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Malliavin differentiability of the Heston volatility and applications to option pricing

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  • Alos, Elisa
  • Ewald, Christian-Oliver

Abstract

We prove that the Heston volatility is Malliavin differentiable under the classical Novikov condition and give an explicit expression for the derivative. This result guarantees the applicability of Malliavin calculus in the framework of the Heston stochastic volatility model. Furthermore we derive conditions on the parameters which assure the existence of the second Malliavin derivative of the Heston volatility. This allows us to apply recent results of the first author [3] in order to derive approximate option pricing formulas in the context of the Heston model. Numerical results are given.

Suggested Citation

  • Alos, Elisa & Ewald, Christian-Oliver, 2007. "Malliavin differentiability of the Heston volatility and applications to option pricing," MPRA Paper 3237, University Library of Munich, Germany.
    • Handle: RePEc:pra:mprapa:3237
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    References listed on IDEAS

    as
    1. Elisa Alòs & Jorge A. León & Josep Vives, 2006. "On the short-time behavior of the implied volatility for jump-diffusion models with stochastic volatility," Economics Working Papers 968, Department of Economics and Business, Universitat Pompeu Fabra.
    2. Jérôme Detemple & René Garcia & Marcel Rindisbacher, 2005. "Representation formulas for Malliavin derivatives of diffusion processes," Finance and Stochastics, Springer, vol. 9(3), pages 349-367, July.
    3. Elisa Alòs, 2006. "A generalization of the Hull and White formula with applications to option pricing approximation," Finance and Stochastics, Springer, vol. 10(3), pages 353-365, September.
    4. Hélyette Geman & Marc Yor, 1993. "Bessel Processes, Asian Options, And Perpetuities," Mathematical Finance, Wiley Blackwell, vol. 3(4), pages 349-375, October.
    5. 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.
    6. Hull, John C & White, Alan D, 1987. "The Pricing of Options on Assets with Stochastic Volatilities," Journal of Finance, American Finance Association, vol. 42(2), pages 281-300, June.
    Full references (including those not matched with items on IDEAS)

    Citations

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    Cited by:

    1. Christian-Oliver Ewald & Zhaojun Yang, 2008. "Utility based pricing and exercising of real options under geometric mean reversion and risk aversion toward idiosyncratic risk," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 68(1), pages 97-123, August.
    2. Elisa Alòs & Jorge A. León, 2016. "On the short-maturity behaviour of the implied volatility skew for random strike options and applications to option pricing approximation," Quantitative Finance, Taylor & Francis Journals, vol. 16(1), pages 31-42, January.
    3. Ben Hambly & Nikolaos Kolliopoulos, 2018. "Fast mean-reversion asymptotics for large portfolios of stochastic volatility models," Papers 1811.08808, arXiv.org, revised Feb 2020.
    4. Taiga Saito & Akihiko Takahashi, 2021. "Portfolio Optimization with Choice of a Probability Measure," CIRJE F-Series CIRJE-F-1165, CIRJE, Faculty of Economics, University of Tokyo.
    5. Annalena Mickel & Andreas Neuenkirch, 2021. "The Weak Convergence Rate of Two Semi-Exact Discretization Schemes for the Heston Model," Risks, MDPI, vol. 9(1), pages 1-38, January.
    6. Elisa Alòs & Yan Yang, 2014. "A closed-form option pricing approximation formula for a fractional Heston model," Economics Working Papers 1446, Department of Economics and Business, Universitat Pompeu Fabra.
    7. Elisa Alòs, 2012. "A decomposition formula for option prices in the Heston model and applications to option pricing approximation," Finance and Stochastics, Springer, vol. 16(3), pages 403-422, July.
    8. repec:spr:compst:v:68:y:2008:i:1:p:97-123 is not listed on IDEAS
    9. Ewald, Christian-Oliver & Wang, Wen-Kai, 2010. "Irreversible investment with Cox-Ingersoll-Ross type mean reversion," Mathematical Social Sciences, Elsevier, vol. 59(3), pages 314-318, May.
    10. Elisa Al`os & Michael Coulon, 2018. "On the optimal choice of strike conventions in exchange option pricing," Papers 1807.05396, arXiv.org.
    11. Hyungbin Park, 2021. "Influence of risk tolerance on long-term investments: A Malliavin calculus approach," Papers 2104.00911, arXiv.org.
    12. Almeida, Caio & Vicente, José, 2009. "Identifying volatility risk premia from fixed income Asian options," Journal of Banking & Finance, Elsevier, vol. 33(4), pages 652-661, April.
    13. Elisa Alòs & Thorsten Rheinländer, 2015. "Pricing and hedging Margrabe options with stochastic volatilities," Economics Working Papers 1475, Department of Economics and Business, Universitat Pompeu Fabra, revised Feb 2017.
    14. Bilgi Yilmaz, 2018. "Computation of option greeks under hybrid stochastic volatility models via Malliavin calculus," Papers 1806.06061, arXiv.org.
    15. Romain Bompis & Emmanuel Gobet, 2012. "Asymptotic and non asymptotic approximations for option valuation," Post-Print hal-00720650, HAL.
    16. Tahmasebi, M., 2014. "Smooth density for the solution of scalar SDEs with locally Lipschitz coefficients under Hörmander condition," Statistics & Probability Letters, Elsevier, vol. 85(C), pages 51-62.
    17. Maxim Bichuch & Stephan Sturm, 2014. "Portfolio optimization under convex incentive schemes," Finance and Stochastics, Springer, vol. 18(4), pages 873-915, October.

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    More about this item

    Keywords

    Malliavin calculus; stochastic volatility models; Heston model; Cox- Ingersoll-Ross process; Hull and White formula; Option pricing;
    All these keywords.

    JEL classification:

    • G13 - Financial Economics - - General Financial Markets - - - Contingent Pricing; Futures Pricing
    • G11 - Financial Economics - - General Financial Markets - - - Portfolio Choice; Investment Decisions
    • C02 - Mathematical and Quantitative Methods - - General - - - Mathematical Economics

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