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Singular Perturbation Techniques Applied To Multiasset Option Pricing

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  • Peter W. Duck
  • Chao Yang
  • David P. Newton
  • Martin Widdicks

Abstract

It is well known that option valuation problems with multiple‐state variables are often problematic to solve. When valuing options using lattice‐type techniques such as finite‐difference methods, the curse of dimensionality ensures that additional‐state variables lead to exponential increases in computational effort. Monte Carlo methods are immune from this curse but, despite advances, require a great deal of adaptation to treat early exercise features. Here the multiunderlying asset Black–Scholes problem, including early exercise, is studied using the tools of singular perturbation analysis. This considerably simplifies the pricing problem by decomposing the multi‐dimensional problem into a series of lower‐dimensional problems that are far simpler and faster to solve than the full, high‐dimensional problem. This paper explains how to apply these singular perturbation techniques and explores the significant efficiency improvement from such an approach.

Suggested Citation

  • Peter W. Duck & Chao Yang & David P. Newton & Martin Widdicks, 2009. "Singular Perturbation Techniques Applied To Multiasset Option Pricing," Mathematical Finance, Wiley Blackwell, vol. 19(3), pages 457-486, July.
    • Handle: RePEc:bla:mathfi:v:19:y:2009:i:3:p:457-486
    DOI: 10.1111/j.1467-9965.2009.00373.x
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    References listed on IDEAS

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    1. Leif Andersen & Mark Broadie, 2004. "Primal-Dual Simulation Algorithm for Pricing Multidimensional American Options," Management Science, INFORMS, vol. 50(9), pages 1222-1234, September.
    2. 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.
    3. Stulz, ReneM., 1982. "Options on the minimum or the maximum of two risky assets : Analysis and applications," Journal of Financial Economics, Elsevier, vol. 10(2), pages 161-185, July.
    4. Mark Broadie & Jerome B. Detemple, 2004. "ANNIVERSARY ARTICLE: Option Pricing: Valuation Models and Applications," Management Science, INFORMS, vol. 50(9), pages 1145-1177, September.
    5. Johnson, Herb, 1987. "Options on the Maximum or the Minimum of Several Assets," Journal of Financial and Quantitative Analysis, Cambridge University Press, vol. 22(3), pages 277-283, September.
    6. Boyle, Phelim P & Evnine, Jeremy & Gibbs, Stephen, 1989. "Numerical Evaluation of Multivariate Contingent Claims," Review of Financial Studies, Society for Financial Studies, vol. 2(2), pages 241-250.
    7. D. Andricopoulos, Ari & Widdicks, Martin & Newton, David P. & Duck, Peter W., 2007. "Extending quadrature methods to value multi-asset and complex path dependent options," Journal of Financial Economics, Elsevier, vol. 83(2), pages 471-499, February.
    8. Martin Widdicks & Peter W. Duck & Ari D. Andricopoulos & David P. Newton, 2005. "The Black‐Scholes Equation Revisited: Asymptotic Expansions And Singular Perturbations," Mathematical Finance, Wiley Blackwell, vol. 15(2), pages 373-391, April.
    9. Mark Broadie & Jérôme Detemple, 1997. "The Valuation of American Options on Multiple Assets," Mathematical Finance, Wiley Blackwell, vol. 7(3), pages 241-286, July.
    10. Chen, Ren-Raw & Chung, San-Lin & Yang, Tyler T., 2002. "Option Pricing in a Multi-Asset, Complete Market Economy," Journal of Financial and Quantitative Analysis, Cambridge University Press, vol. 37(4), pages 649-666, December.
    11. Boyle, Phelim P., 1988. "A Lattice Framework for Option Pricing with Two State Variables," Journal of Financial and Quantitative Analysis, Cambridge University Press, vol. 23(1), pages 1-12, March.
    12. Stapleton, Richard C & Subrahmanyam, Marti G, 1984. "The Valuation of Multivariate Contingent Claims in Discrete Time Models," Journal of Finance, American Finance Association, vol. 39(1), pages 207-228, March.
    13. Andricopoulos, Ari D. & Widdicks, Martin & Duck, Peter W. & Newton, David P., 2003. "Universal option valuation using quadrature methods," Journal of Financial Economics, Elsevier, vol. 67(3), pages 447-471, March.
    14. Martin B. Haugh & Leonid Kogan, 2004. "Pricing American Options: A Duality Approach," Operations Research, INFORMS, vol. 52(2), pages 258-270, April.
    15. Boyle, Phelim P. & Tse, Y. K., 1990. "An Algorithm for Computing Values of Options on the Maximum or Minimum of Several Assets," Journal of Financial and Quantitative Analysis, Cambridge University Press, vol. 25(2), pages 215-227, June.
    16. Barraquand, Jérôme & Martineau, Didier, 1995. "Numerical Valuation of High Dimensional Multivariate American Securities," Journal of Financial and Quantitative Analysis, Cambridge University Press, vol. 30(3), pages 383-405, September.
    17. Margrabe, William, 1978. "The Value of an Option to Exchange One Asset for Another," Journal of Finance, American Finance Association, vol. 33(1), pages 177-186, March.
    18. Jèôme Barraquand, 1995. "Numerical Valuation of High Dimensional Multivariate European Securities," Management Science, INFORMS, vol. 41(12), pages 1882-1891, December.
    19. Sam Howison, 2005. "Matched asymptotic expansions in financial engineering," OFRC Working Papers Series 2005mf01, Oxford Financial Research Centre.
    20. Cox, John C. & Ross, Stephen A. & Rubinstein, Mark, 1979. "Option pricing: A simplified approach," Journal of Financial Economics, Elsevier, vol. 7(3), pages 229-263, September.
    21. L. C. G. Rogers, 2002. "Monte Carlo valuation of American options," Mathematical Finance, Wiley Blackwell, vol. 12(3), pages 271-286, July.
    22. Ho, Teng-Suan & Stapleton, Richard C & Subrahmanyam, Marti G, 1995. "Multivariate Binomial Approximations for Asset Prices with Nonstationary Variance and Covariance Characteristics," Review of Financial Studies, Society for Financial Studies, vol. 8(4), pages 1125-1152.
    23. 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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    3. Adi Ben-Meir & Jeremy Schiff, 2012. "The Variance of Standard Option Returns," Papers 1204.3452, arXiv.org.

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