1. Survey of Local Volatility Models Lunch at the lab Greg Orosi University of Calgary November 1, 2006

2. Outline Volatility Smile and Practitioner’s Approach Polynomial model for Local Volatility Spline Representation Penalized Spline Genetic algorithm Conclusion

3. Assumptions of the Black-Scholes model: Black-Scholes assumes constant volatilities across all strikes and expiry But implied volatilities from market exhibit a dependence on 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)

4. Implied Volatility Surface Implied volatility surface for S&P 500:

5. Explaining the Smile Many attempts to explain the Smile by modifying the Black-Scholes assumptions on dynamics of underlying asset returns. –Jumps [Merton, 1976] –Constant Elasticity of Variance (CEV) [Cox and Ross, 1976] –Stochastic Volatility [Heston, 1993] These provide partial explanations at best

6. Practitioner’s approach Practitioners model the implied volatility surface as a linear function of moneyness and expiry time: This consists of computing implied volatilities and performing an OLS regression The model is inconsistent but it works well for vanilla options. Bruno Dupire: "Implied volatility is the wrong number to put into wrong formulae to obtain the correct price.”

7. Another IV surface example:

8. Local Volatility Model Using IV surface to price path dependent options will lead to arbitrage because of inconsistency Derman, Kani and Kamal ( Goldman Sachs Quantitative Research Notes 1994 ) suggest local volatility approach: Financial perspective: model is preference free Get Generalized BS-PDE:

9. Dupire’s Equation In 1994, Dupire ( ” Pricing with a smile”. Risk Magazine ) showed that if the spot price follows GBM, then local volatilities are given by: Where C is the constant volatility BS option price Therefore, Dupire’s equation provides link between IVS and local volatility surface However, this formula has little practical importance

10. DWF model Therefore, local volatility has to be calculated from option prices by minimizing: In 1998 Dumas, Fleming & Whaley ( Journal of Finance: Implied Volatility Functions: Empirical Tests ) proposed a polynomial model of local volatility:

11. Empirical Performance of DWF model For hedging purposes DWF does not outperform constant volatility 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 BS model to price an exotic option instead of DWF

12. Spline representation Coleman, Verma and Li (1998) and Lagnado and Osher (1997) suggest cubic spline representation in “Reconstructing The Unknown Local Volatility” Function - The Journal of Computational Finance “A technique for calibrating derivative security pricing models: numerical solution of an inverse problem” - Journal of Computational Finance Coleman et al show for long dated options the model beats constant volatility BS in 2001 (Journal of Risk “Dynamic Hedging in a Volatile Market ” )

13. Spline representation A cubic spline is constructed of piecewise third-order polynomias which pass through a set of control points (knots). The second derivative of each polynomial is commonly set to zero at the endpoints and this provides a boundary condition that completes the system of equations.

14. 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

15. Bounded Spline Example =

16. Smoothness Penalization Lagnado and Osher (1997) suggest spline representation and additionally penalizing the smoothness Define new objective with penalty:

17. Smoothness Penalization Implemented by Jackson and Suli -1999 “Computation of Deterministic Volatility Surfaces “ Journal of Computational Finance )

18. Tikhonov Regularization Crepey (2003): ( “Calibration of the local volatility in a trinomial tree using Tikhonov regularization ” –Inverse Problems ) suggest calculating local volatility by Tikhonov regularization: Define new objective:

19. Calibration by Relative Entropy A more general version of Tikhonov regularization is calibration by relative entropy See Cont and Tanakov (“Calibration of Jump-Diffusion Option Pricing Models: A Robust Non-Parametric Approach” Journal of Computational Finance - 2004) This can be applied to other models besides local volatility Prior can be parameters estimated form historical prices (e.g. mean reverting models)

20. Genetic Algorithm for Local volatility Because the objective in option calibration is highly non- linear, gradient based optimization methods perform poorly Cont and Hamida (“Recovering Volatility from Option Prices by Evolutionary Optimization ” - Journal of Computational Finance 2005 ) suggest using Genetic Algorithm and spline representation GA uses an initial population and improves this population in each subsequent generation. Therefore, the initial population can be generated using a prior and the use of penalty function is not necessary.

21. GA based local volatility for DAX

22. Conclusion Local volatility models can provide a consistent theoretical option pricing framework. However retrieving local volatility can pose significant computational challenges.