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

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Listed:
  • Jos'e E. Figueroa-L'opez
  • Yankeng Luo
  • Cheng Ouyang

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.

Suggested Citation

  • Jos'e E. Figueroa-L'opez & Yankeng Luo & Cheng Ouyang, 2011. "Small-time expansions for local jump-diffusion models with infinite jump activity," Papers 1108.3386, arXiv.org, revised Jul 2014.
  • Handle: RePEc:arx:papers:1108.3386
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    File URL: http://arxiv.org/pdf/1108.3386
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    References listed on IDEAS

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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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    Cited by:

    1. Jos'e E. Figueroa-L'opez & Yankeng Luo, 2015. "Small-time expansions for state-dependent local jump-diffusion models with infinite jump activity," Papers 1505.04459, arXiv.org, revised Dec 2015.

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