Brownian motion is continuous — it cannot produce a overnight move in any time step. Yet the 1987 crash, the 2008 financial crisis, and countless earnings shocks are precisely such discontinuities. Lévy processes extend Brownian motion by allowing jumps: random, instantaneous finite moves arriving according to a Poisson-like mechanism. This topic develops the theory and practice of jump processes for option pricing, from the simplest compound Poisson models (Merton, Kou) through infinite-activity Lévy processes (Variance Gamma, NIG, CGMY), to the stochastic calculus for semimartingales and Fourier pricing in exponential Lévy models.
By the end of this topic you should be able to:
- explain why jumps are needed at short maturities and what kind of Lévy measure matches the empirical short-dated skew;
- work with compound Poisson processes and the compensated Poisson integral;
- price European options under Merton and Kou jump-diffusion models;
- describe the Lévy-Itô decomposition and apply the Itô formula for semimartingales;
- use the Carr-Madan FFT approach to price options under any exponential Lévy model;
- calibrate a Lévy model to the full term structure of implied vol smiles.
In this lesson
- Contents
- The big picture
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