Asset only & Vanilla Call Option added
Strike K = 0.9, Hedging Cost=0.21074
Strike K = 0.9, Hedging Cost=0.19097
Path Dependent Price - dynamic trading only
Path Dependent Price - dynamic trading with static call
70
In our new market described in Section 4.2.5 which includes the ‘market derived price’ of a vanilla call option with
strike , we have several alternative expressions for the robust bound that we can use for our numerical
implementation.
As per our discussion in Section 3.6.5, recall that Dolinsky and Soner in [7] derived the result that all these
various formulations for robust pricing and robust hedging were equivalent i.e. as per Theorem 3.6.5 in Chapter 3,
we have:
{ ∫ }
{∫ }
{∫ }
For the moment, we shall assume that the equivalent dualities hold in our simplified discrete market, and we shall
test this assumption through numerical implementations.
Consider firstly the expression for the ‘robust price’ that used by Beiglbock, Henry-Labordere and Penkner in [1],
where the ‘constraint’ of the known market price for the call option is embedded directly into the martingale
measure constraint:
In this instance, represents a set of constrained martingale measures, where the constraint is that under any
martingale measure in this set, the discounted expectation of the payoff of the vanilla call option must equal the
extraneously given market price (for one strike K). In other words:
{ }
We can refer to Table 4.2.2 in Section 4.2.4 for several examples of measures that satisfy this constraint i.e. give
the market price for the vanilla call option. Of course, each of these measures may produce a different price for
the path dependent option as Table 4.2.2 illustrates.
The alternative duality for the robust price that was discussed in Section 3.6.5 was that derived by Galichon,
Henry-Labordere and Touzi in [8], which was of the form:
{∫ }
In this instance, the supremum ranges over all unconstrained martingale measures, and then the infimum adds
the constraint of the market price for the vanilla call option.
We will use this second form of the duality in the numerical implementation methods over subsequent sections to
establish robust price bounds for an exotic option. We use the expression {∫
} from [8], rather than from [1] in our numerical routines, as the constraints
implied by this form of the duality are easier to encode in our numerical implementation. Note that as per
Theorem 3.6.5 from Dolinsky and Soner in [7], we have that the expressions are equivalent (in the continuous
market framework they consider).
In the expression introduced in [8], we look to maximise over an unconstrained set of martingale measures ,
and then subsequently look to find the minimum over this functional by varying . We are able therefore to break
the numerical implemtentation into two steps, firstly to the a maximum for fixed ., and subsequently to find the
minimum by varying In constrast, with the expression we need to encode the constraint of the
71
distribution implied by the call option directly into the set of martingale measures, making this potentially more
complex. As such, we look to evaluate the expression {∫ } through
a variety of methods described below.
Implementation Method 1 - Crude Monte Carlo Simulation
A simple Monte Carlo simulation for a range of potential Lagrange multipliers allows us to derive a numerical
value for the second duality expression {∫ }.
When considering the expression ‘ ’, we need to determine the potential form that will take. In the current
market we are considering, this is simple: the function must be of the form:
where . In other words, we only have the one vanilla call option with strike in this
market, so the function must be a linear multiple of this call option.
Method Description:
‘Crude Monte Carlo’ - Numerical implementation of Galichon, Henry-Labordere and Touzi [8] Duality (the
expression {∫ } for 2 period trinomial model with market price for
call option at one strike; based on Monte Carlo for each fixed
Step 1:
A) For fixed , we generate a set of sample unconstrained martingale measures, based on uniform
distribution and the formula previously described in Section 4.1.1. to generate the full martingale probability
measure
B) For each of these martingale measures, we calculate the value of ,
C) We calculate value of ∫ ; noting that ∫ is simply the market price of
the vanilla call option multiplied by the value of (i.e.
D) We determine the maximum value of ∫ from the sample martingale measures
in our Monte Carlo simulation i.e. {∫ }
Step 2:
A) We vary and repeat Step 1 for different value
B) Across the values of we determine the minimum value of the maximum of the expectation of the payoff
under the martingale measures i.e. we evaluate:
{∫ }
For fixed strike , we use the above method to determine the robust price for the path dependent option.
The below Figure 4.2.10 shows the maximum value of the expectation of the payoff of the path dependent option
for various values of fixed .
72
Figure 4.2.10 – Duality {∫
} for valuing path dependent option. The value
calculated through Monte Carlo (with 30,000 iterations for each value of λ) shows that the minimum value of the expression is
approximately 0.19062, at λ (i.e. as per above description of )
From Figure 4.2.10, we can see that the minimum value of the maximum expectation of the payoff occurs at
approximately , and at this point, the maximum expectation of path dependent option’s payoff under the
martingale measures is approximately . Therefore, we have that in this instance that
{∫ } , and this is the value of the upper martingale price.
This compares to the previous method used to determine the bound for a robust hedge in the previous Section
4.2.5, where through consideration of hedging portfolios possible through dynamic trading of the risky asset and
static trading of the call option we determined that the robust bound was – as shown in Figure 4.2.9. This
Monte Carlo method therefore has not found exactly the robust lower price bound, as we might expect from the
crude approach of sampling random measures and calculating the price of the path dependent option each time.
Implementation Method 2- Linear Programming for different values of
The second implementation method is similar to Method 1, however solves a linear programming problem for
each fixed value of , and then the expression is minimised over the range of . The method is described below:
Method Description:
‘Crude Linear Programming’ - Numerical implementation of {∫ }
for 2 period trinomial model with market price for call option at one strike; based on linear programming for each
fixed
Step 1: A) For fixed , we solve a linear programming problem (in two steps corresponding to and
) to maximise the expression
B) We calculate the value ∫ by adding on constant ∫
Step 2: A) We vary across the range [0, 2] and determine minimum value of
∫ in this range
Using this method, we recreate the graph in Figure 4.2.10, this time using linear programming rather than Monte
Carlo, making the algorithm quicker and more accurate. This is shown in Figure 4.2.11 below. The value
0 0.5 1 1.5 20.19
0.192
0.194
0.196
0.198
0.2
0.202
0.204
0.206
0.208
Method 1 - Robust Price for Path Dependent Call Option, Monte Carlo using Galichon et al. Duality
lambda
Sup o
ver
mart
measure
s
Minimum Price =0.19062
73
produced by this method of is the same as that derived from the linear programming problem for the
value of the robust hedge (shown in Section 4.2.5).
Figure 4.2.11 – Duality {∫
} for valuing path dependent option. The value
calculated through linear programming shows that the minimum value of the expression is approximately 0.19097, at .
Implementation Method 3 – Unconstrained linear optimisation through Matlab
The final method available to us in determining the robust price is to directly use a unconstrained nonlinear
optimisation approximation through Matlab to minimise the equation ∫
over . We describe the method briefly as follows:
Method Description:
‘Unconstrained Nonlinear Optimisation’ – unconstrained optimisation on the nonlinear function
∫
Step 1:
A) Define routine for calculation of ∫ by calculating, for fixed , this
supremum through solving linear programming problem ∫ which we have already
described.
Step 2:
B) Use unconstrained nonlinear optimisation in Matlab to determine minimum of this function, where we embed
the constraint as simply requiring that (as the call option could be either held long or short as part of
the hedging portfolio.
This nonlinear optimisation methodology will be described more fully in Chapter 5, Section 5.1. The table below
summarises the results from this 3rd
method:
0 0.5 1 1.5 2 2.50.19
0.195
0.2
0.205
0.21
0.215Method 2 - Robust Price for Path Dependent Call Option, Lin Prog using Galichon et al. Duality
lambda
Sup o
ver
mart
measure
s
Minimum Price =0.19097
74
Minimum value of {∫ }
Value of that minimum is attained at:
Table 4.2.4 – Summary of nonlinear optimisation methodology for determining {∫
}
We see that the unconstrained nonlinear optimisation has determined the same minimum value for
{∫ } as the previous Method 2, as well as the robust hedging bound
found in Figure 4.2.9 through linear programming. In subsequent Sections 4.3 and Chapter 5, we will build on this
Method 3 when we add additional constraints into our financial model.
We summarise briefly then what we have examined to date:
- We have implemented several methods for determining the minimum super hedging cost and robust pricing
upper bound for an exotic option (the example chosen was a path dependent option) in a simple 2 period
trinomial market setting
- In Section 4.2.1, we used a linear programming method to solve the problem of finding the cost of a
superhedging portfolio for the vanilla call option (i.e. to find
) in a 2 period trinomial model in a market which allowed rebalancing of the stock / bond hedging
portfolio at t=1 (but not trading of any static call options)
- In Section 4.2.2, we used a linear programming method to determine the upper martingale price for a vanilla call
option (i.e. determining ) in a market identical to the previous section (two period trinomial,
rebalancing of the stock at ). We demonstrated numerically that the robust pricing / hedging duality i.e.
held in this market, by showing that the values for the robust price that were calculated were the same as those
calculated for the minimum superhedging cost.
- In Section 4.2.3, we introduced an example exotic call option, a path dependent option whose payoff was
defined by:
We again demonstrated numerically that the pricing / hedging duality of the form
held for this exotic option, in a market consisting of 2 period trinomial model, with trading of the
stock at (but no vanilla call option trading)
- In Section 4.2.4, we introduced our first constraint into the market context. We introduced an extraneously given
price for a vanilla call option with strike . We allowed static trading in this call option, as well as the
previous dynamic trading in the underlying risky asset. We showed even though we introduced an extraneously
given vanilla call option price, the market remained incomplete as there were still multiple equivalent market
measures that under risk neutral pricing formula, gave the same market price for the vanilla call option, but
different market prices for the path dependent exotic option.
- In Section 4.2.5, we demonstrated that in this market setting, we were able to derive the minimum superhedging
cost of the path dependent option using this additional market information as a type of constraint on the robust
bound. We firstly demonstrated through linear programing problem that the minimum superhedging cost was
lower when the additional call option was introduced into the market. Secondly, we used the alternative duality
expression for the upper martingale price introduced in [8] i.e. the expression:
{∫ }
75
implemented through three different numerical methods, to determine robust price bounds in this market. Again,
we demonstrated numerically a form of the robust pricing hedging duality held in this market with the additional
constraint included. In particular we provided a numerical demonstration of the below duality from [8]:
{∫ }
4.2.7 Two period Model – Multiple Constraints and market completeness
In this final section on the two period trinomial model, we will look a market with a fuller set of constraints in terms
of additional extraneously given vanilla call option prices, and consider the impact this has on the minimum
superhedging cost and robust pricing bounds.
We now consider a financial market where we have a fuller set of extraneously given prices for vanilla call
options. In this section, we align the strike prices of these vanilla call options with the possible end values of the
risky asset in our two period trinomial model. We denote the market price of the vanilla call options at by
. In our hedging portfolios, we allow dynamic trading in the risky asset, as well as static trading in a
linear combination of vanilla call options.
The market can be described as per below:
Figure 4.2.12 – Additional Market Call Price Constrains in Two Period Trinomial Model
With this market, we can successively introduce market prices for vanilla call options into our framework and see
the impacts that these additional constraints have on our robust price and hedging bounds.
