Chapter 2: Jump Models

2.1 Introduction

The previous chapter introduced models that are based on Brownian motion of asset prices. Deriving, simulating, and estimating models are much simpler when the stochastic behavior is described by Gaussian distributions. Unfortunately, the natural behavior of markets tends to display a greater number of large jumps than predicted by a Gaussian distribution. The fat tails in the distribution can be modeled by a few techniques. The approach within this chapter is an extension of the earlier Gaussian models by adding a Poisson jump process. The Poisson distribution determines the frequency of jumps within a certain time period. Several choices are available for the jump-amplitude distribution. A one-size jump is the simplest model but is quite insightful for describing the behavior of assets that are vulnerable to large negative movements such as defaults. The uniform jump model randomly selects an amplitude from a uniform range of possible jumps. This large but bounded jump size range logically implies that there is some unfeasible level for an asset price to cross. The following section derives the lognormal diffusion and log-uniform jump model and presents a least-squares method and a multinomial maximum likelihood estimation procedure. The chapter concludes with a discussion of other popular jump size distribution models.

2.2 Jump-Diffusion Model

The form of a geometric jump-diffusion stochastic differential equation is

where the subscript ...

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