Two-sided estimates for stock price distribution densities in jump-diffusion models

TL;DR

This paper derives two-sided estimates for stock price distribution densities in jump-diffusion models, analyzing how perturbations with double exponential jumps affect tail behavior and density decay rates.

Contribution

It provides new bounds for the density in stochastic volatility models with jump perturbations and compares tail behaviors before and after perturbation.

Findings

Small jump tail parameter slows density decay

Large jump tail parameter results in negligible density change

Perturbed models exhibit different tail behaviors based on jump parameters

Abstract

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 stochastic volatility models, we obtain two-sided estimates for the stock price distribution density and compare the tail behavior of this density before and after perturbation. It is shown that if the value of the parameter, characterizing 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.