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Functional Ito Calculus, Path-dependence and the Computation of Greeks

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  • Samy Jazaerli
  • Yuri F. Saporito

Abstract

Dupire's functional It\^o calculus provides an alternative approach to the classical Malliavin calculus for the computation of sensitivities, also called Greeks, of path-dependent derivatives prices. In this paper, we introduce a measure of path-dependence of functionals within the functional It\^o calculus framework. Namely, we consider the Lie bracket of the space and time functional derivatives, which we use to classify functionals accordingly to their degree of path-dependence. We then revisit the problem of efficient numerical computation of Greeks for path-dependent derivatives using integration by parts techniques. Special attention is paid to path-dependent functionals with zero Lie bracket, called locally weakly path-dependent functionals in our classification. Hence, we derive the weighted-expectation formulas for their Greeks. In the more general case of fully path-dependent functionals, we show that, equipped with the functional It\^o calculus, we are able to analyze the effect of the Lie bracket on the computation of Greeks. Moreover, we are also able to consider the more general dynamics of path-dependent volatility. These were not achieved using Malliavin calculus.

Suggested Citation

  • Samy Jazaerli & Yuri F. Saporito, 2013. "Functional Ito Calculus, Path-dependence and the Computation of Greeks," Papers 1311.3881, arXiv.org, revised Jun 2018.
  • Handle: RePEc:arx:papers:1311.3881
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    References listed on IDEAS

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    1. Eric Fournié & Jean-Michel Lasry & Pierre-Louis Lions & Jérôme Lebuchoux & Nizar Touzi, 1999. "Applications of Malliavin calculus to Monte Carlo methods in finance," Finance and Stochastics, Springer, vol. 3(4), pages 391-412.
    2. Eric Fournié & Jean-Michel Lasry & Pierre-Louis Lions & Jérôme Lebuchoux, 2001. "Applications of Malliavin calculus to Monte-Carlo methods in finance. II," Finance and Stochastics, Springer, vol. 5(2), pages 201-236.
    3. Emmanuel Gobet, 2004. "Revisiting the Greeks for European and American Options," World Scientific Book Chapters, in: Jiro Akahori & Shigeyoshi Ogawa & Shinzo Watanabe (ed.), Stochastic Processes And Applications To Mathematical Finance, chapter 3, pages 53-71, World Scientific Publishing Co. Pte. Ltd..
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    Cited by:

    1. Yuri F. Saporito & Zhaoyu Zhang, 2020. "PDGM: a Neural Network Approach to Solve Path-Dependent Partial Differential Equations," Papers 2003.02035, arXiv.org, revised Apr 2020.
    2. Yuri F. Saporito, 2017. "First-Order Asymptotics of Path-Dependent Derivatives in Multiscale Stochastic Volatility Environment," Papers 1712.07320, arXiv.org.

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