Two-Sided Estimates for Distribution Densities in Models with Jumps

The present paper is devoted to applications of mathematical analysis to the study of distribution densities arising in stochastic stock price models. We consider uncorrelated Stein-Stein, Heston, and Hull-White models and their perturbations by compound Poisson processes with jump amplitudes distributed according to a double exponential law. Similar perturbations of the Black-Scholes model were studied by S. Kou. For perturbed models, we obtain two-sided estimates for the stock price distribution density and compare the rate of decay of this density in the original and the perturbed model. It is shown that if the value of the parameter, characterizing the rate of decay of the right tail of the double exponential law, is small, then the stock price density in the perturbed model decays slower than the density in the original model. On the other hand, if the value of this parameter is large, then there are no significant changes in the behavior of the stock price distribution density.