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Small-time asymptotics of stopped L\'evy bridges and simulation schemes with controlled bias

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  • Jos'e E. Figueroa-L'opez
  • Peter Tankov

Abstract

We characterize the small-time asymptotic behavior of the exit probability of a L\'evy process out of a two-sided interval and of the law of its overshoot, conditionally on the terminal value of the process. The asymptotic expansions are given in the form of a first-order term and a precise computable error bound. As an important application of these formulas, we develop a novel adaptive discretization scheme for the Monte Carlo computation of functionals of killed L\'evy processes with controlled bias. The considered functionals appear in several domains of mathematical finance (e.g., structural credit risk models, pricing of barrier options, and contingent convertible bonds) as well as in natural sciences. The proposed algorithm works by adding discretization points sampled from the L\'evy bridge density to the skeleton of the process until the overall error for a given trajectory becomes smaller than the maximum tolerance given by the user.

Suggested Citation

  • Jos'e E. Figueroa-L'opez & Peter Tankov, 2012. "Small-time asymptotics of stopped L\'evy bridges and simulation schemes with controlled bias," Papers 1203.2355, arXiv.org, revised Jul 2014.
  • Handle: RePEc:arx:papers:1203.2355
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    File URL: http://arxiv.org/pdf/1203.2355
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    References listed on IDEAS

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    1. Rosenbaum, Mathieu & Tankov, Peter, 2011. "Asymptotic results for time-changed Lévy processes sampled at hitting times," Stochastic Processes and their Applications, Elsevier, vol. 121(7), pages 1607-1632, July.
    2. Claudia Ribeiro & Nick Webber, 2006. "Correcting for Simulation Bias in Monte Carlo Methods to Value Exotic Options in Models Driven by Levy Processes," Applied Mathematical Finance, Taylor & Francis Journals, vol. 13(4), pages 333-352.
    3. Svetlana I. Boyarchenko & Sergei Z. Levendorskiĭ, 2002. "Barrier options," World Scientific Book Chapters,in: Non-Gaussian Merton-Black-Scholes Theory, chapter 8, pages 185-198 World Scientific Publishing Co. Pte. Ltd..
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    Cited by:

    1. Aleksandar Mijatovic & Martijn Pistorius & Johannes Stolte, 2014. "Randomisation and recursion methods for mixed-exponential Levy models, with financial applications," Papers 1410.7316, arXiv.org.
    2. repec:spr:finsto:v:21:y:2017:i:4:d:10.1007_s00780-017-0341-7 is not listed on IDEAS
    3. Mike Giles & Yuan Xia, 2014. "Multilevel Monte Carlo For Exponential L\'{e}vy Models," Papers 1403.5309, arXiv.org, revised May 2017.

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