Jump diffusion models are special cases of exponential Lévy models in which the frequency of jumps are finite. They are considered as prototypes for a large class of complex models such as the stochastic volatility plus jumps models (see ). They have been used extensively in finance to model option prices (see [127, 128, 151]). General Lévy processes will be studied in details in Chapter 12.
The jump diffusion models comprises two parts, namely, a jump part and a diffusion part. The diffusion term is determined by the driving Brownian motion and the jump term is determined by the Poisson process. The Poisson process causes price changes in the underlying asset and is determined by a distribution function. The jump part enables to model sudden and unexpected price jumps of the underlying asset. Examples of the jump diffusion model include the Merton model (see ), the Black–Scholes models with jumps (see ), the Kou double exponential jump diffusion model (see ), and several others. In this chapter we introduce these models by briefly discussing the Poisson process (jumps) and the compound Poisson process.