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Expansion formulas for European options in a local volatility model

Author

Listed:
  • Eric Benhamou

    (Pricing Partners - Pricing Partners)

  • Emmanuel Gobet

    (MATHFI - Mathématiques financières - LJK - Laboratoire Jean Kuntzmann - UPMF - Université Pierre Mendès France - Grenoble 2 - UJF - Université Joseph Fourier - Grenoble 1 - Grenoble INP - Institut polytechnique de Grenoble - Grenoble Institute of Technology - CNRS - Centre National de la Recherche Scientifique)

  • Mohammed Miri

    (Pricing Partners - Pricing Partners, MATHFI - Mathématiques financières - LJK - Laboratoire Jean Kuntzmann - UPMF - Université Pierre Mendès France - Grenoble 2 - UJF - Université Joseph Fourier - Grenoble 1 - Grenoble INP - Institut polytechnique de Grenoble - Grenoble Institute of Technology - CNRS - Centre National de la Recherche Scientifique)

Abstract

Because of its very general formulation, the local volatility model does not have an analytical solution for European options. In this article, we present a new methodology to derive closed form solutions for the price of any European options. The formula results from an asymptotic expansion, terms of which are Black-Scholes price and related Greeks. The accuracy of the formula depends on the payoff smoothness and it converges with very few terms.

Suggested Citation

  • Eric Benhamou & Emmanuel Gobet & Mohammed Miri, 2010. "Expansion formulas for European options in a local volatility model," Post-Print hal-00325939, HAL.
    • Handle: RePEc:hal:journl:hal-00325939
    DOI: 10.1142/S0219024910005887
    Note: View the original document on HAL open archive server: https://hal.science/hal-00325939v1
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    Cited by:

    1. Eric Benhamou & David Saltiel & Serge Tabachnik & Sui Kai Wong & François Chareyron, 2021. "Distinguish the indistinguishable: a Deep Reinforcement Learning approach for volatility targeting models," Working Papers hal-03202431, HAL.
    2. Stefano Pagliarani & Andrea Pascucci, 2017. "The exact Taylor formula of the implied volatility," Finance and Stochastics, Springer, vol. 21(3), pages 661-718, July.
    3. Emmanuel Gobet & Ali Suleiman, 2013. "New approximations in local volatility models," Post-Print hal-00523369, HAL.
    4. Gobet, Emmanuel & Miri, Mohammed, 2014. "Weak approximation of averaged diffusion processes," Stochastic Processes and their Applications, Elsevier, vol. 124(1), pages 475-504.
    5. Chenxu Li, 2014. "Closed-Form Expansion, Conditional Expectation, and Option Valuation," Mathematics of Operations Research, INFORMS, vol. 39(2), pages 487-516, May.
    6. Julien Hok & Philip Ngare & Antonis Papapantoleon, 2018. "Expansion formulas for European quanto options in a local volatility FX-LIBOR model," Papers 1801.01205, arXiv.org, revised Apr 2018.
    7. Alev{s} v{C}ern'y & Stephan Denkl & Jan Kallsen, 2013. "Hedging in L\'evy Models and the Time Step Equivalent of Jumps," Papers 1309.7833, arXiv.org, revised Jul 2017.
    8. Ning Cai & Chenxu Li & Chao Shi, 2014. "Closed-Form Expansions of Discretely Monitored Asian Options in Diffusion Models," Mathematics of Operations Research, INFORMS, vol. 39(3), pages 789-822, August.
    9. Pagliarani, Stefano & Pascucci, Andrea, 2011. "Analytical approximation of the transition density in a local volatility model," MPRA Paper 31107, University Library of Munich, Germany.
    10. Romain Bompis, 2017. "Weak approximations for arithmetic means of geometric Brownian motions and applications to Basket options," Working Papers hal-01502886, HAL.
    11. Julien Hok & Philip Ngare & Antonis Papapantoleon, 2018. "Expansion Formulas For European Quanto Options In A Local Volatility Fx-Libor Model," International Journal of Theoretical and Applied Finance (IJTAF), World Scientific Publishing Co. Pte. Ltd., vol. 21(02), pages 1-43, March.
    12. Elisa Al`os & `Oscar Bur'es & Rafael de Santiago & Josep Vives, 2025. "Volatility Modeling with Rough Paths: A Signature-Based Alternative to Classical Expansions," Papers 2507.23392, arXiv.org, revised Aug 2025.
    13. 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.
    14. Elisa Alòs & Rafael De Santiago & Josep Vives, 2012. "Calibration of stochastic volatility models via second order approximation: the Heston model case," Economics Working Papers 1346, Department of Economics and Business, Universitat Pompeu Fabra.
    15. Colin Turfus & Alexander Shubert, 2017. "ANALYTIC PRICING OF CoCo BONDS," International Journal of Theoretical and Applied Finance (IJTAF), World Scientific Publishing Co. Pte. Ltd., vol. 20(05), pages 1-26, August.
    16. Matthew Lorig & Stefano Pagliarani & Andrea Pascucci, 2013. "Analytical expansions for parabolic equations," Papers 1312.3314, arXiv.org, revised Nov 2014.
    17. 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.
    18. Cai, Ning & Li, Chenxu & Shi, Chao, 2021. "Pricing discretely monitored barrier options: When Malliavin calculus expansions meet Hilbert transforms," Journal of Economic Dynamics and Control, Elsevier, vol. 127(C).
    19. Romain Bompis & Emmanuel Gobet, 2012. "Asymptotic and non asymptotic approximations for option valuation," Post-Print hal-00720650, HAL.
    20. Pierre Etoré & Emmanuel Gobet, 2012. "Stochastic expansion for the pricing of call options with discrete dividends," Post-Print hal-00507787, HAL.

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