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Recovery of volatility coefficient by linearization

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  • Ilia Bouchouev
  • Victor Isakov
  • Nicolas Valdivia

Abstract

We study the problem of reconstruction of the asset price dependent local volatility from market prices of options with different strikes. For a general diffusion process we apply the linearization technique and we conclude that the option price can be obtained as the sum of the Black-Scholes formula and of an explicit functional which is linear in perturbation of volatility. We obtain an integral equation for this functional and we show that under some natural conditions it can be inverted for volatility. We demonstrate the stability of the linearized problem, and we propose a numerical algorithm which is accurate for volatility functions with different properties.

Suggested Citation

  • Ilia Bouchouev & Victor Isakov & Nicolas Valdivia, 2002. "Recovery of volatility coefficient by linearization," Quantitative Finance, Taylor & Francis Journals, vol. 2(4), pages 257-263.
    • Handle: RePEc:taf:quantf:v:2:y:2002:i:4:p:257-263
    DOI: 10.1088/1469-7688/2/4/302
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    References listed on IDEAS

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    1. Marco Avellaneda & Craig Friedman & Richard Holmes & Dominick Samperi, 1997. "Calibrating volatility surfaces via relative-entropy minimization," Applied Mathematical Finance, Taylor & Francis Journals, vol. 4(1), pages 37-64.
    2. Black, Fischer & Scholes, Myron S, 1973. "The Pricing of Options and Corporate Liabilities," Journal of Political Economy, University of Chicago Press, vol. 81(3), pages 637-654, May-June.
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    Cited by:

    1. Yasushi Ota & Yu Jiang & Daiki Maki, 2022. "Parameters identification for an inverse problem arising from a binary option using a Bayesian inference approach," Papers 2205.11012, arXiv.org.
    2. Michael V. Klibanov & Andrey V. Kuzhuget, 2015. "Profitable forecast of prices of stock options on real market data via the solution of an ill-posed problem for the Black-Scholes equation," Papers 1503.03567, arXiv.org.
    3. Y. Esmaeelzade Aghdam & H. Mesgarani & A. Amin & J. F. Gómez-Aguilar, 2024. "An Efficient Numerical Scheme to Approach the Time Fractional Black–Scholes Model Using Orthogonal Gegenbauer Polynomials," Computational Economics, Springer;Society for Computational Economics, vol. 64(1), pages 211-224, July.

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