Lévy Processes and Jump Calculus — Overview

Brownian motion is the unique Lévy process with continuous paths. The moment we allow paths to jump, a richer class of processes opens up: Poisson processes (unit jumps at random times), compound Poisson processes (arbitrary jump sizes), and the full Lévy family — processes whose increments are independent and stationary but whose paths need not be continuous.

In finance, jumps matter because equity returns exhibit sudden large moves (earnings surprises, default events), volatility smiles cannot be reproduced by any diffusion model (Dupire's theorem), and credit derivatives require models where a company's asset value can jump to zero. The Merton (1976) jump-diffusion model replaces Brownian motion with $B_t + \sum_{i=1}^{N_t} J_i$ (a BM plus a compound Poisson sum) and prices options via a characteristic-function / Fourier transform that the pure-diffusion BS formula cannot reach.

1. Contents
2. Two faces of the same theory
3. Prerequisites
4. Finance reading by lesson