Note that in our simulated market, we must be careful in setting the market prices for vanilla call options. We
must ensure that the principle of risk neutral pricing is preserved. In other words, we must ensure that:
1. There is a probability measure under which the stock price is a martingale
2. Under this probability measure the (discounted) value of the expectation of the payoffs of each of the
vanilla call options for each strike must be equal to the market price
In order to ensure that these conditions are satisfied, we choose a measure at random and generate market
prices for call options using this to ensure that there is a risk neutral measure which prices the call options as
(discounted) expectation of the payoffs.
Robust Hedging - Minimum Superhedging cost
We successively introduce the additional market call option prices into our financial model and use linear
programming principles to derive the minimum superhedging cost.
With the extended family of market call options at our disposal, the inequality that our portfolio must satisfy at t =
2 is as per below:
= 1
= 0.83
1.44
= 1.2
Market Prices for Vanilla Call OptionsInterest rate r = 0
76
∑
{ }
where represent the vanilla call options priced by the market, is the amount of call options that is included in
portfolio at , and the value represents the number of additional call options that we add to the financial
market. Again, { } represents the payoff of a path dependent option.
The value of the corresponding initial portfolio that we look to minimise then becomes:
∑
The linear programming setup is described below:
Linear Programming Problem Set up 4.2.7
Minimum suoerhedging cost in Two Period Trinomial Model with dynamic trading of risky asset and
static position in n call options (with n different strikes)
∑
where represent the below matrices:
[
]
[
]
[
]
And also:
[
]
[
]
[
]
Note that the format of the linear programming problem is not altered significantly by the addition of further call
options to the model. In particular, the further addition of market priced vanilla call options does not impose any
further equality constraints on our linear programming problem, as we have imposed that the call options portfolio
must be held static over the market duration. We don’t therefore have to include any additional equalities relating
to self-financing portfolio rebalancing at time as we did for the dynamic trading of the underlying asset.
We demonstrate below the values of the minimum superhedging cost under the following conditions:
77
Method 1 – Robust Hedging for multiple vanilla call options constraint
A. SET UP
Path Dependent Option payoff defined by:
Set and interest rate
Martingale measure used (chosen at random, denoted ):
0.1241
0.6198
0.9208
0.2781
Table 4.2.5 – Martingale measure used for determining market prices of call options
Then the formula referenced in Section 4.1.1 for determining the full measure can be used i.e.:
[
]
[
]
Market Call Prices priced using this martingale measure:
Figure 4.2.13 – ‘Market derived’ vanilla call option prices for the particular market measure
B. RESULTS
Value of path dependent option priced under the above martingale measure, denoted by :
We might consider this as the ‘fully model dependent’ price, for in setting the price at this level, we have chosen a
specific probability measure to describe our financial market.
In terms of robust bounds, the below table shows the successive minimum superhedging costs for the path
dependent option payoff, derived by solving the above linear programming problem:
0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 1.50
0.05
0.1
0.15
0.2
0.25
0.3
0.35Option price for different strike
Strike
Price
C(1.2)=0.016513
C(1)=0.080524
C(0.83333)=0.1928 C(0.83333)=0.1928
C(0.69444)=0.30556 C(0.69444)=0.30556
C(1.44)=0
78
No. of Call Options available in market to trade
{ ∑
}
i.e. Minimum super hedging cost
(no additional call options) 0.140496
( ) 0.124447
( ) 0.097037
( ) 0.097037
0.097037
0.097037
Table 4.2.6 - Minimum superhedging cost for an example path dependent option for a market in which successively many
vanilla call options are traded at extraneously given market prices.
The key point to note from this table is that the minimum superhedging cost does not decrease further after the
second vanilla call option has been introduced as available to static trade. In fact, at this point, when the
value of the minimum superhedging cost is equal to the value of the path dependent option when evaluated
under the sample measure we used to derive the ‘extraneous prices’ in the first place. In other words, we have:
∑
where and .
Since the robust price is defined as , and certainly we have as is a
martingale measure for the price process and we have by definition of :
. Then it certainly follows that ∑ is a lowest possible bound
for , and adding additional vanilla call options will not reduce the minimum super-hedging cost
any further.
Robust Pricing
Finally then, we shall evaluate a robust pricing bound for this market with the successive introduction of multiple
constraints, and demonstrate numerically that this is equivalent to the robust hedging bound, i.e. the duality
described below holds:
∑
Note here that the set is defined as:
{ }
Strictly speaking we should label this set to represent the fact that for every vanilla call option we
add to the market, we have a different (and more narrow) set of potential probability measures to choose from in
our market.
In order to determine the upper martingale price we shall use the form of the duality used in the previous section
from Galichon, Henry-Labordere and Touzi in [8], in particular:
{∫ }
79
Our aim is to demonstrate numerically that in our sample market, the hedging & pricing duality holds.
Method 2 - Robust Pricing for multiple vanilla call options constraint
A. SETUP
We take the same market assumptions as used in Method 1 i.e. we use the measure to determine set of
call options as shown in Figure 4.2.13 above
We define a nonlinear function through {∫ } and look to minimise this
successively over .
Note that the set varies as we add additional tradeable vanilla call options to the model (so we can denote
to denote the set of allowable functions with n vanilla call options in the model. In particular we have (for
):
{ }
{ }
{ }
We use the unconstrained nonlinear optimisation routine in Matlab to determine the below results:
B. RESULTS
We run unconstrained nonlinear optimisation routines for successive multiple call options.
No. of Call Options available in market
{∫ }
i.e. Maximum robust price
(no additional call options) 0.140496
( ) 0.124447
( ) 0.097037
(
)
0.097037
Table 4.2.7 – Robust price bound for varying number of vanilla call options
It is also instructive to consider the values of that are found to be minimisers for each of the above cases; this is
highlighted in the below table:
No. of Call Options available in market Values of such that
{∫
} is minimised
(no additional call options) N/A (no call options)
( )
( )
(
)
80
Table 4.2.8 – Values of that achieve robust price
Note: initial values used to begin unconstrained linear optimisation routine were set at 0.
Finally, we give a visual demonstration that the unconstrained nonlinear optimisation routine has found the
minimal value of {∫ } in the case (i.e. two additional market priced
vanilla call options), by plotting this function over a suitable area as per below:
Figure 4.2.14 – Visual demonstration of the minimum value of {∫
}.
We can conclude the following from the above results:
- A form of the pricing / hedging duality holds irrespective of the number of constraints that we have added to the
market in terms of additional tradeable vanilla call options. More precisely, we have demonstrated the below
pricing and hedging duality:
{∫ } { ∑
}
- The above duality holds if we have (i.e. no additional call options), (i.e. one additional vanilla call
option), (i.e. two additional vanilla call options) or more.
- As per the results for the robust hedging (i.e. minimum superhedging cost), the robust price doesn’t decrease
any further once . At this stage, the robust price is equal to the , and the
constraints we have introduced have been enough to uniquely determine the martingale measure used to price in
the market.
- When , i.e. we have introduced the two call options and into the market, the values
of and that minimise the expression {∫ } are
We can see this visually in Figure 4.2.14, where the minimum point is at (1,1), with value 0.09074 as we
derived separately from the unconstrained nonlinear optimisation routine. In other words, the minimum value of
{∫ } is obtained when we hold a single additional call option at each strike
0
0.5
1
1.5
2
0
0.5
1
1.5
2
0.08
0.1
0.12
0.14
0.16
0.18
0.2
Value of lambda C(1))
X: 1
Y: 1
Z: 0.09704
Robust Price for Path Dependent option, under two vanilla call option constraints
Value of lambda C(1.2)
81
4.2.8 Summary of two period Trinomial Model
We can now recap the following from our examination of the two period trinomial model:
- We have provided numerical demonstrations of the duality’s discussed in Section 3, in the form in which they
were introduced by Galichon, Henry-Labordere and Touzi in [8]. In particular, we have provided a simple
numerical demonstration of the duality:
{∫ }
- We have also given a numerical demonstration that this is equal to the cost required to super hedge the payoff,
in other words we have:
{∫ } ∑
- We have demonstrated numerically that, as expected, the robust price of the path dependent option is
determined by the instruments in the market that are tradeable. The robust price is highest when there are no
vanilla call options available to trade in the market, and the market participant is only able to trade the underlying
risky asset (and the money market account, with )
- The robust price subsequently reduces for each new ‘constraint’ we introduce into the model, where the
constraints take the form of the market prices for vanilla call options at particular strikes. In the case of the two
period trinomial model with the path dependent option considered, adding two separate additional call options
meant that the robust price was equal to the price under a particular martingale measure i.e. we had the
equation:
where we have
In this case, the set of eligible martingale measures contains the single martingale measure .
- We have used two primary optimisation techniques to demonstrate numerically these dualities. The challenge of
finding the minimum robust hedge i.e.
∑
was translated into a linear programming set up, as a form of constrained linear optimisation. The constraints are
in the form of both inequality and equality constraints. The inequality constraints are determined by the
requirement that the portfolio hedges the payoff of the path dependent option. The equality constraints for the
robust hedging problem are driven by the dynamic trading of the stock that allows rebalancing of the hedging
portfolio at ; this translates into an equality of the form:
- The robust pricing problem formulated as:
{∫ }
is solved numerically through use firstly of solving a linear optimisation problem i.e. for fixed we solve
{∫ } . The equality constraints are driven by consideration that is a
martingale measure for the stock price process, and by principle of risk neutral pricing i.e. that the price of an
option is the (discounted) expectation of the payoff under a risk neutral measure. The second stage of the robust
pricing numerical method is to use unconstrained nonlinear optimisation to determine
{∫ }
82
5 N-period Trinomial Model & Conclusions
In Chapter 4, we saw applications of financial interpretations of Monge-Kantorovich type dualities to a two period
recombining trinomial model. We demonstrated numerically that some of the robust hedge & robust price
dualities developed in the literature (and reviewed in Chapter 3) held in this two-period setting, and used these to
price a simple path dependent option (a Lookback option) using some extraneously given market prices for
vanilla call options.
In this Chapter 5, we look to extend the application of these principles to an n-period trinomial financial market
with greater flexibility in setting market parameters and price a wider range of exotic, path-dependent market
options. We will construct in Matlab an implementation that allows us to do this more generally, though we will
quickly find that the n period model requires more sophisticated optimisation techniques.
In the first section, we will provide an overview of the mathematical approach to the period model. The second
section will provide some detail on implementation on Matlab, along with some key features of the model that
was implemented. The third section will provide some results (for low values of ) across a various range of
parameters, exotic options and strikes.
5.1 Mathematical extension to N-period Model
There are limited barriers from a mathematical perspective to extending the 2 period model developed in Chapter
4 to a more general framework. Below we briefly discuss some points of consideration, starting with the robust
hedging problem i.e. the minimum initial capital invested in the stock, bond and vanilla call options required to
superhedge the exotic option payoff under all possible paths of the stock.
1. Self-Financing Portfolios and Equality Constraints
In a two period model, there is only one opportunity for rebalancing the portfolio, at . In an period model,
there are obviously – opportunities for rebalancing the portfolio. In the case of robust hedging, this manifests
itself in the financial model in terms of additional variables and equality constraints that need to be incorporated
into the linear programming problem.
