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

  • 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)

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    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.

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    Article provided by INFORMS in its journal Management Science.

    Volume (Year): 52 (2006)
    Issue (Month): 12 (December)
    Pages: 1930-1944

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    Handle: RePEc:inm:ormnsc:v:52:y:2006:i:12:p:1930-1944
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    1. Peter Carr & Helyette Geman, 2002. "The Fine Structure of Asset Returns: An Empirical Investigation," The Journal of Business, University of Chicago Press, vol. 75(2), pages 305-332, April.
    2. Madan, Dilip B & Seneta, Eugene, 1990. "The Variance Gamma (V.G.) Model for Share Market Returns," The Journal of Business, University of Chicago Press, vol. 63(4), pages 511-24, October.
    3. Fredrik Åkesson & John P. Lehoczky, 2000. "Path Generation for Quasi-Monte Carlo Simulation of Mortgage-Backed Securities," Management Science, INFORMS, vol. 46(9), pages 1171-1187, September.
    4. Dilip B. Madan & Peter P. Carr & Eric C. Chang, 1998. "The Variance Gamma Process and Option Pricing," Review of Finance, European Finance Association, vol. 2(1), pages 79-105.
    5. Dilip B. Madan & Frank Milne, 1991. "Option Pricing With V. G. Martingale Components," Mathematical Finance, Wiley Blackwell, vol. 1(4), pages 39-55.
    6. Boyle, Phelim & Broadie, Mark & Glasserman, Paul, 1997. "Monte Carlo methods for security pricing," Journal of Economic Dynamics and Control, Elsevier, vol. 21(8-9), pages 1267-1321, June.
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