American Institute of Mathematical Sciences

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

Regularity of the free boundary for the American put option

Xinfu Chen 1, and  Huibin Cheng 2,

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 $C^{\infty}$.
Keywords: Regularity, free boundary problem, American put option..
Mathematics Subject Classification: Primary: 35R35, 35B65; Secondary: 91G5.
Citation: Xinfu Chen, Huibin Cheng. Regularity of the free boundary for the American put option. Discrete & 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.  Google Scholar

[2]

A. Bensoussan, On the theory of option pricing, Acta Appl. Math., 2 (1984), 139-158.  Google Scholar

[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.  Google Scholar

[4]

X. Chen and J. Chadam, A mathematical analysis of the optimal boundary for American put options,, SIAM J. Math. Anal., 38 (): 1613.   Google Scholar

[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.  Google Scholar

[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, ().   Google Scholar

[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.  Google Scholar

[8]

A. Friedman, "Variational Principles and Free Boundary Problems," A Wiley-Interscience Publication, Pure and Applied Mathematics, John Wiley & Sons, Inc., New York, 1982.  Google Scholar

[9]

A. Friedman, "Partial Diffferential Equations of Parabolic Type," Prentice-Hall, 1964.  Google Scholar

[10]

A. Friedman, Analyticity of the free boundary for the Stefan problem, Arch. Rational Mech. Anal., 61 (1976), 97-125. doi: 10.1007/BF00249700.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[15]

P. van Moerbeke, On optimal stopping and free boundary problems,, Arch. Rational Mech. Anal., 60 (): 101.   Google Scholar

[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.  Google Scholar

[17]

P. Wilmott, J. Dewynne and S. Howison, "The Mathematics of Financial Derivatives. A Student Introduction," Cambridge University Press, Cambridge, 1995.  Google Scholar

show all references

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.  Google Scholar

[2]

A. Bensoussan, On the theory of option pricing, Acta Appl. Math., 2 (1984), 139-158.  Google Scholar

[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.  Google Scholar

[4]

X. Chen and J. Chadam, A mathematical analysis of the optimal boundary for American put options,, SIAM J. Math. Anal., 38 (): 1613.   Google Scholar

[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.  Google Scholar

[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, ().   Google Scholar

[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.  Google Scholar

[8]

A. Friedman, "Variational Principles and Free Boundary Problems," A Wiley-Interscience Publication, Pure and Applied Mathematics, John Wiley & Sons, Inc., New York, 1982.  Google Scholar

[9]

A. Friedman, "Partial Diffferential Equations of Parabolic Type," Prentice-Hall, 1964.  Google Scholar

[10]

A. Friedman, Analyticity of the free boundary for the Stefan problem, Arch. Rational Mech. Anal., 61 (1976), 97-125. doi: 10.1007/BF00249700.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[15]

P. van Moerbeke, On optimal stopping and free boundary problems,, Arch. Rational Mech. Anal., 60 (): 101.   Google Scholar

[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.  Google Scholar

[17]

P. Wilmott, J. Dewynne and S. Howison, "The Mathematics of Financial Derivatives. A Student Introduction," Cambridge University Press, Cambridge, 1995.  Google Scholar

[1]

Xinfu Chen, Bei Hu, Jin Liang, Yajing Zhang. Convergence rate of free boundary of numerical scheme for American option. Discrete & Continuous Dynamical Systems - B, 2016, 21 (5) : 1435-1444. doi: 10.3934/dcdsb.2016004
[2]

Junkee Jeon, Jehan Oh. Valuation of American strangle option: Variational inequality approach. Discrete & Continuous Dynamical Systems - B, 2019, 24 (2) : 755-781. doi: 10.3934/dcdsb.2018206
[3]

Carlos E. Kenig, Tatiana Toro. On the free boundary regularity theorem of Alt and Caffarelli. Discrete & Continuous Dynamical Systems, 2004, 10 (1&2) : 397-422. doi: 10.3934/dcds.2004.10.397
[4]

Toyohiko Aiki. A free boundary problem for an elastic material. Conference Publications, 2007, 2007 (Special) : 10-17. doi: 10.3934/proc.2007.2007.10
[5]

