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

Jos\'e E. Figueroa-L\'opez, Yankeng Luo and Cheng Ouyang

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Abstract: We consider a Markov process $X$, which is the solution of a stochastic differential equation driven by a L\'{e}vy 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 L\'{e}vy density of $Z$ outside any neighborhood of the origin, we obtain a small-time second-order polynomial expansion for the tail distribution and the transition density of the process $X$. Our method of proof combines a recent regularizing technique for deriving the analog small-time expansions for a L\'{e}vy process with some new tail and density estimates for jump-diffusion processes with small jumps based on the theory of Malliavin calculus, flow of diffeomorphisms for SDEs, and time-reversibility. 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.

Date: 2011-08, Revised 2014-07
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Published in Bernoulli 2014, Vol. 20, No. 3, 1165-1209

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Persistent link: https://EconPapers.repec.org/RePEc:arx:papers:1108.3386

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