At every time t in the model such that the below self-financing constraint must be satisfied:
where denotes the amount of stock purchased at time , when the stock price is at level . Equally,
denotes the amount invested in the bond at time . Also, denotes the price of the stock at time , and
is related to by one of three equations: , ; , each occurring with
probability , , where denotes the probality of moving up, remaining at the same level, or
moving down at time , when the stock price is at level .
2. Additional Variables introduced at each time step
It is important to note that in the above equation there are different values of , for every single level of the
stock at each time . In other words, how we choose to rebalance our superhedging portfolio at any time is
dependent on the current value of the stock, so we need to introduce sufficient variables to model this
requirement. The below table shows the number of variables we need to model the self-financing constraint at
different times , where at the stock price is .
Potential Stock Values
(27 possible values)
Variables Required to model self-financing constraint:
(54 new
variables)
83
etc.
Total Additional number of
Variables: 2 6 18 54
Table 5.1.1 – Additional variables required for increasing number of time periods in trinomial model
If we exclude the variables introduced by vanilla call options for the moment, we can see therefore the number of
additional variables that we introduce at each time step is . This is intuitively clear as we are introducing 2
new variables (an amount we invest in the stock to hedge, and an amount we invest in the bond to hedge)
for each additional set of stock prices we add, which increases by a factor of 3 each time as we are working in a
trinomial model.
If we were aiming to be parsimonious in our model, we need not introduce a second variable representing the
amount invested in the bond. This is because once the amount invested in the stock is fixed, then the self-
financing constraint will determine the value to be put into the bond as the total wealth of the portfolio is known at
each stage, though this potentially involves a marginally more complex form of the linear programming problem in
Matlab. For ease of implementation, we continue to follow the approach followed in Chapter 4 which involves
explicitly including the variable for the bond price and using this to solve the linear programming problem.
Before then we consider hedging with vanilla call options, we can easily derive a formula for the total number of
variables required in our linear programming problem. For an period model (so opportunities to rebalance,
the total number of variables will be:
∑
As a consideration then, we note that we will have (at least) 80 variables in a 4-period trinomial model; 728
variables in a 6-period trinomial model and 59,048 variables in a 10-period trinomial model.
3. Call Options for every strike
As we increase the number of periods, the potential values that the stock can take at the maturity of the exotic
option increases. For a 1-period recombining model, there are 3 possible final stock prices; for a 2-period
recombining model, there are 5; and in general terms there are possible values of the stock price. Note if
the model is not recombining, then we will grow the potential number of end values for the stock price, denoted
, much more quickly, i.e. there will be possible values. We therefore substantially simplify our model by
assuming that the model is recombining.
In the -period trinomial model, we assume we have extraneously given market prices for vanilla call options for
a particular number of strikes, where the strikes are potential values of the stock price . We do not allow strikes
at an ‘intermediate’ point between potential values of i.e. if and , we do not allow vanilla call
options with strikes . We will then introduce flexibility into the model by allowing the user to ‘choose’ how
many vanilla call options are ‘tradeable’ in the sense that they can be statically traded when setting up a
superhedging portfolio.
As per the 2-period trinomial model, when considering how to introduce extraneously given market prices for
vanilla call options into the model, we must be careful to preserve the principle of risk neutral pricing. More
precisely, we require that any market prices for vanilla call options satisfy the constraint:
where is some martingale measure for the stock price process. The simplest way to ensure this condition
is met is to generate a martingale measure at random, and use this to generate the extraneously given
market call prices. In the next section, we give a simple algorithm for doing this based on the method used in the
2-period trinomial model.
Finally, the most natural assumption to make is for extraneously given market prices for vanilla call options to be
available for all potential values of . However, we note that for certain strikes, at the extreme end of the
84
trinomial tree, knowledge of market call prices gives us no useful ‘information’ about the market measure. For
these extremal points, these market prices therefore do not act as constraints at all and can be discarded. In
particular, the value of the call option which has strike equal to the maximum value of will be 0, as the payoff
will be 0 for all values of . Similarly, for the minimum value of , the payoff at any point will always be
{ } and can therefore be statically replicated with purchase of a single stock at for and
investment of { } in a bond.
A simple example highlights this – the below Matlab graphs in Figure 5.1.1 show a 3 period trinomial model, with
sample market prices for vanilla call options calculated by choosing a martingale measure at random.
The ‘extremal’ points of the model are { } and { } ; and at these values the market
prices for the call options (regardless of the chosen martingale measure ) will be and
{ } . We do not therefore introduce any new information into the market by
adding these constraints, and they can be discarded.
Figure 5.1.1 - Market prices for call options for a 3 period trinomial model. Note that vanilla call options with strikes at the
extremal levels do not provide any additional constraints in the market.
4. Call Options in Linear Programming Set up - Robust Hedging
As per the 2-period trinomial model, the introduction of vanilla call options introduces additional variables into the
linear programming problem that is used to determine the minimum superhedging cost for the exotic option. The
inequality representing the superhedging portfolio will be of the form:
∑
where this inequality must hold pathwise i.e. for all possible stock price paths; note , denote the value held
in stock and bonds at time (which will be dependent on ; denotes the stock price at maturity T (after
periods); represents the number of available vanilla call option constraints and represents the
payoff of an exotic option that may be path dependent.
We note that since the call options are statically traded at the inception of the superhedging portfolio, we are not
rebalancing them throughout the time period; and therefore no additional equality constraints are introduced into
the linear programming set up by additional vanilla call options being included in the superhedging portfolio.
5. Additional Path Dependent Exotic Call Options
In the n period trinomial model, there is a stock price path of length from the initial stock price to final stock
price . In a discrete time model, it is relatively simple to record and use these paths individually to define more
exotic path dependent payoffs, including options such as Asians, Lookbacks, Barrier options, and Rachet
(Cliquet) options. In the n-period setting we will consider this wider range of path dependent options, with payoffs
defined as per below:
A. Asian Option Payoff
40 60 80 100 120 140 160 1800
5
10
15
20
25
30
35
40
45
X: 57.87
Y: 42.13
Market Prices of Call Options
Vanilla Call Option Strike
Price o
f V
anill
a C
all
Option
X: 172.8
Y: 0
1 1.5 2 2.5 3 3.5 4
60
70
80
90
100
110
120
130
140
150
160
1.0 1.21.00.8 1.41.21.01.21.00.81.00.80.7 1.71.41.21.41.21.01.21.00.81.41.21.01.21.00.81.00.80.71.21.00.81.00.80.70.80.70.6
Time Period
Sto
ck P
rice
Stock Price Over Time
100.0
120.0
100.0
83.3
144.0
120.0
100.0
120.0
100.0
83.3
100.0
83.3
69.4
172.8
144.0
120.0
144.0
120.0
100.0
120.0
100.0
83.3
144.0
120.0
100.0
120.0
100.0
83.3
100.0
83.3
69.4
120.0
100.0
83.3
100.0
83.3
69.4
83.3
69.4
57.9
85
( )
for parameter K strike, and variable the stock price process with end value T.
B. Lookback Option Payoff
( )
for parameter K strike, and variable the stock price process with end value T.
C. Barrier – Down and Out Option Payoff
( )
where B represents the Barrier at which the option loses its value
D. Barrier – Up and In Option Payoff
( )
E. Cliquet Option Payoff
( )
where represents an intermediate time at which profits are ‘locked in’ and the strike reset to be the same level
as the existing call price.
6. Nonlinear Optimisation & Robust Price
As discussed in Chapter 4 Section 4.2.6 and 4.2.7, in order to derive an accurate robust price, we used the form
of the duality developed by Galichon, Henry-Labordere and Touzi in [8], repeated below:
Δ
∞{∫ Ε
} Μ
Ε
In terms of methodology, we devised a function that calculated for fixed the value of ∞{∫
Ε } (a nonlinear function), then used Matlab nonlinear optimisation routines to find the minimum
of this function.
In an n-period trinomial model, we take a similar approach. In the two dimensional case with one vanilla call
option as a constraint, the Δ were of the form (as described in Section 4.2.6):
In our general -period trinomial model, where we allow the user to set the number of overall call option
constraints, the function Δ will be of the form below:
86
∑
where represents the number of vanilla call options that we use as constraints, represent the amount of
the call option with strike that we will hold in our semi-static hedging portfolio.
As per the above discussion in ‘3. Call Options for every strike’, the maximum number of vanilla call options we
are able to add as constraints will vary depending on the number of periods in our trinomial model. In a two
period model, there are 5 possible values of , so we allow 5 possible values of strikes . However, as noted
above, call options with strikes at the extremal values of only provide new information about the market
measure. Given that we have for period trinomial model vanilla call options, we require that:
The nonlinear optimisation then in the period trinomial model involves optimising ∞{∫
Ε } over variables (i.e. the coefficients of the call options in the function
described above). We thus run an unconstrained nonlinear optimisation over these variables. The full
implementation algorithm used is detailed in the next Section 5.2.
We finally briefly describe the mathematical technique behind the Matlab implementation of the nonlinear
optimisation function (‘fminsearch’ or ‘fminunc‘) that we use to provide some context for the results that
we produce. Following overview in Nocedal and Wright in [26], general unconstrained optimisation routines
(linear and nonlinear) proceed from an initial point and generate a series of iterates { } that terminate
when a solution point is arrived at (within a defined level of accuracy) or no progress is made.
In moving from to there are two key strategies generally used, the line search strategy and trust region
strategy. We assume in below that we are looking to minimise some objective function
Firstly, the line search strategy involves the algorithm choosing a direction and searching in that direction from
existing iterate to determine a point at a certain distance where the objective function value is lower than its
value at . Clearly, the choice of the search direction is critical in these line search methods, and different
algorithms use different directions – for example, the steepest descent method sets (i.e. the gradient
of at ), while the Newton search direction is derived from a 2nd
order Taylor series
approximation which requires computation of the Hessian .
In the trust region strategy, a model function is constructed which has behaviour near point similar to the
objective function . We then look to solve the problem:
where lies inside the trust region. The model is typically defined to be a quadratic function of the form
where is some matrix, either the Hessian or an approximation to it.
If there is not a sufficient decrease in the objective function from the candidate solution, we shrink the trust region
and try again to solve . Thus at each step, we firstly set a distance (the radius of the trust region),
and we then look to determine a direction to previous candidate through minimising the model function. This is
opposed to the line search strategy methods, where we first set a search direction and then look to find a suitable
distance along it that produces a decrease in the objective function.
In terms of Matlab implementations, there are two methods available for unconstrained nonlinear optimisation.
The first method used (for the function fminsearch) is the Nelder-Mead method, which is broadly a variant of
the line search strategy method. It belongs to an approach described as ‘Derivative-Free optimisation’ where
rather than trying to approximate the gradients of objective functions being optimised, function values at a set of
sample points are used to determine a new iterate. In the case of Nelder-Mead, a simplex with vertices is
maintained, and at each iteration we look to remove the vertex with the worst function value and replace it with a
point with a more optimal value.
87
Since its introduction in 1965 the Nelder-Mead method has become a popular search method, though Nocedal
and Wright in [26] note that stagnation has been observed at non optimal points, and restarting the algorithm at
those points with set to be the terminal value of the final iteration can be used. We shall actually encounter this
issue in Section 5.3.3, and this ‘restarting’ technique is successively demonstrated in Figure 5.3.2.
