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Tensor train representations of Greeks for Fourier-based pricing of multi-asset options

Author

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  • Rihito Sakurai
  • Koichi Miyamoto
  • Tsuyoshi Okubo

Abstract

Efficient computation of Greeks for multi-asset options remains a key challenge in quantitative finance. While Monte Carlo (MC) simulation is widely used, it suffers from the large sample complexity for high accuracy. We propose a framework to compute Greeks in a single evaluation of a tensor train (TT), which is obtained by compressing the Fourier transform (FT)-based pricing function via TT learning using tensor cross interpolation. Based on this TT representation, we introduce two approaches to compute Greeks: a numerical differentiation (ND) approach that applies a numerical differential operator to one tensor core and an analytical (AN) approach that constructs the TT of closed-form differentiation expressions of FT-based pricing. Numerical experiments on a five-asset min-call option in the Black-Sholes model show significant speed-ups of up to about $10^{5} \times$ over MC while maintaining comparable accuracy. The ND approach matches or exceeds the accuracy of the AN approach and requires lower computational complexity for constructing the TT representation, making it the preferred choice.

Suggested Citation

  • Rihito Sakurai & Koichi Miyamoto & Tsuyoshi Okubo, 2025. "Tensor train representations of Greeks for Fourier-based pricing of multi-asset options," Papers 2507.08482, arXiv.org.
    • Handle: RePEc:arx:papers:2507.08482
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    References listed on IDEAS

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    1. Rihito Sakurai & Haruto Takahashi & Koichi Miyamoto, 2024. "Learning parameter dependence for Fourier-based option pricing with tensor trains," Papers 2405.00701, arXiv.org, revised Apr 2025.
    2. Robert C. Merton, 2005. "Theory of rational option pricing," World Scientific Book Chapters, in: Sudipto Bhattacharya & George M Constantinides (ed.), Theory Of Valuation, chapter 8, pages 229-288, World Scientific Publishing Co. Pte. Ltd..
    3. Rihito Sakurai & Haruto Takahashi & Koichi Miyamoto, 2025. "Learning Parameter Dependence for Fourier-Based Option Pricing with Tensor Trains," Mathematics, MDPI, vol. 13(11), pages 1-16, May.
    4. Ernst Eberlein & Kathrin Glau & Antonis Papapantoleon, 2010. "Analysis of Fourier Transform Valuation Formulas and Applications," Applied Mathematical Finance, Taylor & Francis Journals, vol. 17(3), pages 211-240.
    5. Madan, Dilip B & Seneta, Eugene, 1990. "The Variance Gamma (V.G.) Model for Share Market Returns," The Journal of Business, University of Chicago Press, vol. 63(4), pages 511-524, October.
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