ABSTRACT
Let us start by some general considerations on Lévy processes, their use as models for price dynamics and different ways to specify such models.
4.1.1 “Jump-diffusions” s. infinite activity Lévy processes
Financial models with jumps fall into two categories. In the first category, called jumpdiffusion models, the “normal” evolution of prices is given by a diffusion process, punctuated by jumps at random intervals. Here the jumps represent rare events-crashes and large drawdowns. Such an evolution can be represented by modelling the (log-)price as a Lévy process with a nonzero Gaussian component and a jump part, which is a compound Poisson process with finitely many jumps in every time interval. Examples of such models are the Merton jump-diffusion model with Gaussian jumps [291] and the Kou model with double exponential jumps [238]. In Chapter 15, we will see a model combining compound Poisson jumps and stochastic volatility: the Bates model [41]. In these models, the dynamical structure of the process is easy to understand and describe, since the distribution of jump sizes is known. They are easy to simulate and efficient Monte Carlo methods for pricing pathdependent options can be used. Models of this type also perform quite well for the purposes of implied volatility smile interpolation (see Chapter 13). However they rarely lead to closed-form densities: statistical estimation and computation of moments or quantiles may be quite difficult.
