Jump-Diffusion Processes: Volatility Smile Fitting and Numerical Methods for Pricing

Abstract

The standard approach (e.g. Dupire (1994) and Rubinstein (1994)) to fitting stock processes to observed option prices models the underlying stock price as a one-factor diffusion process with state- and time-dependent volatility. While this approach is attractive in the sense that market completeness is maintained, the resulting model is often highly non-stationary, difficult to fit to steep volatility smiles, and generally is not well supported by empirical evidence. In this paper, we attempt to overcome some of these problems by overlaying the diffusion dynamics with a jump-process, effectively assuming that a large part of the observed volatility smiles can be explained by fear of sudden large market movements ("crash-o-phobia"). The first part of this paper derives a forward PIDE (Partial Integro-Differential Equation) satisfied by European call option prices and demonstrates how the resulting equation can be used to fit the model to the observed volatility smile/skew. In the second part of the paper, we discuss efficient methods of applying the calibrated model to the pricing of contingent claims. In particular, we develop an ADI (Alternating Directions Implicit) finite difference method that is shown to be unconditionally stable and, if combined with FFT (Fast Fourier Transform) methods, computationally efficient. The paper also discusses the usage of Monte Carlo methods, and contains several detailed examples from the S&P500 market. We compare pricing results obtained by the jump-diffusion approach with those of pure diffusion, and find significant differences for a range of popular contracts.

JEL Classification: G13, C63