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

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

2011-08-17

arxiv.org/abs/1108.3386v2

Mathematics

Probability

Scientific paper

We consider a Markov process X_t with initial condition X_0=x, which is the solution of a stochastic differential equation driven by a Levy process Z and an independent Wiener process W. Under some regularity conditions, including non-degeneracy of the diffusion and jump components of the process as well as smoothness of the Levy 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 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 our-of-the-money option prices in short maturity under local jump-diffusion model is also derived.

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