Yang Zhang. A free boundary problem of the cancer invasion. Discrete & Continuous Dynamical Systems - B, 2022, 27 (3) : 1323-1343. doi: 10.3934/dcdsb.2021092
[6]

Kai Zhang, Song Wang. Convergence property of an interior penalty approach to pricing American option. Journal of Industrial & Management Optimization, 2011, 7 (2) : 435-447. doi: 10.3934/jimo.2011.7.435
[7]

Kai Zhang, Xiaoqi Yang, Kok Lay Teo. A power penalty approach to american option pricing with jump diffusion processes. Journal of Industrial & Management Optimization, 2008, 4 (4) : 783-799. doi: 10.3934/jimo.2008.4.783
[8]

Hayk Mikayelyan, Henrik Shahgholian. Convexity of the free boundary for an exterior free boundary problem involving the perimeter. Communications on Pure & Applied Analysis, 2013, 12 (3) : 1431-1443. doi: 10.3934/cpaa.2013.12.1431
[9]

Ian Knowles, Ajay Mahato. The inverse volatility problem for American options. Discrete & Continuous Dynamical Systems - S, 2020, 13 (12) : 3473-3489. doi: 10.3934/dcdss.2020235
[10]

Huiqiang Jiang. Regularity of a vector valued two phase free boundary problems. Conference Publications, 2013, 2013 (special) : 365-374. doi: 10.3934/proc.2013.2013.365
[11]

Panagiota Daskalopoulos, Eunjai Rhee. Free-boundary regularity for generalized porous medium equations. Communications on Pure & Applied Analysis, 2003, 2 (4) : 481-494. doi: 10.3934/cpaa.2003.2.481
[12]

Xiaoshan Chen, Fahuai Yi. Free boundary problem of Barenblatt equation in stochastic control. Discrete & Continuous Dynamical Systems - B, 2016, 21 (5) : 1421-1434. doi: 10.3934/dcdsb.2016003
[13]

Naoki Sato, Toyohiko Aiki, Yusuke Murase, Ken Shirakawa. A one dimensional free boundary problem for adsorption phenomena. Networks & Heterogeneous Media, 2014, 9 (4) : 655-668. doi: 10.3934/nhm.2014.9.655
[14]

Yongzhi Xu. A free boundary problem model of ductal carcinoma in situ. Discrete & Continuous Dynamical Systems - B, 2004, 4 (1) : 337-348. doi: 10.3934/dcdsb.2004.4.337
[15]

Anna Lisa Amadori. Contour enhancement via a singular free boundary problem. Conference Publications, 2007, 2007 (Special) : 44-53. doi: 10.3934/proc.2007.2007.44
[16]

Shihe Xu. Analysis of a delayed free boundary problem for tumor growth. Discrete & Continuous Dynamical Systems - B, 2011, 15 (1) : 293-308. doi: 10.3934/dcdsb.2011.15.293
[17]

Hiroshi Matsuzawa. A free boundary problem for the Fisher-KPP equation with a given moving boundary. Communications on Pure & Applied Analysis, 2018, 17 (5) : 1821-1852. doi: 10.3934/cpaa.2018087
[18]

Micah Webster, Patrick Guidotti. Boundary dynamics of a two-dimensional diffusive free boundary problem. Discrete & Continuous Dynamical Systems, 2010, 26 (2) : 713-736. doi: 10.3934/dcds.2010.26.713
[19]

Chonghu Guan, Xun Li, Rui Zhou, Wenxin Zhou. Free boundary problem for an optimal investment problem with a borrowing constraint. Journal of Industrial & Management Optimization, 2021  doi: 10.3934/jimo.2021049
[20]

Junde Wu. Bifurcation for a free boundary problem modeling the growth of necrotic multilayered tumors. Discrete & Continuous Dynamical Systems, 2019, 39 (6) : 3399-3411. doi: 10.3934/dcds.2019140

2020 Impact Factor: 1.327

Tools

Metrics

  • PDF downloads (45)
  • HTML views (0)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2022 American Institute of Mathematical Sciences | Privacy Policy