A risk-neutral approach to option pricing with jumps and diffusion

ABSTRACT We introduce a risk-neutral option pricing framework that incorporates jumps in the underlying price process while yielding the familiar Black–Scholes pricing equation in the limit of vanishing jump size. Analytical expressions exist for the prices of simple non-path-dependent options, while more complicated option contracts may be handled via an application of standard numerical techniques (eg, tree methods, PDE solvers, Monte Carlo or portfolio replication). In the context of our PDE formulation, we introduce a perturbative treatment of our pricing equation to obtain an analytically tractable procedure for correcting Black–Scholes prices to accommodate the observed volatility smile in the presence of time-dependent model parameters. The existence of jumps in the underlying price process provides a natural mechanism for generating skew in the implied volatility surface and for producing smile dynamics consistent with a number of underlying asset types. Examples are given that demonstrate the ease of calibrating the model to market prices.