IDEAS home Printed from https://ideas.repec.org/a/spr/annopr/v281y2019i1d10.1007_s10479-018-3001-1.html
   My bibliography  Save this article

On the multidimensional Black–Scholes partial differential equation

Author

Listed:
  • 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
    as

    Download full text from publisher

    File URL: http://link.springer.com/10.1007/s10479-018-3001-1
    File Function: Abstract
    Download Restriction: Access to the full text of the articles in this series is restricted.
    File URL: https://libkey.io/10.1007/s10479-018-3001-1?utm_source=ideas
    LibKey link: if access is restricted and if your library uses this service, LibKey will redirect you to where you can use your library subscription to access this item
    ---><---

    As the access to this document is restricted, you may want to search for a different version of it.

    References listed on IDEAS

    as
    1. Tristan Guillaume, 2017. "Computation of the Quadrivariate and Pentavariate normal cumulative distribution functions," Post-Print hal-02980368, HAL.
    2. Tristan Guillaume, 2010. "Step double barrier options," Post-Print hal-00924266, HAL.
    3. 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.
    4. 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.
    5. 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.
    6. Tristan Guillaume, 2008. "Making the best of best-of," Review of Derivatives Research, Springer, vol. 11(1), pages 1-39, March.
    7. Tristan Guillaume, 2008. "Making the best of best-of," Post-Print hal-00924256, HAL.
    8. 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.
    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.
    Full references (including those not matched with items on IDEAS)

    Citations

    Citations are extracted by the CitEc Project, subscribe to its RSS feed for this item.
    as


    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.

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    1. S H Martzoukos, 2009. "Real R&D options and optimal activation of two-dimensional random controls," Journal of the Operational Research Society, Palgrave Macmillan;The OR Society, vol. 60(6), pages 843-858, June.
    2. repec:dau:papers:123456789/5374 is not listed on IDEAS
    3. Mark Broadie & Jérôme Detemple, 1996. "Recent Advances in Numerical Methods for Pricing Derivative Securities," CIRANO Working Papers 96s-17, CIRANO.
    4. Sarah Bryant & Spiros Martzoukos, 1999. "Multi-currency options and financial institutions' hedging: Correlation does matter," International Advances in Economic Research, Springer;International Atlantic Economic Society, vol. 5(4), pages 478-488, November.
    5. Mark Broadie & Jerome B. Detemple, 2004. "ANNIVERSARY ARTICLE: Option Pricing: Valuation Models and Applications," Management Science, INFORMS, vol. 50(9), pages 1145-1177, September.
    6. Tristan Guillaume, 2008. "Making the best of best-of," Review of Derivatives Research, Springer, vol. 11(1), pages 1-39, March.
    7. Bjork, Tomas, 2009. "Arbitrage Theory in Continuous Time," OUP Catalogue, Oxford University Press, edition 3, number 9780199574742.
    8. Jian Guo & Saizhuo Wang & Lionel M. Ni & Heung-Yeung Shum, 2022. "Quant 4.0: Engineering Quantitative Investment with Automated, Explainable and Knowledge-driven Artificial Intelligence," Papers 2301.04020, arXiv.org.
    9. Lim, Terence & Lo, Andrew W. & Merton, Robert C. & Scholes, Myron S., 2006. "The Derivatives Sourcebook," Foundations and Trends(R) in Finance, now publishers, vol. 1(5–6), pages 365-572, April.
    10. repec:uts:finphd:40 is not listed on IDEAS
    11. Dominique Guegan & Jing Zhang, 2009. "Pricing bivariate option under GARCH-GH model with dynamic copula: application for Chinese market," PSE-Ecole d'économie de Paris (Postprint) halshs-00368336, HAL.
    12. Zhu, Ke & Ling, Shiqing, 2015. "Model-based pricing for financial derivatives," Journal of Econometrics, Elsevier, vol. 187(2), pages 447-457.
    13. Shin-Yun Wang & Ming-Che Chuang & Shih-Kuei Lin & So-De Shyu, 2021. "Option pricing under stock market cycles with jump risks: evidence from the S&P 500 index," Review of Quantitative Finance and Accounting, Springer, vol. 56(1), pages 25-51, January.
    14. Detlefsen, Kai & Härdle, Wolfgang Karl & Moro, Rouslan A., 2007. "Empirical pricing kernels and investor preferences," SFB 649 Discussion Papers 2007-017, Humboldt University Berlin, Collaborative Research Center 649: Economic Risk.
    15. Battulga Gankhuu, 2021. "Equity-Linked Life Insurances on Maximum of Several Assets," Papers 2111.04038, arXiv.org, revised Sep 2024.
    16. Dominique Guegan & Jing Zang, 2009. "Pricing bivariate option under GARCH-GH model with dynamic copula: application for Chinese market," The European Journal of Finance, Taylor & Francis Journals, vol. 15(7-8), pages 777-795.
    17. Luiz Vitiello & Ivonia Rebelo, 2015. "A note on the pricing of multivariate contingent claims under a transformed-gamma distribution," Review of Derivatives Research, Springer, vol. 18(3), pages 291-300, October.
    18. Michael C. Fu & Bingqing Li & Guozhen Li & Rongwen Wu, 2017. "Option Pricing for a Jump-Diffusion Model with General Discrete Jump-Size Distributions," Management Science, INFORMS, vol. 63(11), pages 3961-3977, November.
    19. Melenberg, B. & Werker, B.J.M., 1996. "On the Pricing of Options in Incomplete Markets," Discussion Paper 1996-19, Tilburg University, Center for Economic Research.
    20. El-Khatib, Youssef & Goutte, Stephane & Makumbe, Zororo S. & Vives, Josep, 2023. "A hybrid stochastic volatility model in a Lévy market," International Review of Economics & Finance, Elsevier, vol. 85(C), pages 220-235.
    21. Brisset, Nicolas, 2017. "On Performativity: Option Theory And The Resistance Of Financial Phenomena," Journal of the History of Economic Thought, Cambridge University Press, vol. 39(4), pages 549-569, December.
    22. Tavin, Bertrand, 2015. "Detection of arbitrage in a market with multi-asset derivatives and known risk-neutral marginals," Journal of Banking & Finance, Elsevier, vol. 53(C), pages 158-178.

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:spr:annopr:v:281:y:2019:i:1:d:10.1007_s10479-018-3001-1. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: Sonal Shukla or Springer Nature Abstracting and Indexing (email available below). General contact details of provider: http://www.springer.com .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.