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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.

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  • 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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    References listed on IDEAS

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    1. 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.
    2. 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..
    3. Martin Becker, 2010. "Comment on 'Correcting for Simulation Bias in Monte Carlo Methods to Value Exotic Options in Models Driven by Levy Processes' by C. Ribeiro and N. Webber," Applied Mathematical Finance, Taylor & Francis Journals, vol. 17(2), pages 133-146.
    4. 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.
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    Cited by:

    1. Michael B. Giles & Yuan Xia, 2017. "Multilevel Monte Carlo for exponential Lévy models," Finance and Stochastics, Springer, vol. 21(4), pages 995-1026, October.
    2. Mike Giles & Yuan Xia, 2014. "Multilevel Monte Carlo For Exponential L\'{e}vy Models," Papers 1403.5309, arXiv.org, revised May 2017.
    3. Jorge Ignacio Gonz'alez C'azares & Aleksandar Mijatovi'c, 2021. "Monte Carlo algorithm for the extrema of tempered stable processes," Papers 2103.15310, arXiv.org, revised Dec 2022.
    4. Aleksandar Mijatovic & Martijn Pistorius & Johannes Stolte, 2014. "Randomisation and recursion methods for mixed-exponential Levy models, with financial applications," Papers 1410.7316, arXiv.org.
    5. Kyoung-Kuk Kim & Sojung Kim, 2016. "Simulation of Tempered Stable Lévy Bridges and Its Applications," Operations Research, INFORMS, vol. 64(2), pages 495-509, April.

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