equity derivatives (pdf)
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Guillaume BLACHERReech Capital
Advanced Pricing and Risk
Management for Equity Derivatives
Inte
rnat
iona
l D
eriv
ativ
es E
xhib
ition
FOW
Fra
nkfu
rt M
arch
7 -
8th
2000
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Slide 2
GuillaumeBLACHER
ReechCapital
EquityDerivatives
Outline
➢ Presentation of products most actively studied in
Equity Derivatives
➢ Evolution in pricing exotics
➢ Choosing a proper diffusion model
➢ Numerical methods: complexity and performance
➢ Generic product description tools
➢ The Future of quantitative analysis
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products most actively studied inEquity Derivatives
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Slide 4
GuillaumeBLACHER
ReechCapital
EquityDerivatives
Products• High variety of products in Equity
Derivatives• Always new products tailored to:
– new market conditions– new tax regimes– new corporate situation
• Among most popular:– Cliquets– Structured notes– Baskets and multi-underlying products– Hybrid Products
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Slide 5
GuillaumeBLACHER
ReechCapital
EquityDerivatives
Cliquets• Pays the equity positive performance each
year, defined as:
• Pricing:– By Monte-Carlo: most natural but slow– By Tree/PDE: requires a second dimension for
the strike
• Problems:– Almost a pure volatility product– Strong impact of smile and dividends
• Solution: pricing by static replication
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Slide 6
GuillaumeBLACHER
ReechCapital
EquityDerivatives
Structured notes• Client gives up risk-free return, dividends
and/or x% of capital to get some equityexposure via embedded option.
• Often combines many features into oneproduct:– Bond features (coupons, redemption)– Exotic features (digitals, barriers…)– Swap features (equity swaps)– Multi-underlying (best of, basket)– Path dependent features (average)– American style features (puttable)
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Slide 7
GuillaumeBLACHER
ReechCapital
EquityDerivatives
Exotics on basket• Still a lot of business on baskets, mainly with
path-dependent features (asian baskets) andcapped
• Example: 5Y basket option of 3 indices, with aknock-in in the first 2.5Y at 175% to get an extra25% of gearing.
• Modeling correlation realistically is crucial
• Correlation cannot be hedged on a trade by tradebasis (no exchange traded correlation sensitiveoptions): need for creativity to build up acorrelation - diversified portfolio
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Slide 8
GuillaumeBLACHER
ReechCapital
EquityDerivatives
Hybrid products
• More and more products across assetclasses:– Best of equity / interest rate– Interest swaps triggered by equity level– Debt products with FX components
• Requires– Combination of skills for modeling
heterogeneous assets together– Very high structuring and pay-off description
flexibility– Consistent risk management across all asset
classes
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Slide 9
GuillaumeBLACHER
ReechCapital
EquityDerivatives
Volatility products
• Swaps or options on:– realised volatility from start to maturity– Implied volatility
• Example:In 6M, option pays the 1M at-the-money implied
volatility.
• Pricing mechanism differs from standardstructure:– Mix of semi-static option hedge and dynamic
spot hedge– Several un-hedgeable elements that need to be
taken into account.
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Slide 11
GuillaumeBLACHER
ReechCapital
EquityDerivatives
Pricing components
• Several components in a pricing engine:
Numericalmethod
Diffusionmodel
Productdescription
Modelparameters /Calibration
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Slide 13
GuillaumeBLACHER
ReechCapital
EquityDerivatives
Diffusion models
• All pricings should be consistent withmarket prices of hedging instruments:– Current yield– Smile– Credit curve
• The least it should do to achieve this withdeterministic models:– short term rate: r(time)– instantaneous volatility σ(spot, time)– default probability p(spot, time)
• Those functions can be calibrated tomarket instruments
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Slide 14
GuillaumeBLACHER
ReechCapital
EquityDerivatives
Example: volatility modeling
• The simplest diffusion consistent with amarket skew is:
• Always exists if market prices are notarbitrageable
• Would be unique if a continuum of priceswere given
• Significant price impact on:– Barriers– Cliquets– Basket options
( ) ( ) tdWtSdtdivreporSdS ,σ+−−=
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Slide 15
GuillaumeBLACHER
ReechCapital
EquityDerivatives
Example
• Pricing with smile can be critical when theright price cannot be reach with Black-Scholes formula
Up and Out Call option
0
0.5
1
1.5
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0% 20% 40% 60%
Black-Scholes volatility
Bar
rier p
rice
Black-Scholes priceSmile price
25
27
29
31
33
35
37
39
Market smile
Strike Maturity
25
27
29
31
33
35
Flat smile
Strike Maturity
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Slide 16
GuillaumeBLACHER
ReechCapital
EquityDerivatives
Hedging
• What if the smile disappears?