The second Matlab function that can be used for unconstrained nonlinear optimisation is the function fminunc.
For large scale problems, this function uses a trust region method, based on the interior reflective Newton
method. For medium scale problems, the function uses the BGFS method (named after Broyden, Fletcher,
Goldfarb and Shanno who discovered it), a type of Quasi Newton line search method, where Quasi Newton
means that although the Hessian is not computed, the algorithm still achieves a rate of convergence which is
superlinear.
Matlab thus provides a range of algorithms, involving both the line search strategy and the trust region method for
nonlinear unconstrained optimisation. For our general n-period trinomial model, the Nelder-Mead method
(through the function fminsearch) has been used as the primary method of optimisation as it was found
overall to have a quicker convergence rate than alternatives.
5.2 Matlab Implementation
In this section we describe the implementation in Matlab of the period model trinomial model, with the variable
parameters and additional path dependent exotic options described in the previous section. We outline the key
features of the implementation and describe informally the key algorithms used in the model to generate output.
Full details of the Matlab code are given in the Appendix.
5.2.1 Implementing the Model: Varying Parameters
The below user interface demonstrates the key features of the Matlab implementation, with descriptions
subsequent detailing the rationale for the interface:
Figure 5.2.1 - Input GUI for Matlab implementation of an period trinomial model. Numbers correspond to descriptions below
detailing functionality.
Overview of GUI:
1. Market Parameters
1
2
3
5
6
7
8
10
4
9
88
These inputs allow the user to set the initial parameters in the market. In particular, the parameters that can be
set are: Initial Stock Price, Up Parameter, Interest rate and Number of Periods. In the original 2 period trinomial
model in Chapter, these values were set at: (note that the model counts as 3 -
discrete time periods i.e. ); we have previously described this as the 2-period trinomial
model).
2. Tradeable Assets
This field determines the type of assets to consider tradeable in the Matlab implementation. The option shown in
Figure 5.2.1 is for ‘Trading with risky asset and vanilla call options’; the alternative is to just allow trading in the
risky asset itself without including the vanilla call options as additional constraints.
3. Maximum number of call options available
This field relates to the number of vanilla call options that are tradeable in the market i.e. available to form the
super hedging portfolio for the exotic portfolio. This allows us to progressively examine the effects of introducing
additional call options into the market setting. This is as per Chapter 4 in Section 4.2 when we firstly introduced
one call option and then examined the effects on the robust price of progressively adding more constraints. The
default option of this field is set to ‘All’ i.e. consider that market prices for vanilla call options at all strikes are
available.
If less than the maximum number of call options is used as constraints, then the algorithm must ‘choose’ which
call options to use as constraints. For this implementation, the algorithm starts by using the call options with
strikes around the central values of and then successively moves up through each strike if more constraints
are required.
4. Tolerance on using only ‘relevant’ call options
As per discussion in previous section, prices for some call options contain no information i.e. where call options
have strikes at extremal points. In order to improve the robustness of the linear programming problem and reduce
the overall number of variables in the problem, it is helpful to disregard these extremal points. The ‘Tolerance’
field allows us to do this.
More precisely, note that call options with strikes near the extremal points will only provide ‘minimal’ information.
For example, for the next strike immediately less than the maximum value of , there will be only one possible
path for the stock price that will generate a non zero payoff for a vanilla call option set at this strike i.e. if the stock
price increases every time from the initial value . The value of the vanilla call option with this strike will
therefore be close to zero – and in fact depending on the degree of accuracy required in our linear programming
problem, may not generate any significance difference if included or not.
For these vanilla call options near extremal points with ‘minimal’ information, the ‘Tolerance’ field allows the user
to exclude these if the market call prices are within the defined tolerance of either 0 (for strikes close to the
maximum) or { } (for minimum points).
5. Generate Market Prices
This field is used to determine the market prices of the vanilla call options that are used as constraints in the
period trinomial model. The first option (which must be selected the first time the optimisation is run) is to
‘Generate New Prices’. This option randomly generates a martingale measure for the market that is then used to
derive the market prices of the vanilla call options. The algorithm used to do this is described at a high level
below.
The other option, which may be selected after the optimisation programme has run initially, is to ‘Use Existing
Market Prices’. In this case, the martingale measure and market call prices from the previous running of the
optimisation routine are used rather than creating a different set; this allows the user to set the market prices at
beginning of a session and investigate different parameters / exotic options prices using the same set of market
call prices.
89
Algorithm 5.2.1 - Algorithm for Randomly Generating Martingale Measure and set market prices of vanilla call options
Strategy: to generate the ‘middle probabilities’ for each fork in the trinomial model randomly from uniform distribution, and then the already established formulas for the upper and lower probabilities from Chapter 4 Section 4.1.1 can be used to establish the full probabilities for each fork. When combined across all forks, these probabilities constitute the fully specified measure.
Step 1. Use the uniform distribution generator from Matlab to generate a set of probabilities – one value for each
fork in the trinomial tree.
The below example, taken from the algorithm illustrates the generation of these probabilities in the required format.
T = 1 T = 2 T = 3
0.6160 0.4733 0.5853
- 0.3517 0.5497
- 0.8308 0.9172
- - 0.2858
- - 0.7572
- - 0.7537
- - 0.3804
- - 0.5678
- - 0.0759
Figure 5.2.3 – LHS: Example three period trinomial tree with various ‘trinomial forks. RHS: Example: Randomly Generated Measures corresponding to trinomial forks. Each entry corresponds to the intermediate probability for the corresponding fork.
Step 2. Set the generated probabilities as the ‘middle probabilities’, and for each trinomial fork, use the below
relationships derived in Section 4.1 to derive the probabilities for the upper and lower probabilities in the fork:
[
]
[
]
Step 3: For each possible final value of the Stock Price we set the strike K, and use the formula
to iteratively determine the interim values and ultimately final values of the Call Market Prices.
6. Choosing an Exotic Option
This field is used to set the path dependent exotic option that is evaluated. Payoffs are described in the previous
section for this fixed range of exotic options.
7. Additional parameters for exotic options
These fields are used to set the strike for the exotic option; as well as capture the additional parameters required
to price the option. In particular, the barrier for the Down and Out options as well as the Up and In options are
specified here; as well as the time of reset for the Cliquet option.
8. Optimisation Method – Robust Hedge or Robust Price
These checkboxes are used to determine whether to calculate a robust hedge for the path dependent option or a
robust price.
1.5 2 2.5 3 3.5 440
60
80
100
120
140
160
Time Period
Sto
ck P
rice
Stock Price Over Time
100.0
120.0
100.0
83.3
144.0
120.0
100.0
120.0
100.0
83.3
100.0
83.3
69.4
172.8
144.0
120.0
144.0
120.0
100.0
120.0
100.0
83.3
144.0
120.0
100.0
120.0
100.0
83.3
100.0
83.3
69.4
120.0
100.0
83.3
100.0
83.3
69.4
83.3
69.4
57.9
90
∑
Both the robust hedges and robust prices optimisation algorithms are straightforward extensions of the methods
used in Chapter 4 in the 2 period trinomial model, and described briefly below:
Algorithm 5.2.2 - Algorithm for Calculating Robust Hedge
Strategy: Algorithm solves a single linear programming problem of the form:
∑
Matrices are of the form:
A: a (X x Z) matrix encoding the inequality constraints; where X = and ∑ , where is the
number of call option constraints, and is the number of periods. Individual rows are of the form:
i.e. the values of stock and bond at maturity for a particular stock price path that form the superhedging
portfolio
f: a (Z x 1) column vector with number of rows Z = ∑ i.e. a row for each variable representing the
asset and bond allocation at each stock price and time; and call option constraints. Entries in this column
represent the amounts invested in stock / bonds / call options at any given period for the superhedging portfolio.
B: a (X x 1) column vector with number of rows X = with row entries representing the payoff of the path
dependent option for each of the potential paths of the stock.
: a (Z’ x X) matrix representing the LHS of the equation for the equality constraints representing the self-
financing constraints, where ∑ . For the linear programming problem, these are of the form:
where the represent the values of the stock and the bond at some , and represent
the value of at the preceding time . The zeros on the RHS of the matrix are required to codify the fact that the
vanilla call options are not dynamically traded.
: a (Z x 1) matrix representing the RHS of the equation for the equality constraints representing the self-
financing constraints.
Algorithm 5.2.3 - Algorithm for Calculating Robust Price
Strategy: Use the below form of the duality:
{∫ }
Firstly, construct a function that evaluates {∫ } for fixed . Then, having
chosen a suitable start point for the algorithm, use unconstrained nonlinear optimisation to determine the
minimum over all .
91
Step 1: Fix , where will be of the form
∑
i.e. a linear combination of tradeable vanilla call options
For this fixed function , determine through backwards induction from maturity T the value of
. More precisely:
a) At maturity n, for each stock price path, is known (as is the defined path dependent option
being evaluated, and is a linear combination of vanilla call options which have defined payoffs.
b) For each trinomial fork in the final stock price tree, solve a linear programming to determine the maximum
value at the preceding time of an option with payoff at time . The linear programming
problem is of the form:
where represents the expectation at time , and is an (unconstrained) probability measure for that
trinomial fork. As such, the equality constraints in the linear programming problem are of the form:
[
]
[
] [
]
where , and is the value at time if the stock has moved up since
etc.
Note the first row of the equality constraint matrix above is driven by risk neutral pricing principle i.e. that the
value of the path dependent option at time period is the discounted expectation of the option values at time
under a risk neutral measure. The second row codifies the requirement that the stock prices must be a
martingale under the risk neutral measure. The third row states that , , form a probability measure.
c) Repeat step (b) for each of the intermediate time steps, solving a linear programming problem for each
trinomial fork at each time period to derive the maximum interim value of the option.
The final value then will be as required:
d) The final stage is to add the fixed ∫ to the value determined in step (c) and determine
{∫ }
Step 2: Having defined, through Step 1, the value of for fixed , run a nonlinear
optimisation routine to minimise the value of {∫ } over all .
In order to run the nonlinear optimisation routine, the first step is to determine an appropriate start point for the
algorithm.
a) Determining a start point
In order to determine an appropriate start point, one option is to firstly evaluate the value of
{∫ } for a certain pre-defined set of potential , and then choose the
minimal value over this pre-defined range as a start point.
In the current Matlab implementation, the pre-defined set of potential is the set defined by:
92
∑
where { }. The starting point of the nonlinear optimisation is then the minimal value of
{∫ } over this restricted set of .
b) Running the nonlinear optimisation
Once the start point is determined, then the non-linear optimisation routine is called using the Matlab function
‘fminsearch’ with the start point from step a).
The final value of this non-linear optimisation is then as required:
{∫ }
5.2.2 Implementation Output: Results
Once the inputs and parameters have been set as required, the below output screen is produced from the model,
which is briefly described below.
Figure 5.2.2 – Output GUI for Matlab implementation of an period trinomial model. Numbers correspond to descriptions
below detailing functionality.
1. Path Dependent Option parameters
These fields describe the exotic option that has been priced – in this example, a Down and Out Barrier option
with Barrier and strike .
