robust estimation of shape constrained state price density...

Robust Estimation of Shape Constrained State Price Density Surfaces

Markus LudwigUniversity of Zurich

Options Markets - Baruch College

Markus Ludwig (UZH) State Price Surfaces October 8, 2014 1 / 44

http://ssrn.com/abstract=2138911

mailto:[email protected]

http://faculty.baruch.cuny.edu/lwu/890/FIN890Fall2014.html

http://vimeo.com/markusludwig/

Introduction

Option-Implied Measures

• Since the payoff of an option depends on the future states of the underlying asset,option prices reflect expectations regarding the evolution of that asset, as well asrisk preferences of market participants.

• A widely used measure of market sentiment, calculated from observable prices ofplain vanilla options, is implied volatility (IV).

• A related concept is the state price density (SPD), which captures the risk-neutralprobabilities that the market assigns to the various possible states of the underlyingasset upon expiry of the option.

• [Breeden and Litzenberger, 1978] show that an explicit expression for the SPD canbe obtained as the second partial derivative of the option pricing function withrespect to strike prices.



Introduction

Overview

• This paper revolves around the question of how to approximate well-behaved SPDsover a uniform state space from a snapshot of option prices.

• This is a non-trivial exercise and poses several challenges:

• smoothness despite noisy and sparse data• accurate fits, both in-sample and out-of-sample• support beyond the range of observable strike prices• compliance with shape constraints following from no-arbitrage arguments

• We propose a novel approach that, while based on neural networks, is deeplyrooted in ideas on heuristic optimization.

• The resulting densities are genuinely conditional and forward-looking, and not onlyenable us to investigate how risk-neutral measures evolve over time, but alsoprovide a basis for the recovery of real-world probabilities via [Ross, 2013].



Modeling Option Prices

Connecting the Dots

• Parametric methods rely on specific assumptions about the functional form of therelationship between inputs and outputs.

+ calibration+ incorporation of shape constraints− may fail to capture salient properties of the data (risk of misspecification)

• Nonparametric methods, in contrast, are data-driven and typically only assumethat the unknown function exhibits some kind of smoothness.

+ flexibility− small sample properties− limited by support of observable data




Challenges I

• One of the key challenges in modeling option data arises from the highly irregulardata design. At any given time, there are only a limited number of maturities witha discrete set of strikes available.

• Options appear in strings that are not evenly distributed and advance along thematurity dimension as time passes.

• The data is furthermore noisy, which poses a significant challenge since takingderivatives amplifies even minor irregularities in an estimator.

• In order to mitigate problems arising from finite observations, nonparametricmethods often resort to aggregating data over time.

• While data aggregation alleviates problems related to small data samples, it opensthe door to nonstationarity and regime shift issues. It also limits the accuracy withwhich current observations can be modeled.




Challenges II

• The range of observable strikes is typically not sufficient to recover true probabilitydensities, i.e., densities with converging tails.

• While most authors neglect to address this issue, others assume implied volatilityto remain constant outside the range of observable strikes.

• Extrapolation based solely on the information provided by one string will likelyimply globally inconsistent shapes.

• No-arbitrage arguments impose constraints on the estimated function.

• positivity of IVs• monotonicity and convexity of prices• increasing implied total variance along maturities




Video - Consistent Surfaces


http://vimeo.com/markusludwig/consistency



Video - True Densities


http://vimeo.com/markusludwig/density


Neural Networks

Neural Networks I

• Just like linear models, neural networks approximate functions as a linearcombination of features constructed from the original inputs.

• However, instead of fixed nonlinear transformations, the parameters of the basisexpansions are learned at the same time as the coefficients of the linear model.

• Consider modeling implied volatility σ for a fixed maturity τ as a function offorward-moneyness ψ ≡ K/F , using polynomials of up to second order.

• The resulting function is additive in the features derived from the basis expansionsgi(x) = x

i and can be compactly written as:

σ(ψ) =

2∑i=0

βi · gi(ψ) (1)

where βi corresponds to the coefficient of the i th feature.



Neural Networks

Neural Networks II

• Now, consider a feedforward neural network that models total implied varianceν(ψ, τ) ≡ σ2(ψ, τ) · τ as a function of moneyness and time to maturity:

ν(ψ, τ) = β0 +

H∑i=1

βi · hi(ψ, τ) (2)

hi(ψ, τ) =1

1 + e−ci (ψ,τ)

ci(ψ, τ) = α0i + α1iψ + α2i√τ

here the basis function hi(x, τ) is a logistic sigmoid.

• The weights α1i and α2i control the steepness and thus modulate the amount ofnonlinearity introduced by each expansion.



