Short-time asymptotics for marginal distributions of semimartingales

Abstract

We study the short-time asymptotics of conditional expectations of smooth and non-smooth functions of a (discontinuous) Ito semimartingale; we compute the leading term in the asymptotics in terms of the local characteristics of the semimartingale. We derive in particular the asymptotic behavior of call options with short maturity in a semimartingale model: whereas the behavior of out-of-the-money options is found to be linear in time, the short time asymptotics of at-the-money options is shown to depend on the fine structure of the semimartingale.

Small time central limit theorems for semimartingales with applications

“…In the works [2], [3], [4], [6], [13], [14], [21] and the references therein the authors study the behavior of the random variables E[f (X t 0 +δ )|F X t 0 ] for small values of δ > 0, where X is a finite-dimensional (jump-)diffusion process, a Lévy process or more generally a semimartingale, (F X t ) t≥0 is the filtration it generates and the function f is taken from a space of suitable real-valued test functions. In [4], this program is carried out for general finite-dimensional semimartingales and under appropriate continuity assumptions on the characteristics of X as well as smoothness assumptions on the function f , the almost sure limit (1.1) lim…”