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Application of the Fast Gauss Transform to Option Pricing

Author

Listed:
  • Mark Broadie

    (Graduate School of Business, Columbia University, 3022 Broadway, New York, New York, 10027-6902)

  • Yusaku Yamamoto

    (Central Research Laboratory, Hitachi, Ltd., Tokyo, Japan)

Abstract

In many of the numerical methods for pricing American options based on the dynamic programming approach, the most computationally intensive part can be formulated as the summation of Gaussians. Though this operation usually requiresO(NN') work when there areN' summations to compute and the number of terms appearing in each summation isN, we can reduce the amount of work toO(N+N') by using a technique called the fast Gauss transform. In this paper, we apply this technique to the multinomial method and the stochastic mesh method, and show by numerical experiments how it can speed up these methods dramatically, both for the Black-Scholes model and Merton's lognormal jump-diffusion model. We also propose extensions of the fast Gauss transform method to models with non-Gaussian densities.

Suggested Citation

  • Mark Broadie & Yusaku Yamamoto, 2003. "Application of the Fast Gauss Transform to Option Pricing," Management Science, INFORMS, vol. 49(8), pages 1071-1088, August.
    • Handle: RePEc:inm:ormnsc:v:49:y:2003:i:8:p:1071-1088
    DOI: 10.1287/mnsc.49.8.1071.16405
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    References listed on IDEAS

    as
    1. Longstaff, Francis A & Schwartz, Eduardo S, 2001. "Valuing American Options by Simulation: A Simple Least-Squares Approach," Review of Financial Studies, Society for Financial Studies, vol. 14(1), pages 113-147.
    2. S. G. Kou, 2002. "A Jump-Diffusion Model for Option Pricing," Management Science, INFORMS, vol. 48(8), pages 1086-1101, August.
    3. Amin, Kaushik I, 1993. "Jump Diffusion Option Valuation in Discrete Time," Journal of Finance, American Finance Association, vol. 48(5), pages 1833-1863, December.
    4. Darrell Duffie & Jun Pan & Kenneth Singleton, 2000. "Transform Analysis and Asset Pricing for Affine Jump-Diffusions," Econometrica, Econometric Society, vol. 68(6), pages 1343-1376, November.
    5. Heston, Steven L, 1993. "A Closed-Form Solution for Options with Stochastic Volatility with Applications to Bond and Currency Options," Review of Financial Studies, Society for Financial Studies, vol. 6(2), pages 327-343.
    6. Steve Heston & Guofu Zhou, 2000. "On the Rate of Convergence of Discrete‐Time Contingent Claims," Mathematical Finance, Wiley Blackwell, vol. 10(1), pages 53-75, January.
    7. Broadie, M. & Glasserman, P., 1997. "A Sotchastic Mesh Method for Pricing High-Dimensional American Options," Papers 98-04, Columbia - Graduate School of Business.
    8. Jonathan Alford and Nick Webber, 2001. "Very High Order Lattice Methods for One Factor Models," Computing in Economics and Finance 2001 26, Society for Computational Economics.
    9. 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.
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    Cited by:

