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Advanced Monte Carlo methods for barrier and related exotic options

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  • Emmanuel Gobet

    (MATHFI - Mathématiques financières - LJK - Laboratoire Jean Kuntzmann - UPMF - Université Pierre Mendès France - Grenoble 2 - UJF - Université Joseph Fourier - Grenoble 1 - Grenoble INP - Institut polytechnique de Grenoble - Grenoble Institute of Technology - CNRS - Centre National de la Recherche Scientifique)

Abstract

In this work, we present advanced Monte Carlo techniques applied to the pricing of barrier options and other related exotic contracts. It covers in particular the Brownian bridge approaches, the barrier shifting techniques (BAST) and their extensions as well. We leverage the link between discrete and continuous monitoring to design efficient schemes, which can be applied to the Black-Scholes model but also to stochastic volatility or Merton's jump models. This is supported by theoretical results and numerical experiments.

Suggested Citation

  • Emmanuel Gobet, 2009. "Advanced Monte Carlo methods for barrier and related exotic options," Post-Print hal-00319947, HAL.
    • Handle: RePEc:hal:journl:hal-00319947
    DOI: 10.1016/S1570-8659(08)00012-4
    Note: View the original document on HAL open archive server: https://hal.science/hal-00319947
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    References listed on IDEAS

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    1. Robert C. Merton, 2005. "Theory of rational option pricing," World Scientific Book Chapters, in: Sudipto Bhattacharya & George M Constantinides (ed.), Theory Of Valuation, chapter 8, pages 229-288, World Scientific Publishing Co. Pte. Ltd..
    2. Gobet, Emmanuel & Menozzi, Stéphane, 2004. "Exact approximation rate of killed hypoelliptic diffusions using the discrete Euler scheme," Stochastic Processes and their Applications, Elsevier, vol. 112(2), pages 201-223, August.
    3. Emmanuel Gobet, 2004. "Revisiting the Greeks for European and American Options," World Scientific Book Chapters, in: Jiro Akahori & Shigeyoshi Ogawa & Shinzo Watanabe (ed.), Stochastic Processes And Applications To Mathematical Finance, chapter 3, pages 53-71, World Scientific Publishing Co. Pte. Ltd..
    4. L.C.G. Rogers & E.J. Stapleton, 1997. "Fast accurate binomial pricing," Finance and Stochastics, Springer, vol. 2(1), pages 3-17.
    5. Mark Broadie & Paul Glasserman & Steven Kou, 1997. "A Continuity Correction for Discrete Barrier Options," Mathematical Finance, Wiley Blackwell, vol. 7(4), pages 325-349, October.
    6. Phelim Boyle & Yisong Tian, 1998. "An explicit finite difference approach to the pricing of barrier options," Applied Mathematical Finance, Taylor & Francis Journals, vol. 5(1), pages 17-43.
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    Cited by:

    1. Lee, Hangsuck & Lee, Minha & Ko, Bangwon, 2022. "A semi-analytic valuation of two-asset barrier options and autocallable products using Brownian bridge," The North American Journal of Economics and Finance, Elsevier, vol. 61(C).
    2. Yuri F. Saporito & Zhaoyu Zhang, 2020. "PDGM: a Neural Network Approach to Solve Path-Dependent Partial Differential Equations," Papers 2003.02035, arXiv.org, revised Apr 2020.
    3. Bougias, Alexandros & Episcopos, Athanasios & Leledakis, George N., 2022. "The role of asset payouts in the estimation of default barriers," International Review of Financial Analysis, Elsevier, vol. 81(C).
    4. Fernández Lexuri & Hieber Peter & Scherer Matthias, 2013. "Double-barrier first-passage times of jump-diffusion processes," Monte Carlo Methods and Applications, De Gruyter, vol. 19(2), pages 107-141, July.
    5. Hangsuck Lee & Gaeun Lee & Seongjoo Song, 2022. "Multi‐step reflection principle and barrier options," Journal of Futures Markets, John Wiley & Sons, Ltd., vol. 42(4), pages 692-721, April.
    6. M. Krivko & M. V. Tretyakov, 2012. "Application of simplest random walk algorithms for pricing barrier options," Papers 1211.5726, arXiv.org.
    7. Gobet, Emmanuel & Menozzi, Stéphane, 2010. "Stopped diffusion processes: Boundary corrections and overshoot," Stochastic Processes and their Applications, Elsevier, vol. 120(2), pages 130-162, February.
    8. Hangsuck Lee & Gaeun Lee & Seongjoo Song, 2021. "Multi-step Reflection Principle and Barrier Options," Papers 2105.15008, arXiv.org.

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