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A smooth estimator for MC/QMC methods in finance

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  • Han, Chuan-Hsiang
  • Lai, Yongzeng

Abstract

We investigate the effect of martingale control as a smoother for MC/QMC methods. Numerical results of estimating low-biased solutions for American put option prices under the Black–Scholes model demonstrate that using QMC methods can be problematic. But it can be fixed by adding a (local) martingale control variate into the least-squares estimator to gain accuracy and efficiency. In examples of estimating European option prices under multi-factor stochastic volatility models, randomized QMC methods improve the variance by merely a single digit. After adding a martingale control, the variance reduction ratio raise up to 700 times for randomized QMC and about 50 times for MC simulations. When the delta estimation problem is considered, the efficiency of the martingale control variate method decreases. We propose an importance sampling method which performs better particularly in the presence of rare events.

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  • Han, Chuan-Hsiang & Lai, Yongzeng, 2010. "A smooth estimator for MC/QMC methods in finance," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 81(3), pages 536-550.
    • Handle: RePEc:eee:matcom:v:81:y:2010:i:3:p:536-550
    DOI: 10.1016/j.matcom.2010.07.013
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    1. Corwin Joy & Phelim P. Boyle & Ken Seng Tan, 1996. "Quasi-Monte Carlo Methods in Numerical Finance," Management Science, INFORMS, vol. 42(6), pages 926-938, June.
    2. Haynes H. M. Yung & Hua Zhang, 2003. "An empirical investigation of the GARCH option pricing model: Hedging performance," Journal of Futures Markets, John Wiley & Sons, Ltd., vol. 23(12), pages 1191-1207, December.
    3. Longstaff, Francis A & Schwartz, Eduardo S, 2001. "Valuing American Options by Simulation: A Simple Least-Squares Approach," University of California at Los Angeles, Anderson Graduate School of Management qt43n1k4jb, Anderson Graduate School of Management, UCLA.
    4. Longstaff, Francis A & Schwartz, Eduardo S, 2001. "Valuing American Options by Simulation: A Simple Least-Squares Approach," The Review of Financial Studies, Society for Financial Studies, vol. 14(1), pages 113-147.
    5. Heston, Steven L, 1993. "A Closed-Form Solution for Options with Stochastic Volatility with Applications to Bond and Currency Options," The Review of Financial Studies, Society for Financial Studies, vol. 6(2), pages 327-343.
    6. Pierre L'Ecuyer & Christiane Lemieux, 2000. "Variance Reduction via Lattice Rules," Management Science, INFORMS, vol. 46(9), pages 1214-1235, September.
    7. Jean-Pierre Fouque & Chuan-Hsiang Han, 2004. "Variance reduction for Monte Carlo methods to evaluate option prices under multi-factor stochastic volatility models," Quantitative Finance, Taylor & Francis Journals, vol. 4(5), pages 597-606.
    8. Sassan Alizadeh & Michael W. Brandt & Francis X. Diebold, 2002. "Range‐Based Estimation of Stochastic Volatility Models," Journal of Finance, American Finance Association, vol. 57(3), pages 1047-1091, June.
    9. Black, Fischer & Scholes, Myron S, 1973. "The Pricing of Options and Corporate Liabilities," Journal of Political Economy, University of Chicago Press, vol. 81(3), pages 637-654, May-June.
    10. L. C. G. Rogers, 2002. "Monte Carlo valuation of American options," Mathematical Finance, Wiley Blackwell, vol. 12(3), pages 271-286, July.
    11. Phelim Boyle & Yongzeng Lai & Ken Seng Tan, 2005. "Pricing Options Using Lattice Rules," North American Actuarial Journal, Taylor & Francis Journals, vol. 9(3), pages 50-76.
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    Cited by:

    1. Han, Chuan-Hsiang & Molina, German & Fouque, Jean-Pierre, 2014. "McMC estimation of multiscale stochastic volatility models with applications," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 103(C), pages 1-11.

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