Black-Scholes assumption would then apply. Our
mark-to-market position would be showing a
big loss (if we bought the barrier) or a big
profit (if we sell the barrier)
• Only makes sense if a consistent vega
hedge is used.
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Slide 18
GuillaumeBLACHER
ReechCapital
EquityDerivatives
Vega analysis
Volatility risk can be measured at 3 levels:
1) Single vega number:– Parallel shift of the
whole smile surface
– No protection againstdeformations
– Which Europeanoptions to buy as ahedge?
-1%
0%
1%
2%
Strike Maturity
Classic Vega perturbation
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Slide 19
GuillaumeBLACHER
ReechCapital
EquityDerivatives
-1%
0%
1%
2%
Strike Maturity
Bucket Vega perturbation
Vega analysis (2)
2) Vega buckets:– Parallel shift of the skew at each tenor
– Protection against time deformations only
– Which strike should we buy as a hedge?Vega bucket for an exotic
1M 2M 3M 4M 5M 6M 7M
Tenor
Vega
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Slide 20
GuillaumeBLACHER
ReechCapital
EquityDerivatives
Vega
StrikeMat
Superbucket: Up&Out Call
-1%
0%
1%
2%
Strike Maturity
Superbucket perturbation
Vega analysis (3)
3) Superbuckets:– Shift of each individual point on the surface
– Protection against any deformation
– Provides strikes and maturities to hedge with.
barrier
strike
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Slide 21
GuillaumeBLACHER
ReechCapital
EquityDerivatives
Global risk management
• Each individual exotic has got its superbuckettopography.
• What is really relevant is the superbucket of thewhole book, aggregating exotics and vanillas on thesame underlying:
Vega
StrikeMat
Portfolio superbucket
Sensitivities tobe hedged out(by tradingcorrespondingEuropeans)
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Slide 22
GuillaumeBLACHER
ReechCapital
EquityDerivatives
Risk Reports
Spot
Time Vega
Classic Report:
Strike
Mat
urity
Vega
CORRECT Report:
Traders will try to flatten the Classic Report where
they should be trying to flatten the Correct Report:
• Equivalent for European options
• Different for Exotics
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Slide 23
GuillaumeBLACHER
ReechCapital
EquityDerivatives
Global risk management (2)
• This (correct) analysis should be available at
portfolio level on all model parameters, whether
they be:
– hedgeable (credit term structure, smile surface)
– un-hedgeable (instantaneous forward correlation)
• It gives today’s hedge but may not be static: need
for readjusting in the future
• Question: if readjusting is necessary, what is the
cost?
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Slide 25
GuillaumeBLACHER
ReechCapital
EquityDerivatives
Cost of hedging
Some options exhibit a vega that varies with:
• Spot
• Volatility itself
Rebalancing vega hedge will produce systematic
cost/profit that can only be estimated will a
stochastic volatility model.