Neural Networks

Neural Networks III

x

f(x)

α0 = 0α0 = 1α0 = 2

x

f(x)

α1 = 0.6α1 = 1.0α1 = 2.0

x

f(x)

β = 0.90β = 0.20β = 0.15

β

α

τψ

ν(·)

h3(·)h2(·)h1(·) h4(·)

ci(ψ, τ) = α0i + α1iψ + α2i√τ

hi(ψ, τ) =1

1 + e−ci (ψ,τ)

ν(ψ, τ) = β0 +

H∑i=1

βi · hi(ψ, τ)



Neural Networks

Neural Networks IV

• Multilayer networks have been shown to be universal approximators, i.e., given asufficient number of hidden nodes H, they can approximate any continuousfunction on a compact input domain up to an arbitrary degree of accuracy.

• The parameters α and β are typically estimated by minimizing the residualsum-of-squares between output and target values.

• Since the basis functions are adaptive, the optimization has no closed-formsolution and needs to be solved by means of iterative numerical methods, typicallygradient descent.

• Due to symmetries in the loss function, the number of local minima in the errorsurface is high, which causes solutions to be sensitive to the initial starting valuesof the optimization.



Neural Networks

Model Complexity and Generalization

• As for all flexible methods, the key challenge is to determine the optimal modelcomplexity, i.e., to optimize the bias-variance trade-off.

• Overly simplistic models, while robust to spurious patterns in the data, typicallyexhibit a lack of fit due to a high bias.

• Overly complex models on the other hand will start to fit the idiosyncratic noise andgeneralize poorly to previously unseen data.

• In neural networks, model complexity can be controlled both through the choice ofthe architecture and through regularization.

• A similar effect can be achieved through early stopping. Since coefficients areinitialized at random starting values near zero, the model starts out nearly linear.



Neural Networks

Model Selection

• Going back to [Hutchinson et al., 1994], research on modeling options with neuralnetworks uses cross-validation, which evaluates the generalization performance ofan estimator on a subset of observations that were omitted during training.

• However, due to the irregular data design of exchange traded equity options,cross-validation is of limited usefulness, as it can only check locations for which weactually observe prices.

• This means we cannot assess the quality of a surface estimator between maturities,or for the sizable regions outside of traded strikes.

• Prior research addressed this issue by using large quantities of historical pricespartitioned into non-overlapping data sets. Several architectures are then trainedon 6 to 12 months of data and the best performing network is selected based onthe pricing error on a validation set spanning 1 to 3 months.



Neural Networks

Shape Constrained Networks

• For our approach, we adapt the perspective of heuristic optimization and performboth parameter estimation and model selection by generating a population ofsolutions, that we subsequently evaluate based on properties other than the errormetrics defining the loss function used to compute them.

• First, we create several networks of the form in (2), each of which varies both withrespect to the architecture and the initial weights.

• Second, we explore the neighborhoods of these initial solutions by running aregularized gradient descent algorithm for only 10 steps.

• Third, we check the resulting surfaces for violations of static arbitrage constraints.

• Once we have a subset of 25 valid solutions, we rank them based on their fit in IVspace and obtain the final surface as an average of the three best candidates.

• In contrast to cross-validation, no-arbitrage constraints allow us to assess thequality of the entire surface. The irregular data design now becomes an advantageand allows us to ferret out estimators that generalize poorly into the tails orbetween traded maturities, as an excess in model complexity will ipso factomanifest in spurious nonlinearities in regions for which we have no training data.



Neural Networks

Video - Training


http://vimeo.com/markusludwig/training


Empirical Analysis

Data and Benchmark Models

• This section demonstrates both the pricing performance and robustness of ourshape constrained network (SCN) approach by estimating total implied variancesurfaces over a fixed domain, with ψ ∈ [0.5, 1.5] and τ ∈ [20, 365]. The estimatorsare then mapped into implied volatility, price, and state price density space.

• We contrast the IV pricing performance with two ad hoc Black-Scholes models.The string-wise polynomial in (1) and the widely used global specification:

σ(ψ, τ) = β0 + β1ψ + β2ψ2 + β3τ + β4τ

2 + β5ψτ (3)

• Option data and interest rates were obtained from OptionMetrics. We take themean of bid and ask prices as option prices and discard observations with bidsbelow $0.50, or outside the moneyness-maturity domain of our surfaces. We alsoexclude options that violate general price bounds or strike arbitrage constraints.



