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A power penalty method for a 2D fractional partial differential linear complementarity problem governing two-asset American option pricing

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  • Chen, Wen
  • Wang, Song

Abstract

In this paper we propose a power penalty method for a linear complementarity problem (LCP) involving a fractional partial differential operator in two spatial dimensions arising in pricing American options on two underlying assets whose prices follow two independent geometric Lévy processes. We first approximate the LCP by a nonlinear 2D fractional partial differential equation (fPDE) with a penalty term. We then prove that the solution to the fPDE converges to that of the LCP in a Sobolev norm at an exponential rate depending on the parameters used in the penalty term. The 2D fPDE is discretized by a 2nd-order finite difference method in space and Crank–Nicolson method in time. Numerical experiments on a model Basket Option pricing problem were performed to demonstrate the convergent rates and the effectiveness of the penalty method.

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  • Chen, Wen & Wang, Song, 2017. "A power penalty method for a 2D fractional partial differential linear complementarity problem governing two-asset American option pricing," Applied Mathematics and Computation, Elsevier, vol. 305(C), pages 174-187.
    • Handle: RePEc:eee:apmaco:v:305:y:2017:i:c:p:174-187
    DOI: 10.1016/j.amc.2017.01.069
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    References listed on IDEAS

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    1. Wen Li & Song Wang, 2014. "A numerical method for pricing European options with proportional transaction costs," Journal of Global Optimization, Springer, vol. 60(1), pages 59-78, September.
    2. Leif Andersen & Jesper Andreasen, 2000. "Jump-Diffusion Processes: Volatility Smile Fitting and Numerical Methods for Option Pricing," Review of Derivatives Research, Springer, vol. 4(3), pages 231-262, October.
    3. Robert C. Merton, 2005. "Theory of rational option pricing," World Scientific Book Chapters, in: Sudipto Bhattacharya & George M Constantinides (ed.), Theory Of Valuation, chapter 8, pages 229-288, World Scientific Publishing Co. Pte. Ltd..
    4. S. Wang & X. Q. Yang & K. L. Teo, 2006. "Power Penalty Method for a Linear Complementarity Problem Arising from American Option Valuation," Journal of Optimization Theory and Applications, Springer, vol. 129(2), pages 227-254, May.
    5. 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.
    6. 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.
    7. Y. Zhou & S. Wang & X. Yang, 2014. "A penalty approximation method for a semilinear parabolic double obstacle problem," Journal of Global Optimization, Springer, vol. 60(3), pages 531-550, November.
    8. Peter Carr & Helyette Geman, 2002. "The Fine Structure of Asset Returns: An Empirical Investigation," The Journal of Business, University of Chicago Press, vol. 75(2), pages 305-332, April.
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    Cited by:

    1. Kirkby, J. Lars & Nguyen, Dang H. & Nguyen, Duy, 2020. "A general continuous time Markov chain approximation for multi-asset option pricing with systems of correlated diffusions," Applied Mathematics and Computation, Elsevier, vol. 386(C).
    2. Chen, Wen & Wang, Song, 2020. "A 2nd-order ADI finite difference method for a 2D fractional Black–Scholes equation governing European two asset option pricing," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 171(C), pages 279-293.
    3. Wen Li & Song Wang & Volker Rehbock, 2019. "Numerical Solution of Fractional Optimal Control," Journal of Optimization Theory and Applications, Springer, vol. 180(2), pages 556-573, February.

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