2. Model Dependent Price
This field gives the value of the path dependent option when evaluated under the same market measure
that was used to generate the market prices for the vanilla call options. In other words, it is the value of:
( ) [ ( )]
It is a model dependent view of the price, in that a particular market measure has been chosen to price the path
dependent option.
3. Market Call Prices
2
3 4
5 6 7
1
93
This graph shows the market prices of call options for different strikes using the martingale measure .
These are the vanilla call options that have been used as part of the robust hedging or robust pricing algorithms.
4. Stock Price Paths
This graph shows the potential stock price paths based on the input parameters – the example shown is of a 3
period trinomial model . The data labels show the stock prices at each level.
5. Model Independent Price & Hedge – Risky Assets Only
This panel describes the robust hedge and robust price when considering trading in just the risky asset i.e.
without adding any additional call options as a constraint. It displays both the robust price and robust hedge
separately, and these are populated depending on the inputs selected.
6. Model Independent Hedge – Trading in Risky Assets and Vanilla Call options
This panel describes the robust hedge and robust price when trading is permissible in both the risky asset as well
as a set of vanilla call options. It contains the Number of Call options used as constraints; and the next line down
is the Robust Hedge calculated based on this number of constraints. The final line is then the Robust Hedge
calculated based on the tolerance level set by the user i.e. using as call options only those that are deemed
‘relevant’ as described in the previous section.
7. Model Independent Price – Trading in Risky Assets and Vanilla Call options
This final panel displays the robust price based on the nonlinear optimisation routine. The first display shows the
starting value used for the nonlinear optimisation routine; the second field shows the status of this optimisation.
The final field then displays the Robust Price, based on the ‘relevant’ call options.
5.3 Model Results
Having built a flexible framework in Matlab, we can now use this implementation to derive results for robust
hedges and prices for a variety of parameters and financial instruments. These are shown over the subsequent
pages.
We provide results for a set of path dependent options in a two, three, four and five period trinomial model across
a range of strikes. We note subsequently some of the limitations in the construction of the implementation that
prevent the model producing results for higher values (although results are possible, but not shown, for values of
up to approximately 9)
94
5.3.1 Two period Trinomial Model – Exotic Options
0. Overview – Two period trinomial model with wider range of exotics
Robust pricing and hedging for a range of path dependent options for a range of strikes in the two period trinomial model already examined in Chapter 4, with zero interest rate
and initial stock price of 100; and using an example set of Market prices for vanilla call options.
1. Market Parameters – Call Option prices & Stock Price over Time
Probability Measure –
‘Middle Probabilities’ for each trinomial fork
n=1 n=2
0.043 0.64
0.28
0.54
2. Exotic Option Prices – Values of Path Dependent Options for range of strikes
Exotic Option Q Market Measure Robust Hedge - Risky Asset Only Robust Hedge - 1 call Option Only Robust Hedge - All call options Robust Price - All call options
Strike: 80 90 100 110 120
80 90 100 110 120
80 90 100 110 120
80 90 100 110 120
80 90 100 110 120
Asian 20.000 11.067 5.895 1.739 0.096
20.000 11.708 6.061 2.342 0.275
20.000 11.086 5.988 2.313 0.272
20.000 11.086 5.988 1.786 0.096
20.009 11.094 6.062 1.804 0.103
Lookback 30.713 20.713 10.713 6.223 1.732
34.050 24.050 14.050 9.504 4.959
33.881 23.881 13.881 9.390 4.899
30.713 20.713 10.731 6.223 1.732
30.713 20.713 10.731 6.231 1.732
Barrier - Dow n and Out, Barrier: 85 18.204 13.593 8.982 5.357 1.732
20.000 13.636 9.091 7.025 4.959
18.204 13.593 8.982 6.940 4.899
18.204 13.593 8.982 5.357 1.732
19.376 13.638 8.982 5.412 1.732
Barrier - Up and In,
Barrier: 120 4.618 3.897 3.175 2.453 1.732
13.223 11.157 9.091 7.025 4.959
13.064 11.023 8.982 6.940 4.899
4.618 3.897 3.175 2.453 1.732
4.626 3.902 3.179 2.455 1.732
Cliquet, reset at T = : 2 23.836 17.317 12.539 8.187 3.836
29.091 22.727 18.182 13.636 9.091
28.982 22.574 17.963 13.472 8.982
28.523 20.129 12.539 10.531 8.523
28.982 20.252 12.539 10.569 8.982
Explanation of terms: 1. Q-Market Measure: price derived for path dependent option under assumption of specific market measure (the one described in table above and used to price vanilla call
options; 2. Robust Hedge – Risky Asset Only: minimum capital required for robust hedge if only permissible tradeable instrument is the risky asset; 3. Robust Hedge – 1 Call option only: minimum
capital required for robust hedge if permitted to trade risky asset and one vanilla call option; 4. Robust Hedge – All Call Options: minimum capital required if permitted to trade risky asset and all
vanilla call options; 5. Robust Price – All Call Options: value of {∫
} with all call options potentially tradeable
60 70 80 90 100 110 120 130 140 1500
5
10
15
20
25
30
35
X: 83.33
Y: 18.49
Market Prices of Call Options
Vanilla Call Option Strike
Price o
f V
anill
a C
all
Option
X: 120
Y: 1.732
X: 100
Y: 8.982
X: 69.44
Y: 30.56
X: 144
Y: 0
1 1.5 2 2.5 3 3.570
80
90
100
110
120
130
140
100.0
120.0
100.0
83.3
144.0
120.0
100.0
120.0
100.0
83.3
100.0
83.3
69.4Time Period
Sto
ck P
rice
Stock Price Over Time
100.0
120.0
100.0
83.3
144.0
120.0
100.0
120.0
100.0
83.3
100.0
83.3
69.4
100.0
120.0
100.0
83.3
144.0
120.0
100.0
120.0
100.0
83.3
100.0
83.3
69.4
100.0
120.0
100.0
83.3
144.0
120.0
100.0
120.0
100.0
83.3
100.0
83.3
69.4
100.0
120.0
100.0
83.3
144.0
120.0
100.0
120.0
100.0
83.3
100.0
83.3
69.4
100.0
120.0
100.0
83.3
144.0
120.0
100.0
120.0
100.0
83.3
100.0
83.3
69.4
95
5.3.2 Three period Trinomial Model – Exotic Options
0. Overview – Three period trinomial model and range of exotics
Robust pricing and hedging for a range of path dependent options for a range of strikes in a three period trinomial model, with zero interest rate and initial stock price of 100;
and using an example set of Market prices for vanilla call options.
1. Market Parameters – Call Option prices & Stock Price over Time
Probability Measure – ‘Middle
Probabilities’ for each trinomial fork
n=1 n=2 n=3
0.74 0.24 0.77
- 0.92 0.19
- 0.27 0.29
- - 0.09
- - 0.58
- - 0.68
- - 0.55
- - 0.43
- - 0.64
2. Exotic Option Prices – Values of Path Dependent Options for range of strikes
Exotic Option 1. Q Market Measure 2. Robust Hedge - Risky Asset Only 3. Robust Hedge - 1 call Option Only 4. Robust Hedge - All call options 5. Robust Price - All call options
Strike 80 90 100 110 120
80 90 100 110 120
80 90 100 110 120
80 90 100 110 120
80 90 100 110 120
Asian 20.026 10.660 2.721 0.909 0.297
20.379 12.810 6.912 3.512 1.446 20.253 11.872 4.542 2.340 0.963
20.026 11.271 4.542 1.892 0.441
20.028 11.361 4.542 2.121 0.442
Lookback 27.376 17.376 7.376 4.507 1.638
39.008 29.008 19.008 13.336 7.663 31.161 21.161 11.161 8.133 5.105
27.694 17.694 7.694 4.666 1.638
27.694 17.694 7.694 6.456 1.638
Barrier - Dow n and Out, Barrier: 85 18.828 12.344 5.860 3.686 1.512
20.000 14.538 11.345 8.152 4.959
19.030 12.422 6.056 4.679 3.303
18.902 12.422 6.056 3.784 1.512
19.065 13.678 6.056 4.679 1.512
Barrier - Up and In, Barrier: 120 4.424 3.696 2.968 2.240 1.512
17.731 14.538 11.345 8.152 4.959
9.465 7.760 6.056 4.679 3.303
5.164 4.251 3.338 2.425 1.512
5.167 5.213 6.056 2.428 1.512
Cliquet, reset at T = : 2 25.359 16.310 7.737 6.548 5.359
29.091 22.727 18.182 13.636 9.091 26.056 18.478 12.111 9.084 6.056
26.056 16.968 8.605 7.219 6.056
26.056 16.968 8.605 7.368 6.056
Table 5.3.2 – Model results for five path dependent options for range of strikes in three period trinomial model
40 60 80 100 120 140 160 1800
5
10
15
20
25
30
35
40
45
X: 69.44
Y: 30.68
Market Prices of Call Options
Vanilla Call Option Strike
Price o
f V
anill
a C
all
Option
X: 57.87
Y: 42.13
X: 83.33
Y: 17.7
X: 100
Y: 6.056
X: 120
Y: 1.512 X: 144
Y: 0.1256 1 1.5 2 2.5 3 3.5 4
60
70
80
90
100
110
120
130
140
150
160
100.0
120.0
100.0
83.3
144.0
120.0
100.0
120.0
100.0
83.3
100.0
83.3
69.4
172.8
144.0
120.0
144.0
120.0
100.0
120.0
100.0
83.3
144.0
120.0
100.0
120.0
100.0
83.3
100.0
83.3
69.4
120.0
100.0
83.3
100.0
83.3
69.4
83.3
69.4
57.9
Time Period
Sto
ck P
rice
Stock Price Over Time
96
3. Analysis and Points of Note
Prices for path dependent options under the ‘Q-Market measure’ are always the smallest compared to other methods for deriving a price. This is as expected – we
have chosen a particular martingale measure and priced the path dependent option using this measure, whereas in the other cases we have not assumed this level of
knowledge of the Q-Market measure, but instead inferred facts about it from prices of vanilla call options
Robust hedging with the risky asset only in all instances gives the highest price, as expected. Adding an additional call option will improve the accuracy of the hedge
i.e. the value will be lower; and adding all call options in general further improves the accuracy of the robust hedge
In general, we can see numerically that the form of the robust hedging / pricing duality we have been evaluating holds. More particularly, we can see that in most
cases we have:
∑
{∫ }
For specific examples of this, compare the sections of the table entitled ‘4. Robust Hedge - All call options’ and ‘5. Robust Price - All call options’. We see that the
values are identical for example for all strikes except for Asian, Lookback and Cliquet options in particular.
In cases however, where the duality does not seem to hold, e.g. for Asian, Lookback and Cliquet options, in all cases the robust price is greater than the
robust hedge. However as seen in Chapter 3, a simple no-arbitrage argument concludes that the cost of the robust superhedge must be greater than the upper
martingale price. This indicates therefore, an issue in this particular numerical implementation. In the next set of results, we show how the algorithm can be improved
to give a more accurate result for cases where the duality does not appear to hold
Finally, we observe that in all cases, even adding a full set of call options the robust hedges and prices calculated are not the same as the model dependent prices.
This contrasts with the results in Chapter 4, where we saw that the addition of multiple call options was enough to determine a robust hedge that was identical to the
price under the ‘Q-Market measure’.
