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Option valuation and hedging using asymmetric risk function: asymptotic optimality through fully nonlinear Partial Differential Equations

Author

Listed:
  • Emmanuel Gobet

    (CMAP - Centre de Mathématiques Appliquées - Ecole Polytechnique - X - École polytechnique - CNRS - Centre National de la Recherche Scientifique)

  • Isaque Pimentel

    (CMAP - Centre de Mathématiques Appliquées - Ecole Polytechnique - X - École polytechnique - CNRS - Centre National de la Recherche Scientifique, EDF - EDF)

  • Xavier Warin

    (EDF - EDF)

Abstract

Discrete time hedging produces a residual risk, namely, the tracking error. The major problem is to get valuation/hedging policies minimizing this error. We evaluate the risk between trading dates through a function penalizing asymmetrically profits and losses. After deriving the asymptotics within a discrete time risk measurement for a large number of trading dates, we derive the optimal strategies minimizing the asymptotic risk in the continuous time setting. We characterize the optimality through a class of fully nonlinear Partial Differential Equations (PDE). Numerical experiments show that the optimal strategies associated with discrete and asymptotic approach coincides asymptotically.

Suggested Citation

  • Emmanuel Gobet & Isaque Pimentel & Xavier Warin, 2018. "Option valuation and hedging using asymmetric risk function: asymptotic optimality through fully nonlinear Partial Differential Equations," Working Papers hal-01761234, HAL.
    • Handle: RePEc:hal:wpaper:hal-01761234
    Note: View the original document on HAL open archive server: https://hal.science/hal-01761234
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    References listed on IDEAS

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    1. repec:dau:papers:123456789/4273 is not listed on IDEAS
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    4. Föllmer, H. & Schweizer, M., 1988. "Hedging by Sequential Regression: An Introduction to the Mathematics of Option Trading," ASTIN Bulletin, Cambridge University Press, vol. 18(2), pages 147-160, November.
    5. Schweizer, Martin, 1999. "A guided tour through quadratic hedging approaches," SFB 373 Discussion Papers 1999,96, Humboldt University of Berlin, Interdisciplinary Research Project 373: Quantification and Simulation of Economic Processes.
    6. N. El Karoui & S. Peng & M. C. Quenez, 1997. "Backward Stochastic Differential Equations in Finance," Mathematical Finance, Wiley Blackwell, vol. 7(1), pages 1-71, January.
    7. Benoit Pochart & Jean-Philippe Bouchaud, 2004. "Option pricing and hedging with minimum local expected shortfall," Quantitative Finance, Taylor & Francis Journals, vol. 4(5), pages 607-618.
    8. Emmanuel Temam & Emmanuel Gobet, 2001. "Discrete time hedging errors for options with irregular payoffs," Finance and Stochastics, Springer, vol. 5(3), pages 357-367.
    9. Potters, Marc & Bouchaud, Jean-Philippe & Sestovic, Dragan, 2001. "Hedged Monte-Carlo: low variance derivative pricing with objective probabilities," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 289(3), pages 517-525.
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    Cited by:

    1. Simon F'ecamp & Joseph Mikael & Xavier Warin, 2019. "Risk management with machine-learning-based algorithms," Papers 1902.05287, arXiv.org, revised Aug 2020.
    2. Clémence Alasseur & Emmanuel Gobet & Isaque Pimentel & Xavier Warin, 2019. "A power plant valuation under an asymmetric risk criterion taking into account maintenance costs," Working Papers hal-02077740, HAL.

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    More about this item

    Keywords

    hedging; asymmetric risk; fully nonlinear parabolic PDE; regression Monte Carlo;
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