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Efficient Monte Carlo and Quasi-Monte Carlo Option Pricing Under the Variance Gamma Model

Author

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  • Athanassios N. Avramidis

    (Département d'Informatique et de Recherche Opérationnelle, Université de Montréal, C.P. 6128, Succ. Centre-Ville, Montréal, H3C 3J7, Canada)

  • Pierre L'Ecuyer

    (Département d'Informatique et de Recherche Opérationnelle, Université de Montréal, C.P. 6128, Succ. Centre-Ville, Montréal, H3C 3J7, Canada)

Abstract

We develop and study efficient Monte Carlo algorithms for pricing path-dependent options with the variance gamma model. The key ingredient is difference-of-gamma bridge sampling, based on the representation of a variance gamma process as the difference of two increasing gamma processes. For typical payoffs, we obtain a pair of estimators (named low and high) with expectations that (1) are monotone along any such bridge sampler, and (2) contain the continuous-time price. These estimators provide pathwise bounds on unbiased estimators that would be more expensive (infinitely expensive in some situations) to compute. By using these bounds with extrapolation techniques, we obtain significant efficiency improvements by work reduction. We then combine the gamma bridge sampling with randomized quasi-Monte Carlo to reduce the variance and thus further improve the efficiency. We illustrate the large efficiency improvements on numerical examples for Asian, lookback, and barrier options.

Suggested Citation

  • Athanassios N. Avramidis & Pierre L'Ecuyer, 2006. "Efficient Monte Carlo and Quasi-Monte Carlo Option Pricing Under the Variance Gamma Model," Management Science, INFORMS, vol. 52(12), pages 1930-1944, December.
    • Handle: RePEc:inm:ormnsc:v:52:y:2006:i:12:p:1930-1944
    DOI: 10.1287/mnsc.1060.0575
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    References listed on IDEAS

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    Cited by:

    1. Vladimir K. Kaishev & Dimitrina S. Dimitrova, 2009. "Dirichlet Bridge Sampling for the Variance Gamma Process: Pricing Path-Dependent Options," Management Science, INFORMS, vol. 55(3), pages 483-496, March.
    2. Ye, Zhi-Sheng, 2013. "On the conditional increments of degradation processes," Statistics & Probability Letters, Elsevier, vol. 83(11), pages 2531-2536.
    3. Dingeç, Kemal Dinçer & Hörmann, Wolfgang, 2012. "A general control variate method for option pricing under Lévy processes," European Journal of Operational Research, Elsevier, vol. 221(2), pages 368-377.
    4. Kenichiro Shiraya & Hiroki Uenishi & Akira Yamazaki, 2019. "A General Control Variate Method for Lévy Models in Finance (Published in European Journal of Operational Research.)," CARF F-Series CARF-F-455, Center for Advanced Research in Finance, Faculty of Economics, The University of Tokyo, revised Jan 2020.
    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.
    6. Hatem Ben‐Ameur & Rim Chérif & Bruno Rémillard, 2020. "Dynamic programming for valuing American options under a variance‐gamma process," Journal of Futures Markets, John Wiley & Sons, Ltd., vol. 40(10), pages 1548-1561, October.
    7. Genin, Adrien & Tankov, Peter, 2020. "Optimal importance sampling for Lévy processes," Stochastic Processes and their Applications, Elsevier, vol. 130(1), pages 20-46.
    8. Madan, Dilip B. & Schoutens, Wim, 2013. "Systemic risk tradeoffs and option prices," Insurance: Mathematics and Economics, Elsevier, vol. 52(2), pages 222-230.
    9. Shiraya, Kenichiro & Uenishi, Hiroki & Yamazaki, Akira, 2020. "A general control variate method for Lévy models in finance," European Journal of Operational Research, Elsevier, vol. 284(3), pages 1190-1200.
    10. Baldeaux Jan, 2008. "Quasi-Monte Carlo methods for the Kou model," Monte Carlo Methods and Applications, De Gruyter, vol. 14(4), pages 281-302, January.
    11. Xie, Fei & He, Zhijian & Wang, Xiaoqun, 2019. "An importance sampling-based smoothing approach for quasi-Monte Carlo simulation of discrete barrier options," European Journal of Operational Research, Elsevier, vol. 274(2), pages 759-772.
    12. Pierre L’Ecuyer & Florian Puchhammer & Amal Ben Abdellah, 2022. "Monte Carlo and Quasi–Monte Carlo Density Estimation via Conditioning," INFORMS Journal on Computing, INFORMS, vol. 34(3), pages 1729-1748, May.
    13. Klößner, Stefan & Becker, Martin & Friedmann, Ralph, 2012. "Modeling and measuring intraday overreaction of stock prices," Journal of Banking & Finance, Elsevier, vol. 36(4), pages 1152-1163.
    14. Roman V. Ivanov, 2023. "The Semi-Hyperbolic Distribution and Its Applications," Stats, MDPI, vol. 6(4), pages 1-21, October.
    15. Wang, Chou-Wen & Wu, Chin-Wen & Tzang, Shyh-Weir, 2012. "Implementing option pricing models when asset returns follow an autoregressive moving average process," International Review of Economics & Finance, Elsevier, vol. 24(C), pages 8-25.
    16. Adrien Genin & Peter Tankov, 2016. "Optimal importance sampling for L\'evy Processes," Papers 1608.04621, arXiv.org.
    17. Xiaoqun Wang & Ken Seng Tan, 2013. "Pricing and Hedging with Discontinuous Functions: Quasi-Monte Carlo Methods and Dimension Reduction," Management Science, INFORMS, vol. 59(2), pages 376-389, July.
    18. Gunther Leobacher, 2017. "A short introduction to quasi-Monte Carlo option pricing," Papers 1707.04293, arXiv.org, revised Jul 2017.

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