PIDEs for Option Pricing Under Jumps

In a diffusion model, the option pricing PDE is local: the value at $(S, t)$ is determined by a differential relationship with neighbouring values. Add jumps, and the equation becomes an integro-differential equation — the value at $(S, t)$ depends on the value at all possible jump destinations $(Se^x, t)$ via an integral against the Lévy measure. This is the PIDE (partial integro-differential equation) for option pricing under jumps.

The PIDE generalises the Black-Scholes PDE exactly as the Lévy-Khintchine formula generalises the characteristic exponent of Brownian motion. It is the deterministic pricing tool for jump models, complementing the Fourier approach (which works for European options) with a method that handles path-dependent and early-exercise features.

Learning objectives. By the end of this note you should be able to: