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

Abstract

We consider a Markov process $X$, which is the solution of a stochastic differential equation driven by a Lévy 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 Lévy 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 Lévy 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.

Published in at http://dx.doi.org/10.3150/13-BEJ518 the Bernoulli (http://isi.cbs.nl/bernoulli/) by the International Statistical Institute/Bernoulli Society (http://isi.cbs.nl/BS/bshome.htm)

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• Testing for jumps in a discretely observed process
• Is Brownian motion necessary to model high-frequency data?
• Small-Time Asymptotics of Option Prices and First Absolute Moments
• Small-time expansions of the distributions, densities, and option prices of stochastic volatility models with Lévy jumps
• Small-time expansions for local jump-diffusion models with infinite jump activity

Cited by in corpus (1)

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