Small-time expansions for local jump-diffusion models with infinite jump activity
AbstractWe consider a Markov process X which is the solution of a stochastic differential equation driven by a L\'evy 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\'evy 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\'evy 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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Bibliographic InfoPaper provided by arXiv.org in its series Papers with number 1108.3386.
Date of creation: Aug 2011
Date of revision: Jan 2013
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- Jose E. Figueroa-Lopez & Martin Forde, 2011. "The small-maturity smile for exponential Levy models," Papers 1105.3180, arXiv.org, revised Dec 2011.
- Peter Carr & Liuren Wu, 2002.
"What Type of Process Underlies Options? A Simple Robust Test,"
- 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.
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