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Convergence of a fitted finite volume method for pricing two dimensional assets with stochastic volatilities

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  • Nyoumbi, Christelle Dleuna
  • Tambue, Antoine

Abstract

In this article, we provide the rigorous mathematical convergence proof both in space and time of the two dimensional Black Scholes equation with stochastic volatility. The spatial approximation of this three dimensional problem is performed using the finite volume method coupled with a fitted technique to tackle the degeneracy in the Black Scholes operator, while the temporal discretization is performed using implicit Euler method. We provide a mathematical rigorous convergence proof in space and time of the full discretized scheme. Numerical results are presented to validate our theoretical results.

Suggested Citation

  • Nyoumbi, Christelle Dleuna & Tambue, Antoine, 2023. "Convergence of a fitted finite volume method for pricing two dimensional assets with stochastic volatilities," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 207(C), pages 388-416.
    • Handle: RePEc:eee:matcom:v:207:y:2023:i:c:p:388-416
    DOI: 10.1016/j.matcom.2023.01.001
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    1. 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.
    2. Hull, John & White, Alan, 1988. "The Use of the Control Variate Technique in Option Pricing," Journal of Financial and Quantitative Analysis, Cambridge University Press, vol. 23(3), pages 237-251, September.
    3. 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.
    4. 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.
    5. Cox, John C. & Ross, Stephen A. & Rubinstein, Mark, 1979. "Option pricing: A simplified approach," Journal of Financial Economics, Elsevier, vol. 7(3), pages 229-263, September.
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