A Technique for Calibrating Derivative Security Pricing Models: Numerical Solution of an Inverse Problem

Ronald Lagnado - C-ATS Software, Inc. and Stanley Osher - University of California, at Los Angeles

A technique is presented for calibrating derivative security pricing models with respect to market prices, This technique can be applied in a very general multi-factor setting where model parameters such as volatilities and correlations are allowed to be functions of the underlying state variables. These functions are estimated from price observations by solving the inverse problem cited with the parabolic partial differential equation (PDE) governing arbitrage-free derivative security prices. A detailed exposition is given for consistent pricing of equity index options under a stochastic model that treats index volatility as a deterministic function of index level and time.

We consider a class of exchange-traded options on an equity index such an the S&P 500, composed of calls and puts with a variety of maturities and strike prices. A consistent pricing model should match the market prices of the options (at least to within the bid-ask spread) given a single assignment of model parameters. Such consistency between model and market is desirable for hedging portfolios of options or for pricing and hedging path-dependent and other exotic options on the same underlying index. It is well known that the Black-Scholes model, based upon the assumption of constant index volatility, typically falls to consistently price index options across the entire range of expiration dates and strike prices. This shortcoming is exemplified by exemplified by the so-called implied volatility "smile". However, consistent pricing can be obtained in principal by extending the model so that index volatility is allowed to be a deterministic function of index level S and time t. This local volatility function can be uniquely determined for (S, t) E [0, oo] X [0,Tmax], given the arbitrage-free prices of European options for all conceivable strike prices K > 0 and expiration dates 0 < T < Tmax. In practice, though, the determination of this function is an under-specified problem given only what the market provides — prices of exchange-traded options for a finite number of expiration dates and strikes.

The local volatility function appears as a coefficient of the second-order partial derivative in the PDE governing arbitrage-free derivative security prices. We show who to find this coefficient as the solution of an inverse problem, thereby determining the local volatility function from a finite number of observed option prices. The problem of under-specification is alleviated by a regularization technique. Here we minimize the L2 norm of the gradient of the local volatility over an appropriate space of smooth functions subject to a constraint that ensures that solutions of the pricing PDE watch observed market prices. This minimization is carried out numerically using a gradient descent procedure implemented in a finite-difference framework. The procedure requires values of the variational derivative of the option pricing function with respect to the local volatility function. We present a novel numerical approach for evaluating this variational derivative.

The viability of our technique is supported with results from two types of numerical experiments. The first type of experiment uses observed prices that are generated with a known form of the local volatility function (i.e., constant elasticity of variance model). Here we are able to observe not only how well the calibrated model reproduces the price observations, but the degree to which the local volatility function can be recovered. The effect of grid resolution and the density of price observations on this recovery is discussed. The second type of experiment uses actual market prices of S&P 500 index options.

Finally, we discuss the relations of our calibration technique with other popular approaches based on so-called implied trees. While our techniques is more computationally demanding, if offers a number of advantages. First, no explicit interpolation or extrapolation of observed market prices is requires. Second, the observations used for the calibration are not limited to European option prices — they could be prices of American or other exotic options. Third, we can extend the technique in a straightforward way to more complex settings involving, for example, options on multiple underlying assets or interest rate derivatives. Fourth, theoretical and empirical evidence indicates that our technique is well-posed, whereas some of the other techniques may not always be. For example, our technique ensures continuous dependence of the model parameters as functions of the price observations.

Scheduled for Session 5.4 Computational Finance