Empirical Analysis

Pricing Performance

2000 2001 2002 2003 2004 2005 2006 2007 2008 2009 2010 2011 2012600

800

1000

1200

1400

1600

S&

P 5

00

2000 2001 2002 2003 2004 2005 2006 2007 2008 2009 2010 2011 20120

0.5

1

1.5

2

2.5

3

IV R

MS

E

2000 2001 2002 2003 2004 2005 2006 2007 2008 2009 2010 2011 20120

0.2

0.4

0.6

0.8

1

Out−

of−

Sam

ple

IV

R2



Empirical Analysis

SCN Surface Cross-Sections

Empirical Analysis


Empirical Analysis

SCN Surface Cross-SectionsEmpirical Analysis


0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 1.510

15

20

25

30

35

40

45

50

K/F

IV

0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 1.50

0.5

1

1.5

2

2.5

3

3.5

4x 10

−3

K/F

SP

D

Markus Ludwig (UZH) State Price Surfaces October 8, 2014 23 / 45Markus Ludwig (UZH) State Price Surfaces October 8, 2014 22 / 44Markus Ludwig (UZH) State Price Surfaces October 8, 2014 22 / 44Markus Ludwig (UZH) State Price Surfaces October 8, 2014 22 / 44


Empirical Analysis

State Price Density Evolution

2000−2003

2004−2007

2008−2011


http://vimeo.com/markusludwig/evolution


Conclusion

Concluding Remarks

• We propose a novel approach to approximate arbitrage-free implied volatility, price,and state price density surfaces.

• We show that our method is robust enough to carry out both model selection andparameter calibration using only a snapshot of option prices, and demonstrate thein-sample and out-of-sample quality of our estimators over a period of 12 years.

• Our state price density surfaces provide a comprehensive snapshot of currentmarket sentiment and, unlike maturity-wise estimators, enable us to trace theevolution of investors’ expectations and risk perceptions along a continuum offuture spot trajectories and maturity horizons.



An Empirical Analysis of the Ross Recovery Theorem

Markus LudwigUniversity of Zurich

Joint work with Francesco Audrino and Robert Huitema

Options Markets - Baruch College

Markus Ludwig (UZH) Empirical Ross Recovery October 8, 2014 25 / 44

http://ssrn.com/abstract=2433170

mailto:[email protected]

http://faculty.baruch.cuny.edu/lwu/890/FIN890Fall2014.html


Introduction

Overview

• Recently, [Ross, 2013] has shown that the market’s risk aversion can be recoveredfrom the risk-neutral transition matrix of a Markovian state variable.

• Knowledge of the pricing kernel allows us to disentangle the market’s assessmentof real-world probabilities from their risk-neutral counterparts.

• This makes the information embedded in option prices directly accessible toapplications such as risk management, portfolio optimization and market timing.

• Ross’ recovery theorem is intriguing because, in contrast to previous literature, itdoes neither rely on historical returns nor restrictions on the shape of the kernel.



Introduction

Setup and Assumptions

• In a dynamically complete arbitrage-free market, the price of an option is given bythe expected present value of the payoff under the risk-neutral density (RND).

• [Breeden and Litzenberger, 1978] show that the discounted RND is equal to thesecond derivative of option prices with respect to strike.

• [Ross, 2013] requires two additional conditions to uniquely disentangle risk aversionfrom what the market believes the real-world probabilities to be.

• The risk-neutral process evolves as a:

• discrete• time-homogeneous• irreducible Markov chain• on a finite state space

• The pricing kernel is path independent.



The Recovery Theorem [Ross, 2013]

Ross Recovery I

• Consider a time-homogeneous process Xt , Xt+1, . . . on a finite state space withvalues in 1, . . . , n. Since calendar time is irrelevant, the transition probability ofmoving from state i at time t to state j at time t + 1 is given by:

Pi ,j = P r(Xt+1 = j |Xt = i) (4)

where Pn×n denotes the one-step ahead transition matrix and is clearlyelement-wise non-negative. Since we also assume that P is irreducible, that is,P t > 0 for some t, all states can be reached from all other states.

• From the Perron-Frobenius theorem we know that a non-negative irreduciblematrix has a unique positive eigenvalue λ and corresponding dominant left- andright-eigenvectors that are unique and strictly positive.

v>P = λv> (5)

P z = λz (6)




Ross Recovery II

• The elements Pi ,j of the state price transition matrix correspond to the prices ofsingle period Arrow-Debreu securities, indexed by the starting and ending state andcan be decomposed as:

Pi ,j = δMi ,jFi ,j (7)

where δ is the market’s average discount rate, Mn×n is the pricing kernel, and Fn×ndenotes the real-world transition matrix.

• The assumption that the pricing kernel is transition independent means that it onlydepends on the marginal rate of substitution between the future and currentconsumption, which allows us to write (4) as:

Pi ,j = δdjdiFi ,j (8)

where d = (d1, . . . , dn)> is a vector of marginal utilities.




