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First-Order Asymptotics Of Path-Dependent Derivatives In Multiscale Stochastic Volatility Environment

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  • YURI F. SAPORITO

    (Escola de Matemática Aplicada (EMAp), Fundação Getulio Vargas, Praia de Botafogo, 190, Rio de Janeiro, RJ 22250-900, Brazil)

Abstract

In this paper, we extend the first-order asymptotics analysis of Fouque et al. to general path-dependent financial derivatives using Dupire’s functional Itô calculus. The main conclusion is that the market group parameters calibrated to vanilla options can be used to price to the same order exotic, path-dependent derivatives as well. Under general conditions, the first-order condition is represented by a conditional expectation that could be numerically evaluated. Moreover, if the path-dependence is not too severe, we are able to find path-dependent closed-form solutions equivalent to the first-order approximation of path-independent options derived in Fouque et al. Additionally, we exemplify the results with Asian options and options on quadratic variation.

Suggested Citation

  • 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.
    • Handle: RePEc:wsi:ijtafx:v:21:y:2018:i:03:n:s0219024918500243
    DOI: 10.1142/S0219024918500243
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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.
    2. Heston, Steven L, 1993. "A Closed-Form Solution for Options with Stochastic Volatility with Applications to Bond and Currency Options," The Review of Financial Studies, Society for Financial Studies, vol. 6(2), pages 327-343.
    3. Jean-Pierre Fouque & Matthew Lorig & Ronnie Sircar, 2016. "Second order multiscale stochastic volatility asymptotics: stochastic terminal layer analysis and calibration," Finance and Stochastics, Springer, vol. 20(3), pages 543-588, July.
    4. 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.
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    Cited by:

    1. Ofelia Bonesini & Antoine Jacquier & Chloe Lacombe, 2020. "A theoretical analysis of Guyon's toy volatility model," Papers 2001.05248, arXiv.org, revised Nov 2022.
    2. Yuri F. Saporito, 2020. "Pricing Path-Dependent Derivatives under Multiscale Stochastic Volatility Models: a Malliavin Representation," Papers 2005.04297, arXiv.org.

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