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On the multidimensional Black–Scholes partial differential equation

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  • Tristan Guillaume

    (Université de Cergy-Pontoise)

Abstract

In this article, two general results are provided about the multidimensional Black–Scholes partial differential equation: its fundamental solution is derived, and it is shown how to turn it into the standard heat equation in whatever dimension. A fundamental connection is established between the multivariate normal distribution and the linear second order partial differential operator of parabolic type. These results allow to compute new closed form formulae for the valuation of multiasset options, with possible boundary crossing conditions, thus partially alleviating the « curse of dimensionality », at least in moderate dimension.

Suggested Citation

  • Tristan Guillaume, 2019. "On the multidimensional Black–Scholes partial differential equation," Annals of Operations Research, Springer, vol. 281(1), pages 229-251, October.
    • Handle: RePEc:spr:annopr:v:281:y:2019:i:1:d:10.1007_s10479-018-3001-1
    DOI: 10.1007/s10479-018-3001-1
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    References listed on IDEAS

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    1. Tristan Guillaume, 2008. "Making the best of best-of," Post-Print hal-00924256, HAL.
    2. Tristan Guillaume, 2017. "Computation of the Quadrivariate and Pentavariate normal cumulative distribution functions," Post-Print hal-02980368, HAL.
    3. Tristan Guillaume, 2010. "Step double barrier options," Post-Print hal-00924266, HAL.
    4. Harrison, J. Michael & Pliska, Stanley R., 1981. "Martingales and stochastic integrals in the theory of continuous trading," Stochastic Processes and their Applications, Elsevier, vol. 11(3), pages 215-260, August.
    5. J. Michael Harrison & Stanley R. Pliska, 1981. "Martingales and Stochastic Integrals in the Theory of Continous Trading," Discussion Papers 454, Northwestern University, Center for Mathematical Studies in Economics and Management Science.
    6. 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.
    7. 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.
    8. Tristan Guillaume, 2008. "Making the best of best-of," Review of Derivatives Research, Springer, vol. 11(1), pages 1-39, March.
    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. Mauricio Contreras & Alejandro Llanquihu'en & Marcelo Villena, 2015. "On the Solution of the Multi-asset Black-Scholes model: Correlations, Eigenvalues and Geometry," Papers 1510.02768, arXiv.org.
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    Cited by:

    1. Tianchen Zhao & Chuhao Sun & Asaf Cohen & James Stokes & Shravan Veerapaneni, 2022. "Quantum-inspired variational algorithms for partial differential equations: Application to financial derivative pricing," Papers 2207.10838, arXiv.org.
    2. Shih-Hsien Tseng & Tien Son Nguyen & Ruei-Ci Wang, 2021. "The Lie Algebraic Approach for Determining Pricing for Trade Account Options," Mathematics, MDPI, vol. 9(3), pages 1-9, January.
    3. Chaeyoung Lee & Jisang Lyu & Eunchae Park & Wonjin Lee & Sangkwon Kim & Darae Jeong & Junseok Kim, 2020. "Super-Fast Computation for the Three-Asset Equity-Linked Securities Using the Finite Difference Method," Mathematics, MDPI, vol. 8(3), pages 1-13, February.
    4. Darko Mitrovic, 2023. "Pre-electoral coalition agreement from the Black-Scholes point of view," Papers 2310.16424, arXiv.org, revised Feb 2024.

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