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Fast Estimation of True Bounds on Bermudan Option Prices under Jump-diffusion Processes

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  • Helin Zhu
  • Fan Ye
  • Enlu Zhou
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    Abstract

    Fast pricing of American-style options has been a difficult problem since it was first introduced to financial markets in 1970s, especially when the underlying stocks' prices follow some jump-diffusion processes. In this paper, we propose a new algorithm to generate tight upper bounds on the Bermudan option price without nested simulation, under the jump-diffusion setting. By exploiting the martingale representation theorem for jump processes on the dual martingale, we are able to explore the unique structure of the optimal dual martingale and construct an approximation that preserves the martingale property. The resulting upper bound estimator avoids the nested Monte Carlo simulation suffered by the original primal-dual algorithm, therefore significantly improves the computational efficiency. Theoretical analysis is provided to guarantee the quality of the martingale approximation. Numerical experiments are conducted to verify the efficiency of our proposed algorithm.

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    File URL: http://arxiv.org/pdf/1305.4321
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    Paper provided by arXiv.org in its series Papers with number 1305.4321.

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    Date of creation: May 2013
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    Handle: RePEc:arx:papers:1305.4321

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    1. Fang, Fang & Oosterlee, Kees, 2008. "Pricing Early-Exercise and Discrete Barrier Options by Fourier-Cosine Series Expansions," MPRA Paper 9248, University Library of Munich, Germany.
    2. Merton, Robert C., 1975. "Option pricing when underlying stock returns are discontinuous," Working papers 787-75., Massachusetts Institute of Technology (MIT), Sloan School of Management.
    3. Christian Bender, 2011. "Dual pricing of multi-exercise options under volume constraints," Finance and Stochastics, Springer, vol. 15(1), pages 1-26, January.
    4. Leif Andersen & Mark Broadie, 2004. "Primal-Dual Simulation Algorithm for Pricing Multidimensional American Options," Management Science, INFORMS, vol. 50(9), pages 1222-1234, September.
    5. 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.
    6. Evis K´┐Żllezi & Nick Webber, 2004. "Valuing Bermudan options when asset returns are Levy processes," Quantitative Finance, Taylor & Francis Journals, vol. 4(1), pages 87-100.
    7. L. C. G. Rogers, 2002. "Monte Carlo valuation of American options," Mathematical Finance, Wiley Blackwell, vol. 12(3), pages 271-286.
    8. Denis Belomestny & Christian Bender & John Schoenmakers, 2009. "True Upper Bounds For Bermudan Products Via Non-Nested Monte Carlo," Mathematical Finance, Wiley Blackwell, vol. 19(1), pages 53-71.
    9. Nan Chen & Paul Glasserman, 2007. "Additive and multiplicative duals for American option pricing," Finance and Stochastics, Springer, vol. 11(2), pages 153-179, April.
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