Ross Recovery III

• Defining the diagonal matrix Dn×n with d on the main diagonal allows us torearrange for F as follows:

P = δD−1FD

F = δ−1DPD−1 (9)

• Since F is stochastic, F1 = 1, where 1 is a vector of ones, we obtain:

PD−11 = δD−11 (10)

• If we define the vector z to contain the inverse of the diagonal elements of D, weobtain the unique solution from (6), with λ corresponding to the discount factor δ.

P z = λz



From Option Prices to State Prices [Ludwig, 2012]

Data Design S&P 500 Options

050100150200250300350400450500

0.5

1

1.5

τ

K/F




Implied Volatility Surface

50

100

150

200

250

300

350

0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 1.5

0

10

20

30

40

50

60

70

K/F

τ

IV




State Price Density Surface

50

100

150

200

250

300

350

0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 1.5

0

1

2

3

4

5

6

7

x 10−3

K/F

τ

SP

D



From State Prices to Markov Chains

Constructing the Transition Matrix I

• Given a matrix of state prices Sn×m for m equidistantly spaced maturities,[Ross, 2013] shows that if m ≥ n, we can estimate P because it specifies thetime-homogeneous transition from one maturity to the next:

S:,t+1>= S:,t

>P, t = 1, 2, . . . , m − 1 (11)

• Let A>= S:,1:m−1 and B>= S:,2:m contain state prices, equation (11) gives rise toa matrix factorization problem:

minP≥0

‖AP − B‖22 (12)

• Let pj ≡ P:,j and bj ≡ B:,j we have:

minpj≥0

‖Apj − bj‖22 , j = 1, 2, . . . , n (13)




Constructing the Transition Matrix II

• This non-negative least squares problem can be solved with standard algorithms,cf. [Lawson and Hanson, 1974].

• However, active-set methods depend on the inverse of A>A, which renders theminfeasible if A is ill-conditioned, as is the case for our application.

• We therefore consider the damped least squares problem:

minp

‖Ap− b‖22 + ζ ‖p‖22 (14)

where the scalar ζ controls the trade-off between fit and stability of the solution byreducing the impact of small singular values.

• The regularized formulation is also a least squares problem. Augmenting theconstrained least squares problem yields:

P = argminP≥0

n∑j=1

∥∥∥∥∥[A

ζI

]pj −

[bj0

]∥∥∥∥∥2

2

(15)

where I denotes an identity matrix and 0 is a vector of zeros.




Constructing the Transition Matrix III

• A natural approach to determine ζ is to minimize the discrepancy betweenobservable state prices and the unrolled Markov chain.

• Let ιi be a vector with 1 in the i th position and zeros elsewhere, the t-step aheadstate prices implied by a given transition matrix can be computed as:

U:,t> = ι

>i P

t , t = 1, 2, . . . , m (16)

• Since we define our state space symmetrically around the current state, (16) tellsus that the center row of P t contains the t-steps ahead state price approximation.

• We then solve for the smoothing parameter ζ that minimizes:

minζ

DKL (S‖U) (17)

where DKL denotes the generalized Kullback-Leibler divergence, which is defined as:

∑i ,j

Si ,j · log(Si ,jUi ,j

)−∑i ,j

Si ,j +∑i ,j

Ui ,j (18)



Empirical Analysis

State Prices, Implied Volatilities and Pricing Kernels��

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Conclusion

Concluding Remarks

• We present a method based on Tikhonov regularized non-negative least squares toconstruct robust time-homogeneous Markov chains that not only yield an excellentfit to state prices at various maturity horizons, but pricing kernels that exhibit asensible variation over time.

• We investigate the predictive information content of the risk-neutral and recoveredmoments and find that – in line with economic theory – the real-world meanimpounds most of the information of higher-order risk-neutral moments.

• Without any restrictions on the shape of the pricing kernel we always find apositive equity risk premium and a negative variance risk premium. Higher-orderpremia are also consistent with economic intuition.

• We find that the recovered pricing kernels are closely related to state prices andillustrate the possibility for pseudo recovery.



References I

Breeden, D. and Litzenberger, R. (1978).Prices of state-contingent claims implicit in option prices.Journal of Business, 51(4):621–651.

Hutchinson, J., Lo, A., and Poggio, T. (1994).A nonparametric approach to pricing and hedging derivative securities via learningnetworks.Journal of Finance, 49(3):851–889.

Lawson, C. and Hanson, R. (1974).Solving Least Squares Problems.SIAM.

Ludwig, M. (2012).Robust estimation of shape constrained state price density surfaces.Working Paper, University of Zurich.

Ross, S. (2013).The recovery theorem.Journal of Finance (forthcoming).



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