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

Convergence rate of free boundary of numerical scheme for American option

Xinfu Chen 1, , Bei Hu 2, , Jin Liang 3, and  Yajing Zhang 4,

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.
Keywords: explicit finite difference scheme, binomial tree model., Convergence rate, free boundary, error estimate, American options.
Mathematics Subject Classification: 91G20, 91G60, 91G8.
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.

show all references

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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