To price the cost of having... we need to model...
dSpotdVega
dVoldVega
Correlation (Spot, Vol)
Volatility of volatility
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Slide 26
GuillaumeBLACHER
ReechCapital
EquityDerivatives
Stationarity
• Many different ways to match current smile
• Choosing the wrong model means having non-stationary parameters:– Even when market does not change, model parameters
do
– Forward smiles not realistic
• Although the reason for choosing a model isconsistency and not ability to forecast, the cost ofrehedge will be wrongly estimated
• Conclusion: try to find the model with most stableparameters
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Slide 28
GuillaumeBLACHER
ReechCapital
EquityDerivatives
Auto-calibrated models
• Some model can be considered as auto-calibrated:“calibration” is analytical, quasi-instantaneous andhappens automatically inside the numericalmethod
• Examples:– Local volatility in a Black-Scholes framework
– Local volatility in a deterministic volatility model
– Drift of the short term rate in an Heath-Jarrow-Mortonframework
– Instantaneous default probability if deterministic
– etc...
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Slide 29
GuillaumeBLACHER
ReechCapital
EquityDerivatives
Calibration
• More and more advanced models are not auto-calibrated
• Calibration is a long process
• On the other hand, it is done once and for all (untilmarket conditions change significantly)
• Examples:– Parameters of stochastic volatility / jump models
– Volatility in interest rate models
• Systems have to be flexible enough to let thecalibration process happen independently frompricing
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Slide 31
GuillaumeBLACHER
ReechCapital
EquityDerivatives
Trees• Just a way to compute an expectation• Binomial trees obsolete (except for quick
approximations)• Trinomial trees do the job, although they are less
efficient than PDEs.• 2-dimensional trees technology well mastered
Binomial Trinomial 2D Trinomialtime
spot
time
spot
time
S1
S2
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Slide 32
GuillaumeBLACHER
ReechCapital
EquityDerivatives
PDEs• Allows for implicit discretisation• More flexibility on the grid• Better performance in general, much better for some
products (barriers for example)• 2-dimensional PDEs technology well mastered• Provides option price at any time for any spot
Optionprice Time Intrinsic value
Spot
Pay-off for anUp&Out CallPDE gridPrice
profiletoday
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Slide 33
GuillaumeBLACHER
ReechCapital
EquityDerivatives
Monte Carlo• Low discrepancy series (ex: Sobol) provide very high
convergence rate compared to standard Monte Carlo• Unfortunately biased in high dimension• Dimension reduction techniques have to be used in high
dimension (spectral truncation)• Several very efficient techniques to price american style
options within simulations
0
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0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10
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0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
Standard MC sampling Sobol sampling
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Slide 34
GuillaumeBLACHER
ReechCapital
EquityDerivatives
Complexity vs Performance
• More accurate models usually mean greatercomputation time.
• Becomes an issue when thousands of positionshave to be revalued.
• Better understanding of models and numericalspeed-ups should precede the purchase of high-end hardware.
• Using those instead of brutal calculation powerneeds structural changes in trading systemsarchitecture.
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Slide 36
GuillaumeBLACHER
ReechCapital
EquityDerivatives
Equity Derivatives richness
• Structured products and Derivatives have 3 mainproperties:1) Extreme diversity of features
2) All features have to be priced together
3) Product popularity does not last forever
• This means:– A lot of work for quants to implement new features in
the library
– Library contain high quantity of obsolete products (oreven products that have never been dealt)
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Slide 37
GuillaumeBLACHER
ReechCapital
EquityDerivatives
Generic product description
• Quants should:– spend no time implementing new products in a Library
– spend more time modeling the market parameters
– spend more time improving their numerical methods
• Structurers/Marketers should:– spend no time waiting for quants to come up with a new
product in the Library
– spend more time studying the risks of the product
– spend more time trying new ideas
• What’s the solution?
A generic product description language
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Slide 38
GuillaumeBLACHER
ReechCapital
EquityDerivatives
The future of quantitative analysis
• Main challenge:
model more sources of risk while maintainingreasonable computation speed.
• How is it possible?– Better understanding of the products helps identifying
most important factors
– Some risks can be taken into account without increasingthe dimension
– More advanced numerical methods and numericalspeed-ups are to be used
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