American Institute of Mathematical Sciences

July  2016, 21(5): 1435-1444. doi: 10.3934/dcdsb.2016004

Convergence rate of free boundary of numerical scheme for American option

 1 Department of Mathematics, University of Pittsburgh, Pittsburgh, PA 15260 2 Department of Applied and Computational Mathematics and Statistics, University of Notre Dame, Notre Dame, IN 46556 3 Department of Mathematics, Tongji University, Shanghai 200092 4 School of Mathematical Sciences, Shanxi University, Taiyuan, Shanxi 030006

Received  September 2013 Published  April 2016

Based on the optimal estimate of convergence rate $O(\Delta x)$ of the value function of an explicit finite difference scheme for the American put option problem in [6], an $O(\sqrt{\Delta x})$ rate of convergence of the free boundary resulting from a general compatible numerical scheme to the true free boundary is proven. A new criterion for the compatibility of a generic numerical scheme to the PDE problem is presented. A numerical example is also included.
Citation: Xinfu Chen, Bei Hu, Jin Liang, Yajing Zhang. Convergence rate of free boundary of numerical scheme for American option. Discrete and Continuous Dynamical Systems - B, 2016, 21 (5) : 1435-1444. doi: 10.3934/dcdsb.2016004
References:
 [1] X. Chen and J. Chadam, A mathematical analysis of the optimal exercise boundary for American put options, SIAM Journal on Mathematical Analysis, 38 (2007), 1613-1641. doi: 10.1137/S0036141003437708. [2] X. Chen, J. Chadam, L. Jiang and W. Zheng, Convexity of the exercise boundary of the American put option on a zero dividend asset, Mathematical Finance, 18 (2008), 185-197. doi: 10.1111/j.1467-9965.2007.00328.x. [3] J. Cox and M. Rubinstein, Option pricing: A simplified approach, J. Finan. Econ., 7 (1979), 229-263. [4] A. Friedman, Variational Principles and Free Boundary Problems, John Wiley and Sons, New York, 1982. [5] Hall, J., Options, Futures, and Other Derivatives, Prentice-Hall, Inc., New Jersey, 1989. [6] B. Hu, L. Jiang and J. Liang, Optimal convergence rate of the explicit finite difference scheme for american options, J. Comp. Appl. Math., 230 (2009), 583-599. doi: 10.1016/j.cam.2008.12.018. [7] L. Jiang, Mathematical Modeling and Methods for Option Pricing, World Scientific, 2005. doi: 10.1142/5855. [8] L. Jiang and M. Dai, Convergence of the explicit difference scheme and the binomial tree method for American options, J. Comp. Math., 22 (2004), 371-380. [9] L. Jiang and M. Dai, Convergence of binomial tree methods for European/American options path-depedent options, SIAM J Numer. Anal., 42 (2004), 1094-1109. doi: 10.1137/S0036142902414220. [10] J. Liang, B. Hu, L. Jiang and B. Bian, On the rate of convergence of the binomial tree scheme for American options, Numerische Mathematik, 107 (2007), 333-352. doi: 10.1007/s00211-007-0091-0. [11] J. Liang, B. Hu and L. Jiang, Optimal convergence rate of the binomial tree scheme for American options with jump diffusion, SIAM Financial Mathematics, 1 (2010), 30-65. doi: 10.1137/090746239. [12] R. Myneni, The pricing of the American option, The Annals of Applied Probability, 2 (1992), 1-23. doi: 10.1214/aoap/1177005768. [13] X. Qian, C. Xu, L. Jiang and B. Bian, Convergence of the binomial tree method for American options in jump-diffusion model, SIAM J. Numer. Anal., 42 (2005), 1899-1913. doi: 10.1137/S0036142902409744. [14] C. Xu, X. Qian and L. Jiang, Numerical analysis on binomial tree methods for a jump-diffusion model, J. Com. and Appl. Math. 156 (2003), 23-45. doi: 10.1016/S0377-0427(02)00903-2. [15] W. Wilmott, S. Howison and J. Dewyne, The Mathematics of Financial Derivatives, Cambridge University Press, 1995. doi: 10.1017/CBO9780511812545.

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References:
 [1] X. Chen and J. Chadam, A mathematical analysis of the optimal exercise boundary for American put options, SIAM Journal on Mathematical Analysis, 38 (2007), 1613-1641. doi: 10.1137/S0036141003437708. [2] X. Chen, J. Chadam, L. Jiang and W. Zheng, Convexity of the exercise boundary of the American put option on a zero dividend asset, Mathematical Finance, 18 (2008), 185-197. doi: 10.1111/j.1467-9965.2007.00328.x. [3] J. Cox and M. Rubinstein, Option pricing: A simplified approach, J. Finan. Econ., 7 (1979), 229-263. [4] A. Friedman, Variational Principles and Free Boundary Problems, John Wiley and Sons, New York, 1982. [5] Hall, J., Options, Futures, and Other Derivatives, Prentice-Hall, Inc., New Jersey, 1989. [6] B. Hu, L. Jiang and J. Liang, Optimal convergence rate of the explicit finite difference scheme for american options, J. Comp. Appl. Math., 230 (2009), 583-599. doi: 10.1016/j.cam.2008.12.018. [7] L. Jiang, Mathematical Modeling and Methods for Option Pricing, World Scientific, 2005. doi: 10.1142/5855. [8] L. Jiang and M. Dai, Convergence of the explicit difference scheme and the binomial tree method for American options, J. Comp. Math., 22 (2004), 371-380. [9] L. Jiang and M. Dai, Convergence of binomial tree methods for European/American options path-depedent options, SIAM J Numer. Anal., 42 (2004), 1094-1109. doi: 10.1137/S0036142902414220. [10] J. Liang, B. Hu, L. Jiang and B. Bian, On the rate of convergence of the binomial tree scheme for American options, Numerische Mathematik, 107 (2007), 333-352. doi: 10.1007/s00211-007-0091-0. [11] J. Liang, B. Hu and L. Jiang, Optimal convergence rate of the binomial tree scheme for American options with jump diffusion, SIAM Financial Mathematics, 1 (2010), 30-65. doi: 10.1137/090746239. [12] R. Myneni, The pricing of the American option, The Annals of Applied Probability, 2 (1992), 1-23. doi: 10.1214/aoap/1177005768. [13] X. Qian, C. Xu, L. Jiang and B. Bian, Convergence of the binomial tree method for American options in jump-diffusion model, SIAM J. Numer. Anal., 42 (2005), 1899-1913. doi: 10.1137/S0036142902409744. [14] C. Xu, X. Qian and L. Jiang, Numerical analysis on binomial tree methods for a jump-diffusion model, J. Com. and Appl. Math. 156 (2003), 23-45. doi: 10.1016/S0377-0427(02)00903-2. [15] W. Wilmott, S. Howison and J. Dewyne, The Mathematics of Financial Derivatives, Cambridge University Press, 1995. doi: 10.1017/CBO9780511812545.
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