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Quasi-Monte Carol Methods for the Heston Model

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  • Jan Baldeaux
  • Dale Roberts

    (Australian National University)

Abstract

In this paper, we discuss the application of quasi-Monte Carlo methods to the Heston model. We base our algorithms on the Broadie-Kaya algorithm, an exact simulation scheme for the Heston model. As the joint transition densities are not available in closed-form, the Linear Transformation method due to Imai and Tan, a popular and widely applicable method to improve the effectiveness of quasi-Monte Carlo methods, cannot be employed in the context of path-dependent options when the underlying price process follows the Heston model. Consequently, we tailor quasi-Monte Carlo methods directly to the Heston model. The contributions of the paper are threefold: We firstly show how to apply quasi-Monte Carlo methods in the context of the Heston model and the SVJ model, secondly that quasi-Monte Carlo methods improve on Monte Carlo methods, and thirdly how to improve the effectiveness of quasi-Monte Carlo methods by using bridge constructions tailored to the Heston and SVJ models. Finally, we provide some extensions for computing greeks, barrier options, multidimensional and multi-asset pricing, and the 3=2 model.

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Bibliographic Info

Paper provided by Quantitative Finance Research Centre, University of Technology, Sydney in its series Research Paper Series with number 307.

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Length: 21
Date of creation: 01 Jun 2012
Date of revision:
Handle: RePEc:uts:rpaper:307

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Keywords: quasi-Monte Carlo methods; computational finance; stochastic volatility; path-dependent derivatives; bridge sampling; exact simulation;

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  1. Mark Broadie & Paul Glasserman, 1996. "Estimating Security Price Derivatives Using Simulation," Management Science, INFORMS, vol. 42(2), pages 269-285, February.
  2. Jan Baldeaux, 2012. "Exact Simulation Of The 3/2 Model," International Journal of Theoretical and Applied Finance (IJTAF), World Scientific Publishing Co. Pte. Ltd., vol. 15(05), pages 1250032-1-1.
  3. Lin Yuan & John Kalbfleisch, 2000. "On the Bessel Distribution and Related Problems," Annals of the Institute of Statistical Mathematics, Springer, vol. 52(3), pages 438-447, September.
  4. Paul Glasserman & Kyoung-Kuk Kim, 2011. "Gamma expansion of the Heston stochastic volatility model," Finance and Stochastics, Springer, vol. 15(2), pages 267-296, June.
  5. Fred Espen Benth & Martin Groth & Paul C. Kettler, 2006. "A Quasi-Monte Carlo Algorithm For The Normal Inverse Gaussian Distribution And Valuation Of Financial Derivatives," International Journal of Theoretical and Applied Finance (IJTAF), World Scientific Publishing Co. Pte. Ltd., vol. 9(06), pages 843-867.
  6. Alan L. Lewis, 2000. "Option Valuation under Stochastic Volatility," Option Valuation under Stochastic Volatility, Finance Press, number ovsv, July.
  7. 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 637-54, May-June.
  8. Pierre L'Ecuyer & Christiane Lemieux, 2000. "Variance Reduction via Lattice Rules," Management Science, INFORMS, vol. 46(9), pages 1214-1235, September.
  9. Jan Baldeaux & Dale Roberts, 2012. "Quasi-Monte Carlo methods for the Heston model," Papers 1202.3217, arXiv.org, revised May 2012.
  10. Andrey Itkin & Peter Carr, 2010. "Pricing swaps and options on quadratic variation under stochastic time change models—discrete observations case," Review of Derivatives Research, Springer, vol. 13(2), pages 141-176, July.
  11. Bates, David S, 1996. "Jumps and Stochastic Volatility: Exchange Rate Processes Implicit in Deutsche Mark Options," Review of Financial Studies, Society for Financial Studies, vol. 9(1), pages 69-107.
  12. Peter Carr & Jian Sun, 2007. "A new approach for option pricing under stochastic volatility," Review of Derivatives Research, Springer, vol. 10(2), pages 87-150, May.
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Cited by:
  1. Jan Baldeaux & Dale Roberts, 2012. "Quasi-Monte Carol Methods for the Heston Model," Research Paper Series 307, Quantitative Finance Research Centre, University of Technology, Sydney.
  2. Jan Baldeaux & Eckhard Platen, 2012. "Computing Functionals of Multidimensional Diffusions via Monte Carlo Methods," Papers 1204.1126, arXiv.org.

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