Survey of Local Volatility Models

Lunch 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 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)

Implied Volatility Surface

•Implied volatility surface for S&P 500:

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

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.”

Another IV surface example:

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:

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

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:

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

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”)

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.

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 representation and additionally penalizing the smoothness

• Define new objective with penalty:

Smoothness Penalization

• Implemented by Jackson and Suli -1999 • “Computation of Deterministic Volatility Surfaces “ Journal of Computational Finance)

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:

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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)

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.

GA based local volatility for DAX

Conclusion

• Local volatility models can provide a consistent theoretical option pricing framework.

• However retrieving local volatility can pose significant computational challenges.