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

We consider a Markov process {X t }t≥0 with initial condition X (x) t = 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, we obtain a small-time second-order polynomial expansion in t for the tail distribution and the transition density of the process X. The method of proof combines a recent approach for regularizing the tail probability P(X t ≥ x + y) with classical results based on Malliavin calculus for purely-jump processes, which have to be extended here to deal with the mixture model X. 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.