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:
 C ( (S , T ))  C 
 2
i 1:n
i
i
   ( S , T )   0 
2
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 nonlinear, 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.