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Risk-minimization and hedging claims on a jump-diffusion market model, Feynman-Kac Theorem and PIDE

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  • Jacek Jakubowski
  • Mariusz Niew\k{e}g{\l}owski
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    Abstract

    At first, we solve a problem of finding a risk-minimizing hedging strategy on a general market with ratings. Next, we find a solution to this problem on Markovian market with ratings on which prices are influenced by additional factors and rating, and behavior of this system is described by SDE driven by Wiener process and compensated Poisson random measure and claims depend on rating. To find a tool to calculate hedging strategy we prove a Feynman-Kac type theorem. This result is of independent interest and has many applications, since it enables to calculate some conditional expectations using related PIDE's. We illustrate our theory on two examples of market. The first is a general exponential L\'{e}vy model with stochastic volatility, and the second is a generalization of exponential L\'{e}vy model with regime-switching.

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    File URL: http://arxiv.org/pdf/1305.4132
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    Bibliographic Info

    Paper provided by arXiv.org in its series Papers with number 1305.4132.

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    Date of creation: May 2013
    Date of revision: Jul 2013
    Handle: RePEc:arx:papers:1305.4132

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    Web page: http://arxiv.org/

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    1. Rama Cont & Ekaterina Voltchkova, 2005. "Integro-differential equations for option prices in exponential Lévy models," Finance and Stochastics, Springer, vol. 9(3), pages 299-325, 07.
    2. Young Kim & Frank Fabozzi & Zuodong Lin & Svetlozar Rachev, 2012. "Option pricing and hedging under a stochastic volatility Lévy process model," Review of Derivatives Research, Springer, vol. 15(1), pages 81-97, April.
    3. Thomas Møller, 2001. "Risk-minimizing hedging strategies for insurance payment processes," Finance and Stochastics, Springer, vol. 5(4), pages 419-446.
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