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Recovery of Implied Volatility in a Spatial-Fractional Black–Scholes Equation Under a Finite Moment Log Stable Model

Author

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  • Xiaoying Jiang

    (School of Mathematics and Computer Science, Zhejiang A&F University, Hangzhou 311300, China)

  • Chunmei Shi

    (School of Mathematics and Computer Science, Zhejiang A&F University, Hangzhou 311300, China)

  • Yujie Wei

    (School of Mathematics Sciences, Zhejiang University, Hangzhou 310027, China)

Abstract

In this paper, we study direct and inverse problems for a spatial-fractional Black–Scholes equation with space-dependent volatility. For the direct problem, we provide CN-WSGD (Crank–Nicholson and the weighted and shifted Grünwald difference) scheme to solve the initial boundary value problem. The latter aims to recover the implied volatility via observable option prices. Using a linearization technique, we rigorously derive a mathematical formulation of the inverse problem in terms of a Fredholm integral equation of the first kind. Based on an integral equation, an efficient numerical reconstruction algorithm is proposed to recover the coefficient. Numerical results for both problems are provided to illustrate the validity and effectiveness of proposed methods.

Suggested Citation

  • Xiaoying Jiang & Chunmei Shi & Yujie Wei, 2025. "Recovery of Implied Volatility in a Spatial-Fractional Black–Scholes Equation Under a Finite Moment Log Stable Model," Mathematics, MDPI, vol. 13(15), pages 1-14, August.
    • Handle: RePEc:gam:jmathe:v:13:y:2025:i:15:p:2480-:d:1715424
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    1. repec:bla:jfinan:v:58:y:2003:i:2:p:753-778 is not listed on IDEAS
    2. Cartea, Álvaro & del-Castillo-Negrete, Diego, 2007. "Fractional diffusion models of option prices in markets with jumps," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 374(2), pages 749-763.
    3. Peter Carr & Liuren Wu, 2003. "The Finite Moment Log Stable Process and Option Pricing," Journal of Finance, American Finance Association, vol. 58(2), pages 753-777, April.
    4. 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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