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The first passage time problem for mixed-exponential jump processes with applications in insurance and finance

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  • Chuancun Yin
  • Yuzhen Wen
  • Zhaojun Zong
  • Ying Shen
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    Abstract

    This paper stidies the first passage times to constant boundaries for mixed-exponential jump diffusion processes. Explicit solutions of the Laplace transforms of the distribution of the first passage times, the joint distribution of the first passage times and undershoot (overshoot) are obtained. As applications, we present explicit expression of the Gerber-Shiu functions for surplus processes with two-sided jumps, present the analytical solutions for popular path-dependent options such as lookback and barrier options in terms of Laplace transforms and give a closed-form expression on the price of the zero-coupon bond under a structural credit risk model with jumps.

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    File URL: http://arxiv.org/pdf/1302.6762
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    Bibliographic Info

    Paper provided by arXiv.org in its series Papers with number 1302.6762.

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    Date of creation: Feb 2013
    Date of revision: Jun 2014
    Handle: RePEc:arx:papers:1302.6762

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    Web page: http://arxiv.org/

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    1. Merton, Robert C., 1976. "Option pricing when underlying stock returns are discontinuous," Journal of Financial Economics, Elsevier, vol. 3(1-2), pages 125-144.
    2. Bianca Hilberink & L.C.G. Rogers, 2002. "Optimal capital structure and endogenous default," Finance and Stochastics, Springer, vol. 6(2), pages 237-263.
    3. Olivier Le Courtois & François Quittard-Pinon, 2006. "Risk-neutral and actual default probabilities with an endogenous bankruptcy jump-diffusion model," Asia-Pacific Financial Markets, Springer, vol. 13(1), pages 11-39, March.
    4. Ernesto Mordecki, 2002. "Optimal stopping and perpetual options for Lévy processes," Finance and Stochastics, Springer, vol. 6(4), pages 473-493.
    5. S. G. Kou & Hui Wang, 2004. "Option Pricing Under a Double Exponential Jump Diffusion Model," Management Science, INFORMS, vol. 50(9), pages 1178-1192, September.
    6. Goldman, M Barry & Sosin, Howard B & Shepp, Lawrence A, 1979. "On Contingent Claims That Insure Ex-post Optimal Stock Market Timing," Journal of Finance, American Finance Association, vol. 34(2), pages 401-13, May.
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