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Pricing Path-Dependent Derivatives under Multiscale Stochastic Volatility Models: a Malliavin Representation

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  • Yuri F. Saporito

Abstract

In this paper we derive a efficient Monte Carlo approximation for the price of path-dependent derivatives under the multiscale stochastic volatility models of Fouque \textit{et al}. Using the formulation of this pricing problem under the functional It\^o calculus framework and making use of Greek formulas from Malliavin calculus, we derive a representation for the first-order approximation of the price of path-dependent derivatives in the form $\mathbb{E}[\mbox{payoff} \times \mbox{weight}]$. The weight is known in closed form and depends only on the group market parameters arising from the calibration of the multiscale stochastic volatility to the market's implied volatility. Moreover, only simulations of the Black-Scholes model is required. We exemplify the method for a couple path-dependent derivatives.

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  • Yuri F. Saporito, 2020. "Pricing Path-Dependent Derivatives under Multiscale Stochastic Volatility Models: a Malliavin Representation," Papers 2005.04297, arXiv.org.
    • Handle: RePEc:arx:papers:2005.04297
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    References listed on IDEAS

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    1. Fouque,Jean-Pierre & Papanicolaou,George & Sircar,Ronnie & Sølna,Knut, 2011. "Multiscale Stochastic Volatility for Equity, Interest Rate, and Credit Derivatives," Cambridge Books, Cambridge University Press, number 9780521843584, October.
    2. Davis, Mark H.A. & Johansson, Martin P., 2006. "Malliavin Monte Carlo Greeks for jump diffusions," Stochastic Processes and their Applications, Elsevier, vol. 116(1), pages 101-129, January.
    3. 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.
    4. Yuri F. Saporito, 2018. "First-Order Asymptotics Of Path-Dependent Derivatives In Multiscale Stochastic Volatility Environment," International Journal of Theoretical and Applied Finance (IJTAF), World Scientific Publishing Co. Pte. Ltd., vol. 21(03), pages 1-22, May.
    5. 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.
    6. Jazaerli, Samy & F. Saporito, Yuri, 2017. "Functional Itô calculus, path-dependence and the computation of Greeks," Stochastic Processes and their Applications, Elsevier, vol. 127(12), pages 3997-4028.
    7. 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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    Cited by:

    1. Anselm Hudde & Ludger Rüschendorf, 2023. "European and Asian Greeks for Exponential Lévy Processes," Methodology and Computing in Applied Probability, Springer, vol. 25(1), pages 1-24, March.

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