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A moving boundary approach to American option pricing

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  • Muthuraman, Kumar

Abstract

This paper describes a method to solve the free-boundary problem that arises in the pricing of American options. Most numerical methods for American option pricing exploit the representation of the option price as the expected pay-off under the risk-neutral measure and calculate the price for a given time to expiration and stock price. They do not solve the related free-boundary problem explicitly. The advantage of solving the free-boundary problem is that it provides the entire price function as well as the optimal exercise boundary explicitly. Our approach, which we term the Moving Boundary Approach, is based on using a boundary guess and the value associated with the guess to construct an improved boundary. It is also shown that on iteration, the sequence of boundaries converge monotonically to the optimal exercise boundary. Examples illustrating the convergence behavior as well as discussions providing insight into the method are also presented. Finally, we compare runtimes and speeds with other methods that solve the free-boundary problem and compute the optimal boundaries explicitly, like the front-fixing method, penalty method, method based on the integral representations and the method by Brennan and Schwartz [1977. The valuation of American put options. Journal of Finance 32 (2), 449-462].

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  • Muthuraman, Kumar, 2008. "A moving boundary approach to American option pricing," Journal of Economic Dynamics and Control, Elsevier, vol. 32(11), pages 3520-3537, November.
    • Handle: RePEc:eee:dyncon:v:32:y:2008:i:11:p:3520-3537
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    3. Kirkby, J. Lars & Nguyen, Duy & Cui, Zhenyu, 2017. "A unified approach to Bermudan and barrier options under stochastic volatility models with jumps," Journal of Economic Dynamics and Control, Elsevier, vol. 80(C), pages 75-100.
    4. San‐Lin Chung & Jr‐Yan Wang, 2018. "A simple iteration algorithm to price perpetual Bermudan options under the lognormal jump‐diffusion‐ruin process," Journal of Futures Markets, John Wiley & Sons, Ltd., vol. 38(8), pages 898-924, August.
    5. Chockalingam, Arun & Muthuraman, Kumar, 2015. "An approximate moving boundary method for American option pricing," European Journal of Operational Research, Elsevier, vol. 240(2), pages 431-438.
    6. Dabadghao, Shaunak S. & Chockalingam, Arun & Soltani, Taimaz & Fransoo, Jan C., 2021. "Valuing switching options with the moving-boundary method," Other publications TiSEM 45fe7e78-129f-4d41-ac2f-5, Tilburg University, School of Economics and Management.
    7. Lee, Jung-Kyung, 2020. "A simple numerical method for pricing American power put options," Chaos, Solitons & Fractals, Elsevier, vol. 139(C).
    8. Zhongkai Liu & Tao Pang, 2016. "An efficient grid lattice algorithm for pricing American-style options," International Journal of Financial Markets and Derivatives, Inderscience Enterprises Ltd, vol. 5(1), pages 36-55.
    9. Jung-Kyung Lee, 2020. "On a Free Boundary Problem for American Options Under the Generalized Black–Scholes Model," Mathematics, MDPI, vol. 8(9), pages 1-11, September.
    10. Jun Cheng & Jin Zhang, 2012. "Analytical pricing of American options," Review of Derivatives Research, Springer, vol. 15(2), pages 157-192, July.
    11. Arun Chockalingam & Kumar Muthuraman, 2011. "American Options Under Stochastic Volatility," Operations Research, INFORMS, vol. 59(4), pages 793-809, August.
    12. Chinonso Nwankwo & Weizhong Dai, 2020. "An Adaptive and Explicit Fourth Order Runge-Kutta-Fehlberg Method Coupled with Compact Finite Differencing for Pricing American Put Options," Papers 2007.04408, arXiv.org, revised Jul 2021.
    13. Chockalingam, Arun & Feng, Haolin, 2015. "The implication of missing the optimal-exercise time of an American option," European Journal of Operational Research, Elsevier, vol. 243(3), pages 883-896.
    14. Song-Ping Zhu & Nhat-Tan Le & Wen-Ting Chen & Xiaoping Lu, 2015. "Pricing Parisian down-and-in options," Papers 1511.01564, arXiv.org.
    15. Zhu, Song-Ping & Chen, Wen-Ting, 2013. "An inverse finite element method for pricing American options," Journal of Economic Dynamics and Control, Elsevier, vol. 37(1), pages 231-250.

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