We can briefly consider the accuracy of the nonlinear optimisation algorithm that is used to calculate the section of the table entitled ‘5. Robust Price - All call options’.
The graphs on the following page show the convergence of the algorithm from its start point for a selection of the above options. The top set of graphs shows the
value of the objective function for each iteration of the algorithm, and the bottom set of graphs show the values of the call options used to hedge at the termination of
the algorithm.
In the top row, Chart 1 for an Asian option shows that the algorithm takes about 60 iterations to converge to an answer, that in this case is equal to the value
calculated for the robust hedge as expected from the above duality. Chart 2 shows that for strike , the optimisation starts at a value of 3.7 , and converges to
a minimum value of after 160 iterations; indicating that the algorithm does move the objective functions value significantly and can take a significant number of
iterations to do so. Finally, chart 3 shows that the starting point of the algorithm actually is the minimum value of termination. A moment’s reflection and Chart 6
illustrates why this is the case – an up and in barrier option in the three trinomial model with barrier , and strike can be perfectly hedged with the
vanilla call option with strike . Chart 6 shows that our algorithm has indeed found that in fact just one call option (denoted variable 4 in the graph
corresponding to the vanilla call option with strike ) is needed to minimise the function {∫ }.
97
Chart 1: Asian Option – Strike
Objective value of function
Chart 2: Asian Option – Strike
Objective value of function
Chart 3:Barrier Up and In – Strike
Objective value of function
Chart 4: Asian Option – Strike
Amount of call options held at termination
Chart 5: Asian Option – Strike
Amount of call options held at termination
Chart 6:Barrier Up and In – Strike
Amount of call options held at termination
Figure 5.3.1 – Demonstration of convergence of the nonlinear optimisation algorithm evaluating {∫
} for various path dependent options, for a three
period trinomial model
0 10 20 30 40 50 604.4
4.6
4.8
5
5.2
5.4
5.6
5.8
6
6.2
Iteration
Function v
alu
eCurrent Function Value: 4.54218
0 20 40 60 80 100 120 140 160 1802
2.2
2.4
2.6
2.8
3
3.2
3.4
3.6
3.8
Iteration
Function v
alu
e
Current Function Value: 2.12105
0 10 20 30 40 50 600.5
1
1.5
2
2.5
3
Iteration
Function v
alu
e
Current Function Value: 1.51244
1 2 3 4 50
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
Number of variables: 5
Curr
ent
poin
t
Current Point
1 2 3 4 5-0.4
-0.3
-0.2
-0.1
0
0.1
0.2
0.3
Number of variables: 5
Curr
ent
poin
t
Current Point
1 2 3 4 50
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Number of variables: 5
Curr
ent
poin
t
Current Point
98
5.3.3 Four period Trinomial Model – Exotic Options
0. Overview – Four period trinomial model and range of exotics
Robust pricing and hedging for a range of path dependent options for a range of strikes in a four period trinomial model, with zero interest rate and initial stock price of 100; and
using an example set of Market prices for vanilla call options.
1. Market Parameters – Call Option prices & Stock Price over Time
Probability Measure – ‘Middle
Probabilities’ for each trinomial fork
n=1 n=2 n=3 n=4
0.60 0.09 0.24
Not Shown
- 0.34 0.38
- 0.14 0.27
- - 0.22
- - 0.63
- - 0.80
- - 0.21
- - 0.90
- - 0.75
2. Exotic Option Prices – Values of Path Dependent Options for range of strikes
Exotic Option 1. Q Market Measure 2. Robust Hedge - Risky Asset Only 3. Robust Hedge - 1 call Option Only 4. Robust Hedge - All call options 5. Robust Price - All call options
Strike:
80
90
100
110
120
80
90
100
110
120
80
90
100
110
120
80
90
100
110
120
80
90
100
110
120
Asian 20.215
11.510
5.747
2.603
0.982
21.022
13.676
8.200
4.563
2.345
21.022
13.676
8.166
4.294
1.966
20.417
12.904
7.583
4.220
1.966
20.586
13.599
8.153
4.519
2.334
Lookback
35.302
25.302
15.302
10.549
5.796
42.942
32.942
22.942
17.270
11.598
39.704
29.704
19.704
14.032
8.359
36.348
26.348
16.348
11.144
5.941
36.348
26.348
16.348
11.392
5.941
Barrier - Dow n and Out, Barrier: 85
18.067
13.866
9.663
7.055
4.446
20.000
14.538
11.345
9.381
7.418
18.888
14.461
11.153
7.844
4.536
18.215
14.163
10.407
7.471
4.536
18.416
14.537
10.407
7.844
4.536
Barrier - Up and In, Barrier: 120
12.956
10.792
8.628
6.582
4.536
21.010
17.304
13.599
11.123
8.647
16.603
13.586
10.569
7.553
4.536
15.057
12.427
9.796
7.166
4.536
16.243
13.391
9.796
7.400
4.536
Cliquet, reset at T = : 2 29.569
21.007
13.165
11.367
9.569
33.599
27.235
22.690
18.144
13.599
33.407
24.947
19.801
15.503
13.407
31.102
22.834
16.342
13.392
11.102
31.176
23.395
16.342
13.931
11.176
40 60 80 100 120 140 160 180 200 2200
10
20
30
40
50
60Market Prices of Call Options
Vanilla Call Option Strike
Price o
f V
anill
a C
all
Option
X: 48.23
Y: 51.77
X: 69.44
Y: 30.88
X: 83.33
Y: 19.38
X: 100
Y: 10.41
X: 120
Y: 4.536 X: 144
Y: 1.22X: 172.8
Y: 0.185
X: 57.87
Y: 42.16
1 1.5 2 2.5 3 3.5 4 4.5 5 5.550
100
150
200
Time PeriodS
tock P
rice
Stock Price Over Time
100.0
120.0
100.0
83.3
144.0
120.0
100.0
120.0
100.0
83.3
100.0
83.3
69.4
172.8
144.0
120.0
144.0
120.0
100.0
120.0
100.0
83.3
144.0
120.0
100.0
120.0
100.0
83.3
100.0
83.3
69.4
120.0
100.0
83.3
100.0
83.3
69.4
83.3
69.4
57.9
207.4
172.8
144.0
172.8
144.0
120.0
144.0
120.0
100.0
172.8
144.0
120.0
144.0
120.0
100.0
120.0
100.0
83.3
144.0
120.0
100.0
120.0
100.0
83.3
100.0
83.3
69.4
172.8
144.0
120.0
144.0
120.0
100.0
120.0
100.0
83.3
144.0
120.0
100.0
120.0
100.0
83.3
100.0
83.3
69.4
120.0
100.0
83.3
100.0
83.3
69.4
83.3
69.4
57.9
144.0
120.0
100.0
120.0
100.0
83.3
100.0
83.3
69.4
120.0
100.0
83.3
100.0
83.3
69.4
83.3
69.4
57.9
100.0
83.3
69.4
83.3
69.4
57.9
69.4
57.9
48.2
99
3. Analysis and Points of Note
Note that in this extended four period trinomial model, we still have the expected order among the prices that have been calculated. In other words, we have the below
‘ordering’ of prices, with most ‘model dependent’ on the LHS, and most ‘model independent’ on the RHS:
This aligns with the expected order – namely, as we increasingly add more constraints to the model (in the form of market prices for vanilla call options) we have more
information about the price of the path dependent option.
Note that the duality we have been considering i.e. ∑
{∫ } is not so clearly
demonstrated as for the two period or three period model in this Matlab implementation. For the Lookback option, comparing Section 4 and Section 5 of Table 2 above,
we see that the duality in fact numerically holds. However, for the Asian option, the algorithm is less accurate and there is more significant divergence between the
‘Robust Hedge’ and the ‘Robust Price’ (for example, with strike , the Robust hedge is calculated at , whereas the Robust Price is )
One reason for this is the time taken for the nonlinear optimisation algorithm to converge, and the overall accuracy of the ‘fminsearch’ function in Matlab. The
above results were all derived using a maximum iterations figure of 200 and using the start point generated through algorithm 5.2.3. There are therefore two relatively
simple methods that are available to improve the accuracy of the algorithm. Firstly, we can increase the number of maximum iterations of the nonlinear optimisation.
Secondly, as was highlighted in Section 5.1 when discussing the Nelder-Mead algorithm and practical limitations of it, we can rerun the nonlinear optimisation with the
terminal point of the previous non-linear optimisation to ensure a more accurate start point. This steps are demonstrated on the subsequent page for the Asian strike
, which had initial Robust Price of , compared to a Robust Hedge of
From the below diagrams, we see that repeated applications of these steps (increasing the number of initial iterations and using the terminal value of the optimisation
as the starting point of the subsequent iteration), we see that in fact the final value calculated for the Robust Price is 7.5833, exactly equal to that for the Robust
Hedge. Therefore we see that the duality ∑
{∫ } holds; however the Matlab
implementation is not sophisticated enough to quickly determine the value of {∫ }
Finally, in the case of the Asian option with strike , we note from the final diagram Figure X.D below that ultimately, only one constraint i.e. vanilla call option is
used in the final value of {∫ }. This constraint actually corresponds to the vanilla call option with strike ; and so
had we used just this constraint in the optimisation, the algorithm would have converged much quicker. This suggests that the pricing of individual path dependent
options could be achieved more quickly in particular cases by limiting the number of constraints, though the Matlab implementation as it stands offers a wider degree
of flexibility
100
Figure A – Increasing number of maximum iterations to 1000. Algorithm terminates at 8.14
after c. 450 iterations
Figure B – Starting the nonlinear optimisation routine at the terminal value of the previous
nonlinear optimisation shown in Figure A. After 500+ iterations, the optimisation routine now terminates at 7.80965
Figure C – Repeating the process by starting the 3
rd nonlinear optimisation at the terminal
value of the 2nd
optimisation shown in Figure B. The routine now terminates at 7.60132
Figure.D – Repeating this process two more times, we eventually arrive at a nonlinear
optimisation which terminates at 7.5833, the same value as the Robust Hedge
Figure 5.3.2 – Repeated application of non-linear optimisation to numerically demonstrate duality for Asian Option
0 50 100 150 200 250 300 350 400 450 5008.14
8.15
8.16
8.17
8.18
8.19
8.2
8.21
8.22
8.23
Iteration
Function v
alu
e
Current Function Value: 8.14039
1 2 3 4 5 6 7-0.1
0
0.1
0.2
0.3
Number of variables: 7
Curr
ent
poin
t
Current Point
0 100 200 300 400 500 6007.8
8
8.2
8.4
Iteration
Function v
alu
e
Current Function Value: 7.80965
1 2 3 4 5 6 7-0.1
0
0.1
0.2
0.3
Number of variables: 7
Curr
ent
poin
t
Current Point
0 50 100 150 200 250 300 350 400 450 5007.6
7.7
7.8
7.9
Iteration
Function v
alu
e
Current Function Value: 7.60132
1 2 3 4 5 6 7-0.1
0
0.1
0.2
0.3
Number of variables: 7
Curr
ent
poin
t
Current Point
0 50 100 150 200 250 300 3507.583
7.584
7.585
7.586
7.587
Iteration
Function v
alu
e
Current Function Value: 7.58327
101
5.3.4 Five period Trinomial Model – Exotic Options
0. Overview – Five period trinomial model and range of exotics
Selected Robust hedging and Robust Pricing for trinomial models with 5 or more periods for a variety of exotic path dependent options.
