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

Abstract

In this article, we consider a Markov process {X t } t 0 , starting from x ∈ R 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 t } t 0 under the risk-neutral probability measure.

Zooming-in on a Lévy process: failure to observe threshold exceedance over a dense grid

“…where the last limit holds for x, points of continuity of Π + (•), see e.g. [32]. We may assume that γ = 0 since otherwise the statement is obvious.…”

“…Furthermore, there is a large body of literature in risk theory concerned with stochastic observation times, such as the epochs of an independent Poisson process, see [1] providing a link between various exit problems for Poissonian and continuous observations. There is also a large body of literature concerned with the supremum of a Lévy process, see [16,17,38] among many others, and with the small-time behavior of Lévy processes, see [7,22,29,32] and references therein.…”