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

TL;DR

This paper derives second order small-time expansions for tail probabilities and out-of-the-money European call options in state-dependent jump-diffusion models with infinite activity, aiding precise short-term pricing and risk assessment.

Contribution

It introduces a second order expansion for tail probabilities and option prices in complex jump-diffusion models with state-dependent jump intensity and infinite activity.

Findings

Second order tail probability expansion for small time.

Second order expansion for out-of-the-money European call options.

Application of measure change to derive option price expansion.

Abstract

In this article, we consider a Markov process X, starting from x and solving a stochastic differential equation, which is driven by a Brownian motion and an independent pure jump component exhibiting state-dependent jump intensity and infinite jump activity. A second order expansion is derived for the tail probability P[X(t)>x+y] in small time t, for y>0. As an application of this expansion and a suitable change of the underlying probability measure, a second order expansion, near expiration, for out-of-the-money European call option prices is obtained when the underlying stock price is modeled as the exponential of the jump-diffusion process X under the risk-neutral probability measure.