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Pricing and Semimartingale Representations of Vulnerable Contingent Claims in Regime-Switching Markets

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  • Agostino Capponi
  • Jose Figueroa-Lopez
  • Jeffrey Nisen
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    Abstract

    Using a suitable change of probability measure, we obtain a novel Poisson series representation for the arbitrage- free price process of vulnerable contingent claims in a regime-switching market driven by an underlying continuous- time Markov process. As a result of this representation, along with a short-time asymptotic expansion of the claim's price process, we develop an efficient method for pricing claims whose payo?s may depend on the full path of the underlying Markov chain. The proposed approach is applied to price not only simple European claims such as defaultable bonds, but also a new type of path-dependent claims that we term self-decomposable, as well as the important class of vulnerable call and put options on a stock. We provide a detailed error analysis and illustrate the accuracy and computational complexity of our method on several market traded instruments, such as defaultable bond prices, barrier options, and vulnerable call options. Using again our Poisson series representation, we show differentiability in time of the pre-default price function of European vulnerable claims, which enables us to rigorously deduce Feynman-Kac representations for the pre-default pricing function and new semimartingale representations for the price process of the vulnerable claim under both risk-neutral and objective probability measures.

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    File URL: http://arxiv.org/pdf/1110.0403
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    Bibliographic Info

    Paper provided by arXiv.org in its series Papers with number 1110.0403.

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    Date of creation: Oct 2011
    Date of revision: Feb 2012
    Handle: RePEc:arx:papers:1110.0403

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    Web page: http://arxiv.org/

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    1. Tomasz R. Bielecki & Stéphane Crépey & Monique Jeanblanc & Marek Rutkowski, 2008. "Defaultable Options In A Markovian Intensity Model Of Credit Risk," Mathematical Finance, Wiley Blackwell, vol. 18(4), pages 493-518.
    2. Ang, Andrew & Bekaert, Geert, 2002. "Regime Switches in Interest Rates," Journal of Business & Economic Statistics, American Statistical Association, vol. 20(2), pages 163-82, April.
    3. Tomasz Bielecki & Stephane Crepey & Monique Jeanblanc & Marek Rutkowski, 2008. "Arbitrage pricing of defaultable game options with applications to convertible bonds," Quantitative Finance, Taylor and Francis Journals, vol. 8(8), pages 795-810.
    4. Andrew Ang & Geert Bekaert, 2002. "International Asset Allocation With Regime Shifts," Review of Financial Studies, Society for Financial Studies, vol. 15(4), pages 1137-1187.
    5. Tomasz Bielecki & Inwon Jang, 2006. "Portfolio optimization with a defaultable security," Asia-Pacific Financial Markets, Springer, vol. 13(2), pages 113-127, June.
    6. Christophette Blanchet-Scalliet & Monique Jeanblanc, 2004. "Hazard rate for credit risk and hedging defaultable contingent claims," Finance and Stochastics, Springer, vol. 8(1), pages 145-159, January.
    7. Ang, Andrew & Bekaert, Geert, 2002. "Short rate nonlinearities and regime switches," Journal of Economic Dynamics and Control, Elsevier, vol. 26(7-8), pages 1243-1274, July.
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