Integration of Jump Components

Jumps are a feature which may occur both in the underlying and in the subordinated processes. They show typically negative correlation, i.e., a downward jump in the underlying process is associated with an upward jump in the variance process. The negative correlation of Brownian motions in the stochastic volatility case or of jumps in the case of pure jump or jump-diffusion models is known as leverage effect.1 The first approach of incorporating jumps in derivative pricing models traces back to Merton (1976), though he did not work with any form of a subordinated volatility process. Bates (1996b) makes use of the characteristic function approach of Heston (1993) and adds lognormal jumps to a CIR stochastic volatility model. Bakshi, Cao and Chen (1997) allow for jumps in the (log-)underlying process, for stochastic volatility and stochastic interest rates. They provide an empirical analysis of jump-diffusion option pricing models without mean reversion of the underlying. Bates (2000) also tests his jump-diffusion model in an empirical survey.2 The authors point out that the incorporation of jumps is especially important for the correct valuation of short-term options, since they can explain the non-zero prices of options which are out of the money and have only a few days left to maturity. The drawback of continuous diffusion processes in this context is that they have too little time to come back in the money and therefore result in an exercise probability which is virtually zero. In contrast to pure diffusion processes, a jump-diffusion process exhibits a non-zero probability for a jump event even in a small time interval so that the model matches the empirical observation of option prices. The importance of jump components in the valuation model weakens with increasing maturity since in the long run, the jump-diffusion and the pure diffusion model with stochastic volatility both generate a skewed probability distribution of $${X}_{t}$$ with fat tails3 Hilliard and Reis (1998, 1999) apply the framework of Bates (1996a, 1996b) on the valuation of commodity futures and options. However, the commodity price processes in Hilliard and Reis are not mean-reverting.