1. Market Parameters – Call Option prices & Stock Price over Time
Probability Measure – ‘Middle Probabilities’ for each trinomial fork
n=1 n=2 n=3 n=4 n=5
0.18 0.24 0.98
Not Shown
Not Shown
- 0.75 0.71
- 0.20 0.18
- - 0.86
- - 0.91
- - 0.96
- - 0.57
- - 0.56
- - 0.18
2. Exotic Option Prices – Values of Path Dependent Options for range of strikes
Exotic Option - 5 period trinomial model 1. Q Market Measure 2. Robust Hedge - Risky Asset Only 3. Robust Hedge - 1 call Option Only 4. Robust Hedge - All call options 5. Robust Price - 1 call Option Only
Strike: 80
90
100
110
120
80
90
100
110
120
80
90
100
110
120
80
90
100
110
120
80
90
100
110
120
Asian 20.979
13.198
7.576
3.982
1.857
21.561
14.473
9.249
5.531
3.129
21.561
14.428
8.770
5.407
3.129
21.391
14.251
8.770
5.165
2.763
Not Calculated
8.769
Not Calculated
Lookback
37.938
27.938
17.938
12.627
7.316
46.877
36.877
26.877
20.645
14.414
45.051
35.051
25.051
19.203
13.530
39.292
29.292
19.292
13.426
7.560
25.051
Barrier - Dow n and Out,
Barrier:
85
17.858
14.284
10.711
7.950
5.190
20.000
14.985
12.463
9.940
7.418
18.248
14.985
12.463
9.940
7.418
18.130
14.626
11.566
8.506
5.445
12.463
Barrier - Up and In, Barrier:
120
16.803
13.852
10.909
8.177
5.442
24.363
20.266
16.393
12.520
8.647
20.104
16.327
12.550
10.313
8.075
19.695
16.123
12.550
8.998
5.445
12.550
Cliquet, reset at T = :
2
30.278
23.264
17.744
14.011
10.278
33.599
27.235
22.690
18.144
13.599
33.122
26.759
22.213
17.668
13.122
33.086
25.170
19.377
15.860
13.086
22.213
Note: in determing the Robust Price only one call option has been used to ensure convergence of nonlinear optimisation algorithm
0 50 100 150 200 2500
10
20
30
40
50
60
X: 48.23
Y: 51.78
Market Prices of Call Options
Vanilla Call Option Strike
Price o
f V
anill
a C
all
Option
X: 69.44
Y: 32.06
X: 40.19
Y: 59.81
X: 83.33
Y: 21.51
X: 100
Y: 12.55
X: 120
Y: 5.445X: 144
Y: 1.702X: 172.8
Y: 0.412X: 207.4
Y: 0.0008035
X: 57.87
Y: 42.2
1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 6 6.5
60
80
100
120
140
160
180
200
220
240
Time Period
Sto
ck P
rice
Stock Price Over Time
100.0
120.0
100.0
83.3
144.0
120.0
100.0
120.0
100.0
83.3
100.0
83.3
69.4
172.8
144.0
120.0
144.0
120.0
100.0
120.0
100.0
83.3
144.0
120.0
100.0
120.0
100.0
83.3
100.0
83.3
69.4
120.0
100.0
83.3
100.0
83.3
69.4
83.3
69.4
57.9
207.4
172.8
144.0
172.8
144.0
120.0
144.0
120.0
100.0
172.8
144.0
120.0
144.0
120.0
100.0
120.0
100.0
83.3
144.0
120.0
100.0
120.0
100.0
83.3
100.0
83.3
69.4
172.8
144.0
120.0
144.0
120.0
100.0
120.0
100.0
83.3
144.0
120.0
100.0
120.0
100.0
83.3
100.0
83.3
69.4
120.0
100.0
83.3
100.0
83.3
69.4
83.3
69.4
57.9
144.0
120.0
100.0
120.0
100.0
83.3
100.0
83.3
69.4
120.0
100.0
83.3
100.0
83.3
69.4
83.3
69.4
57.9
100.0
83.3
69.4
83.3
69.4
57.9
69.4
57.9
48.2
248.8
207.4
172.8
207.4
172.8
144.0
172.8
144.0
120.0
207.4
172.8
144.0
172.8
144.0
120.0
144.0
120.0
100.0
172.8
144.0
120.0
144.0
120.0
100.0
120.0
100.0
83.3
207.4
172.8
144.0
172.8
144.0
120.0
144.0
120.0
100.0
172.8
144.0
120.0
144.0
120.0
100.0
120.0
100.0
83.3
144.0
120.0
100.0
120.0
100.0
83.3
100.0
83.3
69.4
172.8
144.0
120.0
144.0
120.0
100.0
120.0
100.0
83.3
144.0
120.0
100.0
120.0
100.0
83.3
100.0
83.3
69.4
120.0
100.0
83.3
100.0
83.3
69.4
83.3
69.4
57.9
207.4
172.8
144.0
172.8
144.0
120.0
144.0
120.0
100.0
172.8
144.0
120.0
144.0
120.0
100.0
120.0
100.0
83.3
144.0
120.0
100.0
120.0
100.0
83.3
100.0
83.3
69.4
172.8
144.0
120.0
144.0
120.0
100.0
120.0
100.0
83.3
144.0
120.0
100.0
120.0
100.0
83.3
100.0
83.3
69.4
120.0
100.0
83.3
100.0
83.3
69.4
83.3
69.4
57.9
144.0
120.0
100.0
120.0
100.0
83.3
100.0
83.3
69.4
120.0
100.0
83.3
100.0
83.3
69.4
83.3
69.4
57.9
100.0
83.3
69.4
83.3
69.4
57.9
69.4
57.9
48.2
172.8
144.0
120.0
144.0
120.0
100.0
120.0
100.0
83.3
144.0
120.0
100.0
120.0
100.0
83.3
100.0
83.3
69.4
120.0
100.0
83.3
100.0
83.3
69.4
83.3
69.4
57.9
144.0
120.0
100.0
120.0
100.0
83.3
100.0
83.3
69.4
120.0
100.0
83.3
100.0
83.3
69.4
83.3
69.4
57.9
100.0
83.3
69.4
83.3
69.4
57.9
69.4
57.9
48.2
120.0
100.0
83.3
100.0
83.3
69.4
83.3
69.4
57.9
100.0
83.3
69.4
83.3
69.4
57.9
69.4
57.9
48.2
83.3
69.4
57.9
69.4
57.9
48.2
57.9
48.240.2
102
5.3.5 Some Remarks on the Limits of the algorithm
Performance of Robust Hedging Algorithm 5.2.2
Algorithm 5.2.2 used to determine the robust hedge is not scalable for large numbers of periods in the trinomial
model. The algorithm is based on a solving a single linear programming problem with an increasingly large
number of variables and constraints, and as such, the complexity of solving this single problem rapidly grows too
much.
The below table illustrates the number of variables, constraints and indicative time taken to run for various values
of . The limit of the algorithm is approximately , where the number of variables and constraints involved.
Robust Hedge – Algorithm 5.2.2
No. of
periods
Number of
Variables (Stock
and Bond only)
Size of Stock
Price tree
No. of Linear
Equality
Constraints
No. of Linear
Inequality
Constraints
Indicative time taken to solve
N = 2 8 9 x 3 9 3 < 0.03 seconds
N = 3 26 27 x 4 27 12 < 0.10 seconds
N = 4 80 81 x 5 81 39 < 0.10 seconds
N = 5 242 243 x 6 243 120 < 0.5 seconds
N = 6 728 729 x 7 729 363 < 1 second
N = 7 2186 2187 x 8 2187 1092 < 1 second
N = 8
6560 6561 x 9 6561 3279
c. 5 - 10 seconds
Fails to converge with
additional call option
constraints
N = 9 19,683 19683 x 10 19,683 9,930 Not able to compute
Table 5.3.1 – Table demonstrating performance of algorithm for different periods in trinomial model
In the case of , if we consider the equality matrix required to solve the linear programming problem it would
have 19,684 x 19,683 = 390,943,566 entries, making it impractical to use on a standard computer.
Performance of Robust Pricing Algorithm 5.2.3
Instead of solving one large complex problem, Algorithm 5.2.3 for calculating the Robust Price calculates a large
number of simple problems for a given fixed (i.e. calculating {∫ }), then
looks to vary to determine the overall infimum). In the first instance, for each trinomial fork in the tree, a simple
linear programming problem (with 4 variables and 3 equality constraints) is solved at each step to derive the
maximum value of the exotic option at the previous step. is then varied, and the value
of {∫ } determined again.
As such, the performance of Algorithm 5.2.3 varies greatly depending on the number of constraints (i.e. vanilla
call options that are tradeable). With a full set of call options, at there are 13 constraints and the algorithm
fails to converge. However, with no call option constraints, equivalent to trading in just the risky asset, the
algorithm is robust at (running in c. 3 minutes, iteratively solving approximately 29,524 linear
programming problems in that time). For , using a low number of call options constraints (typically 1 to
3), and by increasing the maximum number of iterations where necessary, the nonlinear optimisation successfully
converges to a robust price bound.
5.3.6 Illustration of Performance of Robust Pricing Algorithm for higher values of n
To illustrate the performance of the Robust Pricing algorithm at higher values of , the table below shows the
value of the Robust Price and Robust Hedge when only trading in the risky asset is allowed. We choose to
examine an Asian Option, with initial stock price , and strike , interest rate for periods
from through to .
103
Number of Periods
Robust Hedge
Robust Price
N = 1 2.27273 2.27273
N = 2 3.78788 3.78788
N = 3 5.10894 5.10894
N = 4 6.26562 6.26562
N = 5 7.2163 7.2163
N = 6 8.08453 8.08453
N = 7 8.92513 8.92513
N = 8 9.68833 9.68833
N = 9 Not available 10.4157
N = 10 Not available 11.0698
Table 5.3.2 – Table numerically demonstrating duality of robust hedge and robust price for an Asian option, strike for
higher values of , up to a ten period trinomial model
We see from the above Table 5.3.2 that the robust pricing / hedging duality of the form below:
holds for all values of up to an 8-period trinomial model. It also demonstrates that Algorithm 5.2.3 for calculating
the Robust Price can handle larger values of than the Robust Hedge Algorithm 5.2.2, though it will take
considerable length of time to determine the value of for larger values of . The below Figure
5.3.3 gives an example of the output from the Matlab implementation for .
Figure 5.3.3 – Example output from the Matlab implementation for six period trinomial model, for calculation of the Robust
Hedge and Robust Price for Asian option, strike where only trading in the risky asset is allowed
104
5.3.7 Varying Parameters – Interest Rate
Up until this point, we have kept the interest rate constant at , as is the assumption in the literature for
example [1] and [7]. In our simple discrete time trinomial model, we can investigate numerically whether the
hedging / pricing duality holds if we relax this assumption for simple path dependent options.
