American Institute of Mathematical Sciences

September  2012, 17(6): 1751-1759. doi: 10.3934/dcdsb.2012.17.1751

Regularity of the free boundary for the American put option

 1 Department of Mathematics, University of Pittsburgh, Pittsburgh, PA 15260 2 Department of Mathematics and Statistics, McMaster University, Hamilton, ON L8S 4K1, Canada

Received  September 2011 Revised  November 2011 Published  May 2012

We show the free boundary of the American put option with dividend payment is $C^{\infty}$.
Citation: Xinfu Chen, Huibin Cheng. Regularity of the free boundary for the American put option. Discrete and Continuous Dynamical Systems - B, 2012, 17 (6) : 1751-1759. doi: 10.3934/dcdsb.2012.17.1751
References:
 [1] E. Bayraktar and H. Xing, Analysis of the optimal exercise boundary of American option for jump diffusions, SIAM. J. Math. Anal., 41 (2009), 825-860. doi: 10.1137/080712519. [2] A. Bensoussan, On the theory of option pricing, Acta Appl. Math., 2 (1984), 139-158. [3] A. Bensoussan & J.-L. Lions, "Application of Variational Inequalities in Stochastic Control," Studies in Mathematics and its Applications, 12, North-Holland Publishing Co., Amsterdam-New York, 1982. [4] X. Chen and J. Chadam, A mathematical analysis of the optimal boundary for American put options, SIAM J. Math. Anal., 38 (2006/07), 1613-1641. [5] Xinfu Chen and J. Chadam, Analytic and numerical approximations for the early exercise boundary for American put options, Dyn. Cont. Disc. and Impulsive Sys., 10 (2003), 649-657. [6] Xinfu Chen, J. Chadam and Huibin Cheng, Non-convexity of the optimal exercise boundary for an American put option on a dividend-paying asset, Mathematical Finance, to appear. Available from: http://www.pitt.edu/~chadam/papers/LargeDNonConvex.pdf. [7] Xinfu Chen, J. Chadam, L. Jiang and W. Zheng, Convexity of the exercise boundary of the American put option on a zero dividend asset, Math. Finance, 18 (2008), 185-197. doi: 10.1111/j.1467-9965.2007.00328.x. [8] A. Friedman, "Variational Principles and Free Boundary Problems," A Wiley-Interscience Publication, Pure and Applied Mathematics, John Wiley & Sons, Inc., New York, 1982. [9] A. Friedman, "Partial Diffferential Equations of Parabolic Type," Prentice-Hall, 1964. [10] A. Friedman, Analyticity of the free boundary for the Stefan problem, Arch. Rational Mech. Anal., 61 (1976), 97-125. doi: 10.1007/BF00249700. [11] L. Jiang, Existence and differentiability of the solution of a two phase Stefan problem for quasi-linear parabolic equations, Chinese Math. Acta, 7 (1965), 481-496. [12] D. Lamberton and M. Mikou, The critical price for the American put in an exponential Lévy model, Finance Stoch., 12 (2008), 561-581. doi: 10.1007/s00780-008-0073-9. [13] P. Laurence and S. Salsa, Regularity of the free boundary of an American option on several assets, Comm. on Pure and Appl. Math., 62 (2009), 969-994. doi: 10.1002/cpa.20268. [14] H. P. McKean, Jr., Appendix: A free boundary problem for the heat equation arising from a problem in mathematical economics, Industrial Management Review, 6 (1965), 32-39. [15] P. van Moerbeke, On optimal stopping and free boundary problems, Arch. Rational Mech. Anal., 60 (1975/76), 101-148. [16] C. Yang, L. Jiang and B. Bian, Free boundary and American option in a jump-diffusion model, Euro. J. of Applied Mathematics, 17 (2006), 95-127. doi: 10.1017/S0956792505006340. [17] P. Wilmott, J. Dewynne and S. Howison, "The Mathematics of Financial Derivatives. A Student Introduction," Cambridge University Press, Cambridge, 1995.

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References:
 [1] E. Bayraktar and H. Xing, Analysis of the optimal exercise boundary of American option for jump diffusions, SIAM. J. Math. Anal., 41 (2009), 825-860. doi: 10.1137/080712519. [2] A. Bensoussan, On the theory of option pricing, Acta Appl. Math., 2 (1984), 139-158. [3] A. Bensoussan & J.-L. Lions, "Application of Variational Inequalities in Stochastic Control," Studies in Mathematics and its Applications, 12, North-Holland Publishing Co., Amsterdam-New York, 1982. [4] X. Chen and J. Chadam, A mathematical analysis of the optimal boundary for American put options, SIAM J. Math. Anal., 38 (2006/07), 1613-1641. [5] Xinfu Chen and J. Chadam, Analytic and numerical approximations for the early exercise boundary for American put options, Dyn. Cont. Disc. and Impulsive Sys., 10 (2003), 649-657. [6] Xinfu Chen, J. Chadam and Huibin Cheng, Non-convexity of the optimal exercise boundary for an American put option on a dividend-paying asset, Mathematical Finance, to appear. Available from: http://www.pitt.edu/~chadam/papers/LargeDNonConvex.pdf. [7] Xinfu Chen, J. Chadam, L. Jiang and W. Zheng, Convexity of the exercise boundary of the American put option on a zero dividend asset, Math. Finance, 18 (2008), 185-197. doi: 10.1111/j.1467-9965.2007.00328.x. [8] A. Friedman, "Variational Principles and Free Boundary Problems," A Wiley-Interscience Publication, Pure and Applied Mathematics, John Wiley & Sons, Inc., New York, 1982. [9] A. Friedman, "Partial Diffferential Equations of Parabolic Type," Prentice-Hall, 1964. [10] A. Friedman, Analyticity of the free boundary for the Stefan problem, Arch. Rational Mech. Anal., 61 (1976), 97-125. doi: 10.1007/BF00249700. [11] L. Jiang, Existence and differentiability of the solution of a two phase Stefan problem for quasi-linear parabolic equations, Chinese Math. Acta, 7 (1965), 481-496. [12] D. Lamberton and M. Mikou, The critical price for the American put in an exponential Lévy model, Finance Stoch., 12 (2008), 561-581. doi: 10.1007/s00780-008-0073-9. [13] P. Laurence and S. Salsa, Regularity of the free boundary of an American option on several assets, Comm. on Pure and Appl. Math., 62 (2009), 969-994. doi: 10.1002/cpa.20268. [14] H. P. McKean, Jr., Appendix: A free boundary problem for the heat equation arising from a problem in mathematical economics, Industrial Management Review, 6 (1965), 32-39. [15] P. van Moerbeke, On optimal stopping and free boundary problems, Arch. Rational Mech. Anal., 60 (1975/76), 101-148. [16] C. Yang, L. Jiang and B. Bian, Free boundary and American option in a jump-diffusion model, Euro. J. of Applied Mathematics, 17 (2006), 95-127. doi: 10.1017/S0956792505006340. [17] P. Wilmott, J. Dewynne and S. Howison, "The Mathematics of Financial Derivatives. A Student Introduction," Cambridge University Press, Cambridge, 1995.
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