Survey of Local Volatility Models Lunch at the lab

1
Survey of Local Volatility ModelsLunch at the
lab

• Greg Orosi
• University of Calgary
• November 1, 2006

2
Outline

• Volatility Smile and Practitioners 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 SP 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
Practitioners 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
Dupires 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, Dupires 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.