Note that when we set the interest rate at non zero, then we have the following form of the duality, where we
consider the discounted expectation [
]:
∑
{∫ [
]}
The below tables illustrates the effect of doing this two example settings: a three period trinomial model for a
Lookback option, and a Cliquet option in a four period trinomial model.
Price of Vanilla Call
option,
Trading with Risky Asset Only Trading with Risky Asset vanilla
call options
Robust Hedge
Robust Price Robust Hedge Robust Price
1. Lookback Option;
8.1629 19.0083 19.0083 10.8687 10.8687
9.6262 20.2225 20.2225 13.8206 13.821
11.1476 21.4346 21.4346 15.3597 15.3597
12.7212 22.6437 22.6437 16.8723 16.8723
14.3412 23.8488 23.8488 18.4692 18.4692
2. Cliquet Option;
8.2772 22.6897 22.6897 18.6067 18.6067
10.2286 24.1693 24.1693 19.2771 19.2772
12.2808 25.6514 25.6514 20.6259 20.6259
14.4200 27.1349 27.1349 21.8138 21.8138
16.6325 28.6188 28.6188 21.9388 22.0286
Table 5.3.3 – Table numerically demonstrating duality of robust hedge and robust price for non-zero interest rate
We can see from the table that in the discrete time model the duality holds with a non-zero interest rate.
105
6 Conclusions and Further Research
6.1 Summary of Dissertation & Conclusions
We end by summarising the material covered and drawing out any appropriate conclusions.
In Chapter 1 we reviewed the problems inherent in the classical financial mathematics approach, which involves
postulating stochastic models and calibrating them to the market, and subsequently being exposed to model risk
with potentially serious consequences either in terms of e.g. risk management (through VaR) or mispricing (of
path dependent exotic options). We discussed an alternative approach to pricing and hedging – the robust
approach which used market information (in the form of prices of market vanilla call options for a given maturity)
and no-arbitrage principles to draw conclusions about the terminal distribution of a stock price. The robust
approach avoids the model risk inherent in the classical financial mathematics approach as it avoids the explicit
postulation of an underlying stochastic model.
Instead, it proceeds by arguing that the Breeden and Litzenberger Lemma imposes constraints on potential risk
neutral probability measures that may be used for pricing. As per Hobson’s suggestion in [16], we consider the
‘extremal elements’ of this set of models by introducing the idea of a ‘upper martingale price’ defined as
i.e. the supremum of the exectation of a (discounted) payoff over a set of now constrained
martingale measures (with the lower martingale price defined similarly as ). We also argued
that no-arbitrage principles imply that minimum superhedging (or subhedging) costs function as robust bounds on
the price of an exotic option, and we were naturally led to the question as to whether the upper (or lower)
martingale prices (i.e. the supremum over martingale measures of the expectation of the payoff) are the same as
any robust hedging bounds (i.e. the minimum superhedging cost for the payoff).
The formulation of the problem as determining a solution to then suggests itself to
treatment through the mathematical framework of Optimal Transportation, which is the subject of Chapter 2.
Optimal Transportation centres around minimising the transportation cost between two measure spaces X and Y
with measures . From a probabilistic perspective, we can interpret finding the minimal transportation cost as
equivalent to finding the minimal expectation under a set of transference plans of the cost function, where the
marginal distributions of the transference plans are equal to . We also reviewed Kantorovich’s’ dual
formulation of the original transportation problem, and introduced the Duality Theorem (Theorem 2.4.1) that
states:
∫
∫
∫
We also briefly introduced and demonstrated a numerical technique, linear programming, that can be used to
solve simple optimal transportation problems and that we would reuse heavily in Chapters 4 and 5.
In Chapter 3 we returned to the Financial Markets, and following Beiglbock, Henry-Labordere, Penkner in [1],
drew out the parallels with the robust approach to financial mathematics and concept of martingale price with the
framework of optimal transportation. Focusing on the lower martingale price, in both cases we are looking for the
minimum of an expectation of some function over a transference plan with known marginals. In the case of the
Financial Markets as opposed to standard Optimal Transportation theory, we have the additional constraint that
we require the measure to be a martingale.
Focusing on a discrete time market framework Beiglbock, Henry-Labordere, Penkner introduced a Duality
formulation for the martingale price. This dual formulation has a natural interpretation as the cost of the super /
sub hedging portfolio for the exotic option. The central result of the paper then is a Duality Theorem (Theorem
3.3) that equates the primal and dual formulations to derive the below duality result (stated as Theorem 3.3 in
Chapter 3 – shown below is the restated version for upper martingale price version Theorem 3.6.3):
{ } { ∑
( ) }
We followed this with a review of other similar results in the literature; including in continuous time settings.
However, we highlighted differences in the Dual formulations given by Galichon, Henry-Labordere and Touzi in
[8], shown below:
106
{ }
{
In the first Duality formulation from Beiglbock, Henry-Labordere and Penkner in [1], we look to maximise
expectations of payoff over a set of constrained martingale measures ( { }). In the
second Duality formulation, we firstly maximise over an unconstrained set of martingale measures, then minimise
over the permissible Lagrange multipliers ( { }. We saw summarised in
Dolinsky and Soner the equivalence of these duality formulations (under certain set of assumptions) along with
the robust superhedging cost (defined both quasi-surely and pathwise) i.e. in Theorem 3.6.5 we saw that:
{ ∫ }
{∫ }
We finished Chapter 3 with a brief comparison back to results previously achieved in the literature for the Robust
approach, which make use of solutions of the Skorokhod Embedding Problem; and saw the equivalence of these
earlier results to the aforementioned Duality results.
From Chapter 4 onwards, we began to test some of these theoretical results in a simple discrete time setting.
Concentrating first on a market where only trading in the risky asset was allowed, we demonstrated numerically
that the robust pricing hedging duality held i.e. we saw that, with no call options included, that:
We then saw how, for a simple path dependent option, the introduction of the ability to statically trade a single
vanilla call option reduced the cost of the robust superhedge, and we used the form of the inequality introduced
by Galichon, Henry-Labordere and Touzi in [8] to demonstrate that a form of the robust pricing hedging duality
holds i.e. we numerically demonstrated that:
{∫ } ∑
When adding additional call option constraints into the model, we saw numerically that the minimum
superhedging cost and robust price reduced further and became equal to the price of the exotic option under the
market martingale measure we had used to determine the prices of the vanilla call options. In other
words, we had, with two vanilla call options introduced into the market:
∑
The final Chapter 5 generalises the concepts and methods used in Chapter 4 to an -period trinomial model, and
builds a Matlab implementation which gives the flexibility to vary market parameters and test a wider variety of
path dependent options. For low values of , we determined a range of robust super hedging costs and upper
martingale prices for these path dependent options, and demonstrated numerically that the duality holds in this
more general setting. We saw however issues with the implementation in particular in terms of the nonlinear
optimisation routine and the Nelder-Mead algorithm, that struggled on occasion to find an accurate value for
{∫ } without significant additional attention (specifically, rerunning
the algorithm at the previous point at which it terminated).
Ultimately in Chapters 4 and 5, we explore numerically the dualities introduced in Chapter 3 in the relatively
parsimonious trinomial model for a series of path dependent options. Our market framework is more restrictive
than that reviewed in the literature in Chapter 3 – most notably, in a trinomial model we severely limit the number
of potential stock price paths (i.e. the stock can only move in one of three ways for each starting position). In
moving to such a restricted setting, we potentially reduce the minimum super-hedging cost, as we now have less
paths for the stock that we need to hedge the exotic option over. At the same time, by reducing the available
paths, we restrict the potential available martingale measures, and we would therefore expect
to reduce also.
107
As such then, it is not immediately a priori obvious that the pricing & hedging dualities, summarised in Theorem
3.6.5 by Dolinsky and Soner in [7], should hold in our restricted trinomial setting. However, our numerical
investigations indicate that they do – firstly in the 2-period trinomial model in Chapter 4, and then in Chapter 5 in
the more general -period trinomial model for a wider series of path dependent options.
Although the trinomial model is relatively simple, though some features might provide a more realistic model for
the market than the fuller financial market frameworks described in the literature. In particular, the trinomial model
described above only assumes finitely many call options strikes are available to trade in the market
6.2 Further Areas of Exploration
There are several other avenues of exploration that could help complete a more full study of the pricing and
hedging duality in a discrete time setting, which we briefly highlight in closing.
Firstly, we consider an aspect of Theorem 3.3 (i.e. the Duality Theorem from Beiglbock, Henry-Labordere and
Penkner in [1]) that we have not examined at length in this paper. The key result we have reviewed is that there
is no duality gap between the martingale price and the super / sub hedging cost; however in addition to this,
Theorem 3.3 also states that the ‘primal value P is attained i.e. there exists a martingale measure
such that { } ’ (in the case of the upper martingale
price. An alternative methodology for a numerical implementation might therefore involve determining a
description of this risk neutral measure and using this to price path dependent exotic options, with the
knowledge that the resulting value will be equal to the upper martingale price and the robust
superhedging bound. Such a strategy is suggested in part by Henry-Labordere in ‘Automated Option Pricing’ [13].
He notes however that:
‘the Martingale measure (i.e. arbitrage-free model), which achieves the super-replication strategy, can be very different from
those generated by (stochastic volatility) diffusive models traders commonly use.’
He suggests this difficultly could be circumvented by using an entropy measure to minimise the distance between
the maximising measure and a particular favourite trading model. A further investigation then of this maximising
measure, and any ‘adjustments’ required to it to ensure that is remains aligned with market recognised models,
might provide alternative and more efficient ways to calculate robust prices in a discrete time model.
Secondly, a natural next step would to increase the number of steps in the trinomial model, while reducing the
time period (and up parameter as required) so that the model tends towards a continuous time limit. In doing
this, it would be natural to test some of the robust bounds for particular path dependent options derived against
results from the Skorokhod Embedding Problem approach, where expressions for robust bounds of exotics such
as Lookback and Double Touch barrier options have been established.
To do this, more efficient numerical methods would have to be determined to ensure that the implementation runs
correctly for larger values of . In particular the nonlinear optimisation routine used to determine the value of
{∫ } would need to be improved, both in terms of overall accuracy
and time to convergence. Given some of the issues encountered with the Nelder-Mead method, a more thorough
investigation of other nonlinear optimisation methods might yield some benefit.
Thirdly, it would be of interest to aim to adapt more elements of the trinomial model to incorporate features of the
market. In particular, one aim would be to use real market prices of call option as data for calibration purposes,
and compare results from the model with real world exotic option prices. To do this would involve determining a
method to match call option prices from the market without introducing arbitrage into the trinomial model. A
similar technique is used in Derman, Kani, and Chriss in [6] to build trinomial trees that match implied volatilities
from the Black Scholes model of market vanilla call options; the challenge would be to do this in a model
independent way. Ultimately the test would be to determine whether bounds generated from the model where
tight enough in practice to be of use in pricing or hedging instruments.
Similarly the assumption of zero, or constant, interest rates could be relaxed by incorporating time dependent
(deterministic or even stochastic) interest rates into the trinomial model and investigating the effect of the pricing /
hedging duality. However, we should be careful of the temptation to construct complex models to capture
particular features of the market – for the original motivation of the robust approach was to precisely avoid the
level of model risk that that approach inherently brings.
108
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