Small-time expansions for local jump-diffusion models with infinite jump activity

TL;DR

This paper derives second-order small-time expansions for the tail distribution and transition density of a jump-diffusion process with infinite jump activity, using advanced probabilistic techniques and Malliavin calculus.

Contribution

It introduces a novel method combining recent regularization techniques with new tail and density estimates for jump-diffusions, extending small-time expansion results to models with infinite jump activity.

Findings

Second-order polynomial expansion for tail distribution and density

Application to short-maturity out-of-the-money option pricing

Method applicable under regularity and non-degeneracy conditions

Abstract

We consider a Markov process $X$, which is the solution of a stochastic differential equation driven by a L\'{e}vy process $Z$ and an independent Wiener process $W$. Under some regularity conditions, including non-degeneracy of the diffusive and jump components of the process as well as smoothness of the L\'{e}vy density of $Z$ outside any neighborhood of the origin, we obtain a small-time second-order polynomial expansion for the tail distribution and the transition density of the process $X$. Our method of proof combines a recent regularizing technique for deriving the analog small-time expansions for a L\'{e}vy process with some new tail and density estimates for jump-diffusion processes with small jumps based on the theory of Malliavin calculus, flow of diffeomorphisms for SDEs, and time-reversibility. As an application, the leading term for out-of-the-money option prices in short maturity under a local jump-diffusion model is also derived.