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A note on the Malliavin differentiability of the Heston volatility

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Abstract

We show that the Heston volatility or equivalently the Cox-Ingersoll-Ross process is Malliavin differentiable and give an explicit expression for the derivative. This result assures the applicability of Malliavin calculus in the framework of the Heston stochastic volatility model and the Cox-Ingersoll-Ross model for interest rates.

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  • Elisa Alòs & Christian-Olivier Ewald, 2005. "A note on the Malliavin differentiability of the Heston volatility," Economics Working Papers 880, Department of Economics and Business, Universitat Pompeu Fabra.
    • Handle: RePEc:upf:upfgen:880
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    1. John C. Cox & Jonathan E. Ingersoll Jr. & Stephen A. Ross, 2005. "A Theory Of The Term Structure Of Interest Rates," World Scientific Book Chapters, in: Sudipto Bhattacharya & George M Constantinides (ed.), Theory Of Valuation, chapter 5, pages 129-164, World Scientific Publishing Co. Pte. Ltd..
    2. Christian-Oliver Ewald & Aihua Zhang, 2006. "A new technique for calibrating stochastic volatility models: the Malliavin gradient method," Quantitative Finance, Taylor & Francis Journals, vol. 6(2), pages 147-158.
    3. 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.
    4. Christian-Oliver Ewald, 2005. "Local volatility in the Heston model: a Malliavin calculus approach," International Journal of Stochastic Analysis, Hindawi, vol. 2005, pages 1-16, January.
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    7. Christian-Oliver Ewald, 2005. "Optimal Logarithmic Utility And Optimal Portfolios For An Insider In A Stochastic Volatility Market," International Journal of Theoretical and Applied Finance (IJTAF), World Scientific Publishing Co. Pte. Ltd., vol. 8(03), pages 301-319.
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    11. 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.
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    14. Eric Benhamou, 2003. "Optimal Malliavin Weighting Function for the Computation of the Greeks," Mathematical Finance, Wiley Blackwell, vol. 13(1), pages 37-53, January.
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    Cited by:

    1. Ben Hambly & Nikolaos Kolliopoulos, 2018. "Fast mean-reversion asymptotics for large portfolios of stochastic volatility models," Papers 1811.08808, arXiv.org, revised Feb 2020.
    2. S. Kuchuk-Iatsenko & Y. Mishura & Y. Munchak, 2016. "Application of Malliavin calculus to exact and approximate option pricing under stochastic volatility," Papers 1608.00230, arXiv.org.

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    More about this item

    Keywords

    Malliavin calculus; stochastic volatility models; Heston model; Cox-Ingersoll-Ross process;
    All these keywords.

    JEL classification:

    • G12 - Financial Economics - - General Financial Markets - - - Asset Pricing; Trading Volume; Bond Interest Rates
    • G19 - Financial Economics - - General Financial Markets - - - Other
    • C19 - Mathematical and Quantitative Methods - - Econometric and Statistical Methods and Methodology: General - - - Other
    • E43 - Macroeconomics and Monetary Economics - - Money and Interest Rates - - - Interest Rates: Determination, Term Structure, and Effects

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