Analysis of Financial Time Series, Third Edition

6.9 Jump Diffusion Models

Empirical studies have found that the stochastic diffusion model based on Brownian motion fails to explain some characteristics of asset returns and the prices of their derivatives [e.g., the “volatility smile” of implied volatilities; see Bakshi, Cao, and Chen (1997) and the references therein]. Volatility smile is referred to as the convex function between the implied volatility and strike price of an option. Both out-of-the-money and in-the-money options tend to have higher implied volatilities than at-the-money options especially in the foreign exchange markets. Volatility smile is less pronounced for equity options. The inadequacy of the standard stochastic diffusion model has led to the developments of alternative continuous-time models. For example, jump diffusion and stochastic volatility models have been proposed in the literature to overcome the inadequacy; see Merton (1976) and Duffie (1995).

Jumps in stock prices are often assumed to follow a probability law. For example, the jumps may follow a Poisson process, which is a continuous-time discrete process. For a given time t, let Xt be the number of times a special event occurs during the time period [0, t]. Then Xt is a Poisson process if

That is, Xt follows a Poisson distribution with parameter λt. The parameter λ governs the occurrence of the special event and is referred to as the rate or ...