Poisson Jumps

In this chapter we focus on Poisson jump models that are very popular in financial modeling since Merton (1976) first derived an option pricing formula based on a stock price process generated by a mixture of a Brownian motion and a Poisson process. This mixed process is also called the jump-diffusion process. The requirement for a jump component in a stock price process is intuitive, and supported by the big crashes in stock markets: The Black Monday on October 17, 1987 and the recent market crashes in the financial crisis since 2008 are two prominent examples. To model jump events, we need two quantities: jump frequency and jump size. The first one specifies how many times jumps happen in a given time period, and the second one determines how large a jump is if it occurs. In a compound jump process, jump arrivals are modeled by Poisson process and jump sizes may be specified by various distributions. Here, we consider three representative jump models that are distinguished from each other solely by the distributions of jump sizes, they are the simple deterministic jumps, the log-normal jumps and the Pareto jumps. Additionally, we will show that Kou’s jump model (2002) with weighted double-exponential jumps is equivalent to the Pareto jump model. The CFs of all these jump models can be derived to value European-style options. Finally, we present the affine jump diffusion model of Duffie, Pan and Singleton (2000), which extends a simple jump-diffusion model to a generalized case. The solution of CFs in the affine jump-diffusion model has an exponential affine form, and may be derived via two ODEs.