Option Pricing with Liquidity Risk Lunch at the lab

Survey of Local Volatility ModelsLunch at the lab

Greg Orosi

University of Calgary

November 1, 2006

Outline

•Volatility Smile and Practitioner’s Approach

•Polynomial model for Local Volatility

•Spline Representation

•Penalized Spline

•Genetic algorithm

•Conclusion

Assumptions of the Black-Scholes model:

•Black-Scholes assumes constant volatilities across allstrikes and expiry

•But implied volatilities from market exhibit a dependenceon strike price and expiry

•Possible reasons for the smile:

-Real prices have fatter tails than GBM

-News events cause jumps

-Supply and demand considerations (investor preference)

Implied Volatility Surface

•Implied volatility surface for S&P 500:

Explaining the Smile

•Many attempts to explain the Smile by modifying theBlack-Scholes assumptions on dynamics of underlyingasset returns.

–Jumps [Merton, 1976]

–Constant Elasticity of Variance (CEV) [Cox andRoss, 1976]

–Stochastic Volatility [Heston, 1993]

These provide partial explanations at best

Practitioner’s approach

•Practitioners model the implied volatilitysurface as a linear function of moneyness andexpiry time:

•This consists of computing implied volatilitiesand performing an OLS regression

•The model is inconsistent but it works well forvanilla options. Bruno Dupire: "Implied volatilityis the wrong number to put into wrong formulaeto obtain the correct price.”

Another IV surface example:

Local Volatility Model

•Using IV surface to price path dependent options willlead to arbitrage because of inconsistency

•Derman, Kani and Kamal (Goldman Sachs QuantitativeResearch Notes 1994) suggest local volatility approach:

•Financial perspective: model is preference free

•Get Generalized BS-PDE:

Dupire’s Equation

•In 1994, Dupire ( ”Pricing with a smile”. Risk Magazine)showed that if the spot price follows GBM, then localvolatilities are given by:

•Where C is the constant volatility BS option price

•Therefore, Dupire’s equation provides link betweenIVS and local volatility surface

•However, this formula has little practical importance

DWF model

•Therefore, local volatility has to be calculated from optionprices by minimizing:

•In 1998 Dumas, Fleming & Whaley (Journal of Finance:Implied Volatility Functions: Empirical Tests) proposed apolynomial model of local volatility:

Empirical Performance of DWF model

•For hedging purposes DWF does not outperform constantvolatility Black-Scholes model

•Overfitting the model leads to worse performance(calibration is not well regularized)

•So a trader is better off using the constant volatility BSmodel to price an exotic option instead of DWF

Spline representation

•Coleman, Verma and Li (1998) and Lagnado and Osher(1997) suggest cubic spline representation in

•“Reconstructing The Unknown Local Volatility” Function - The Journalof Computational Finance

• “A technique for calibrating derivative security pricing models:numerical solution of an inverse problem” - Journal of ComputationalFinance

•Coleman et al show for long dated options the modelbeats constant volatility BS in 2001 (Journal of Risk“Dynamic Hedging in a Volatile Market”)

Spline representation

•A cubic spline is constructed of piecewise third-orderpolynomias which pass through a set of control points(knots).

•The second derivative of each polynomial is commonlyset to zero at the endpoints and this provides a boundarycondition that completes the system of equations.

Bounding

•Note that the spline based calibration is not regularized,meaning more than one possible solution.

•This could lead to poor hedging performance

•Therefore, Coleman et al suggest strict bounding

Bounded Spline Example

•=

Smoothness Penalization

•Lagnado and Osher (1997) suggest spline representationand additionally penalizing the smoothness

•Define new objective with penalty:

Smoothness Penalization

•Implemented by Jackson and Suli -1999

•“Computation of Deterministic Volatility Surfaces “ Journal ofComputational Finance)

Tikhonov Regularization

•Crepey (2003): ( “Calibration of the local volatility in a trinomial treeusing Tikhonov regularization ” –Inverse Problems) suggestcalculating local volatility by Tikhonov regularization:

•Define new objective:

Calibration by Relative Entropy

•A more general version of Tikhonov regularization is calibration byrelative entropy

•See Cont and Tanakov (“Calibration of Jump-Diffusion Option PricingModels: A Robust Non-Parametric Approach” Journal ofComputational Finance - 2004)

•This can be applied to other models besides local volatility

•Prior can be parameters estimated form historical prices (e.g. meanreverting models)

Genetic Algorithm for Local volatility

•Because the objective in option calibration is highly non-linear, gradient based optimization methods performpoorly

•Cont and Hamida (“Recovering Volatility from Option Prices byEvolutionary Optimization ” - Journal of Computational Finance 2005)suggest using Genetic Algorithm and splinerepresentation

•GA uses an initial population and improves this populationin each subsequent generation. Therefore, the initialpopulation can be generated using a prior and the use ofpenalty function is not necessary.

GA based local volatility for DAX

Conclusion

•Local volatility models can provide a consistenttheoretical option pricing framework.

•However retrieving local volatility can pose significantcomputational challenges.