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Pricing American options with the Runge-Kutta-Legendre finite difference scheme

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  • Fabien Le Floc'h

Abstract

This paper presents the Runge-Kutta-Legendre finite difference scheme, allowing for an additional shift in its polynomial representation. A short presentation of the stability region, comparatively to the Runge-Kutta-Chebyshev scheme follows. We then explore the problem of pricing American options with the Runge-Kutta-Legendre scheme under the one factor Black-Scholes and the two factor Heston stochastic volatility models, as well as the pricing of butterfly spread and digital options under the uncertain volatility model, where a Hamilton-Jacobi-Bellman partial differential equation needs to be solved. We explore the order of convergence in these problems, as well as the option greeks stability, compared to the literature and popular schemes such as Crank-Nicolson, with Rannacher time-stepping.

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  • Fabien Le Floc'h, 2021. "Pricing American options with the Runge-Kutta-Legendre finite difference scheme," Papers 2106.12049, arXiv.org.
    • Handle: RePEc:arx:papers:2106.12049
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    References listed on IDEAS

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    1. Brennan, Michael J & Schwartz, Eduardo S, 1977. "The Valuation of American Put Options," Journal of Finance, American Finance Association, vol. 32(2), pages 449-462, May.
    2. Reza Mollapourasl & Ali Fereshtian & Michèle Vanmaele, 2019. "Radial Basis Functions with Partition of Unity Method for American Options with Stochastic Volatility," Computational Economics, Springer;Society for Computational Economics, vol. 53(1), pages 259-287, January.
    3. John C. Cox & Jonathan E. Ingersoll Jr. & Stephen A. Ross, 2005. "A Theory Of The Term Structure Of Interest Rates," World Scientific Book Chapters, in: Sudipto Bhattacharya & George M Constantinides (ed.), Theory Of Valuation, chapter 5, pages 129-164, World Scientific Publishing Co. Pte. Ltd..
    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.
    5. 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.
    6. M. Avellaneda & A. Levy & A. ParAS, 1995. "Pricing and hedging derivative securities in markets with uncertain volatilities," Applied Mathematical Finance, Taylor & Francis Journals, vol. 2(2), pages 73-88.
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