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

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  • Jos\'e E. Figueroa-L\'opez
  • Yankeng Luo
  • 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.

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    File URL: http://arxiv.org/pdf/1108.3386
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    Bibliographic Info

    Paper provided by arXiv.org in its series Papers with number 1108.3386.

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    Date of creation: Aug 2011
    Date of revision: Jul 2014
    Publication status: Published in Bernoulli 2014, Vol. 20, No. 3, 1165-1209
    Handle: RePEc:arx:papers:1108.3386

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    Web page: http://arxiv.org/

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    1. Peter Carr & Liuren Wu, 2003. "What Type of Process Underlies Options? A Simple Robust Test," Journal of Finance, American Finance Association, vol. 58(6), pages 2581-2610, December.
    2. Jose E. Figueroa-Lopez & Martin Forde, 2011. "The small-maturity smile for exponential Levy models," Papers 1105.3180, arXiv.org, revised Dec 2011.
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