    1. Purba Banerjee & Vasudeva Murthy & Shashi Jain, 2024. "Method of Lines for Valuation and Sensitivities of Bermudan Options," Computational Economics, Springer;Society for Computational Economics, vol. 63(1), pages 245-270, January.
    2. Liming Feng & Vadim Linetsky, 2008. "Pricing Discretely Monitored Barrier Options And Defaultable Bonds In Lévy Process Models: A Fast Hilbert Transform Approach," Mathematical Finance, Wiley Blackwell, vol. 18(3), pages 337-384, July.
    3. Barty Kengy & Girardeau Pierre & Strugarek Cyrille & Roy Jean-Sébastien, 2008. "Application of kernel-based stochastic gradient algorithms to option pricing," Monte Carlo Methods and Applications, De Gruyter, vol. 14(2), pages 99-127, January.
    4. Carl Chiarella & Andrew Ziogas, 2009. "American Call Options Under Jump-Diffusion Processes - A Fourier Transform Approach," Applied Mathematical Finance, Taylor & Francis Journals, vol. 16(1), pages 37-79.
    5. Lingfei Li & Vadim Linetsky, 2013. "Optimal Stopping and Early Exercise: An Eigenfunction Expansion Approach," Operations Research, INFORMS, vol. 61(3), pages 625-643, June.
    6. Jérôme Detemple, 2014. "Optimal Exercise for Derivative Securities," Annual Review of Financial Economics, Annual Reviews, vol. 6(1), pages 459-487, December.
    7. Fang, Fang & Oosterlee, Kees, 2008. "A Novel Pricing Method For European Options Based On Fourier-Cosine Series Expansions," MPRA Paper 9319, University Library of Munich, Germany.
    8. Volodymyr Babich, 2006. "Vulnerable options in supply chains: Effects of supplier competition," Naval Research Logistics (NRL), John Wiley & Sons, vol. 53(7), pages 656-673, October.
    9. Tomohisa Yamakami & Yuki Takeuchi, 2022. "Pricing Bermudan Swaption under Two Factor Hull-White Model with Fast Gauss Transform," Papers 2212.08250, arXiv.org.
    10. Grzelak, Lech & Oosterlee, Kees, 2009. "On The Heston Model with Stochastic Interest Rates," MPRA Paper 20620, University Library of Munich, Germany, revised 18 Jan 2010.
    11. Sergei Levendorskiĭ, 2022. "Operators and Boundary Problems in Finance, Economics and Insurance: Peculiarities, Efficient Methods and Outstanding Problems," Mathematics, MDPI, vol. 10(7), pages 1-36, March.
    12. Reza Doostaki & Mohammad Mehdi Hosseini, 2022. "Option Pricing by the Legendre Wavelets Method," Computational Economics, Springer;Society for Computational Economics, vol. 59(2), pages 749-773, February.
    13. Cyrus Ramezani & Yong Zeng, 2007. "Maximum likelihood estimation of the double exponential jump-diffusion process," Annals of Finance, Springer, vol. 3(4), pages 487-507, October.
    14. A. Cassagnes & Y. Chen & H. Ohashi, 2014. "Heterogeneous Computation Of Rainbow Option Prices Using Fourier Cosine Series Expansion Under A Mixed Cpu–Gpu Computation Framework," Intelligent Systems in Accounting, Finance and Management, John Wiley & Sons, Ltd., vol. 21(2), pages 91-104, April.
    15. Hyong-Chol O & Mun-Chol KiM, 2013. "The Pricing of Multiple-Expiry Exotics," Papers 1302.3319, arXiv.org, revised Aug 2013.
    16. M. Broadie & Y. Yamamoto, 2005. "A Double-Exponential Fast Gauss Transform Algorithm for Pricing Discrete Path-Dependent Options," Operations Research, INFORMS, vol. 53(5), pages 764-779, October.
    17. Jing-Sheng Song & Paul Zipkin, 2013. "Supply Streams," Manufacturing & Service Operations Management, INFORMS, vol. 15(3), pages 444-457, July.
    18. Tat Lung & Chan, 2019. "An SFP--FCC Method for Pricing and Hedging Early-exercise Options under L\'evy Processes," Papers 1909.07319, arXiv.org.
    19. 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.
    20. Mark Broadie & Jerome B. Detemple, 2004. "ANNIVERSARY ARTICLE: Option Pricing: Valuation Models and Applications," Management Science, INFORMS, vol. 50(9), pages 1145-1177, September.
    21. Liming Feng & Vadim Linetsky, 2008. "Pricing Options in Jump-Diffusion Models: An Extrapolation Approach," Operations Research, INFORMS, vol. 56(2), pages 304-325, April.

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