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Advanced Derivatives and Exotic Options / Jumps and Lévy Processes

Compound Poisson Processes and the Anatomy of a Jump

A compound Poisson process is the simplest model of a jumpy asset: jumps arrive at random times (Poisson-distributed) with random sizes (i.i.d. from some distribution). This note builds the formal machinery from first principles — Poisson counting processes, random jump sizes, the compensated Poisson integral, and the jump SDE for a stock price — so that the Merton and Kou models in the next lesson are fully grounded.
The note also introduces the Poisson random measure, which is the coordinate-free description of jump arrivals and is needed for the Lévy-Itô decomposition and the full theory of Lévy processes.
Learning objectives. By the end of this note you should be able to:
  • define the Poisson counting process NtN_t and state its distribution at time TT;
  • write the compound Poisson process Jt=∑i=1NtYiJ_t = \sum_{i=1}^{N_t} Y_i and compute its mean and variance;
  • define the compensated Poisson process and state when it is a martingale;
  • write and interpret the Poisson random measure μ(dt,dx)\mu(dt, dx);
  • set up a jump SDE for the log-stock price.

In this lesson

  1. 2.1 The Poisson counting process
  2. 2.2 Compound Poisson processes
  3. 2.3 The compensated Poisson process
  4. 2.4 Poisson random measures
  5. 2.5 The jump SDE for log-stock

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