Some Properties for the American Option-Pricing Model

Hong-Ming Yin

Department of Mathematics, Washington State University, Pullman, WA99164, USA.

DOI: 10.4236/jmf.2012.23027                

Abstract

In this paper we study global properties of the optimal excising boundary for the American option-pricing model. It is shown that a global comparison principle with respect to time-dependent volatility holds. Moreover, we proved a global regularity for the free boundary.

Keywords

American Option Model; Regularity of Free Boundary; Comparison Principle

Conflicts of Interest

The authors declare no conflicts of interest.

References

[1]	J. Hull, “Options, Futures and Other Derivative Securities,” 7th Edition, Prentice-Hall, New York, 2008.
[2]	P. Jaillet, D. Lamberton and B. Lapeyre, “Variational Inequalities and the Pricing of American Options,” Acta Applied Mathematics, Vol. 21, No. 3, 1990, pp. 263-289.
[3]	Y. K. Kwok, “Mathematical Models of Financial Derivatives,” Springer, Berlin, 2008.
[4]	L. S. Jiang, “Mathematical Modelling and Methods of Option Pricing,” World Scientific Publishing Ltd., Singapore City, 2005.
[5]	P. Van Moerbeke, “On Optimal Stopping and Free Boundary Problems,” Archive for Rational Mechanics and Analysis, Vol. 60, 1976, pp. 101-148.
[6]	G. Barles, J. Burdeau, M. Romano and N. Sansoen, “Critical Stock Price near Expiration,” Mathematical Finance, Vol. 5, No. 2, 1995, pp. 77-95.
[7]	P. Carr, R. Jarrow and R. Myneni, “Alternative Characterization of American Put Option,” Mathematical Finance, Vol. 2, 1992, pp. 87-105.
[8]	P. Wilmott, “Derivatives: The Theory and Practice of Financial Engineering,” John Wiley and Sons, Inc., New York, 1998.
[9]	A. Bensoussan, “On the Theory of Option Pricing,” Acta Applied Mathematics, Vol. 2, 1984, pp. 139-158.
[10]	M. Broadie and J. Detemple, “The Valuation of American Options on Multiple Assets,” Mathematical Finance, Vol. 7, No. 3, 1996, pp. 241-286.
[11]	S. Villaneuve, “Exercise Regions of American Option on Several Assets,” Finance and Stochastics, Vol. 3, No. 3, 1999, 295-322.
[12]	C. R. Yang, L. S. Jiang and B. J. Bian, “Free Boundary and American Options in a Jump-Diffusion Model,” European Journal of Applied Mathematics, Vol. 17, No. 1, 2006, pp. 95-127.
[13]	L. S. Jiang and M. Dai, “Convergence of the Explicit Difference Scheme and the Binomial Tree Method for American Options,” Journal of Computational Mathematics, Vol. 22, No. 3, 2004, pp. 371-380.
[14]	I. J. Kim, “The Analytic Valuation of American Options,” Review of Financial Studies, Vol. 3, No. 4, 1990, pp. 547- 572.
[15]	G. H. Meyer, “On Pricing American and Asian Options with PDE Methods,” Acta Mathematica of Univisitatis Comeniane, Vol. 70, No. 1, 2000, pp. 153-165.
[16]	D. M. Salopek, “American Put Options, Pitman Monographs and Surveys in Pure and Applied Mathematics,” Vol. 84, Addison-Wesley Longman, London, 1997.
[17]	D. Sevcovic, “Analysis of the Free Boundary for the Pricing of an American Call Option,” European Journal of Applied Mathematics, Vol. 10, 2001, pp. 1-13.
[18]	D. A. Bunch and H. Johnson, “The American Put Option and Its Critical Stock Price,” Journal of Finance, Vol. 55, No. 5, 2000, pp. 2333-2356.
[19]	E. Bayraktar and Hao Xing, “Analysis of the Optimal Exercise Boundary of American Options for Jump Diffusions,” SIAM Journal of Mathematical Analysis, Vol. 41, No. 2, 2009, pp. 825-860.
[20]	X. F. Chen and J. Chadam, “Analytic and Numerical Approximations for the Early Exercise Boundary for the American Put Options,” Dynamical Continuous and Discrete Impulse Systems, Vol. 10, 2003, pp. 649-657.
[21]	X. F. Chen and J. Chadam, “A Mathematical Analysis of the Optimal Exercise Boundary for American Put Options,” SIAM Journal of Mathematical Analysis, Vol. 38, 2007, pp. 1613-1641.
[22]	S. D. Jacka, “Optimal Stopping and the American Put,” Mathematical Finance, Vol. 1, No. 1, 1991, pp. 1-14.
[23]	C. Knessl, “Asymptotic Analysis of the American Call Option with Dividends,” European Journal of Applied Mathematics, Vol. 13, 2002, pp. 587-616.
[24]	R. A. Kushe and J. B. Keller, “Optimal Exercise Boundary for an American Put Option,” Applied Mathematical Finance, Vol. 5, No. 2, 1998, pp. 107-116.
[25]	R. Stamicar, D. Sevcovic and J. Chadam, “The Early Exercise Boundary for the American Put Near Expiry: Numerical Approximation,” Canadian Appl. Math. Quartely, Vol. 7, No. 4, 1999, pp. 427-444.
[26]	F. Aitsahlia and T. Lai, “Exercise Boundaries and Efficient Approximations to American Option Prices,” Journal of Computational Finance, Vol. 4, 2001, pp. 85-103.
[27]	G. Barone-Adesi and R. E. Whaley, “Efficient Analytic Approximation of American Option Values,” Journal of Finance, Vol. 42, No. 2, 1987, pp. 301-320.
[28]	G. Barone-Adesi and R. Elliott, “Approximations for the Values of American Options,” Stochastic Analysis and Applications, Vol. 9, No. 2, 1991, pp. 115-131.
[29]	X. F. Chen, J. Chadam, L. S. Jiang and W. A. Zheng, “Convexity of the Exercise Boundary of the American Put Option on a Zero Dividend Asset,” Mathematical Finance, Vol. 18, No. 1, 2008, pp. 185-197. doi:10.1111/j.1467-9965.2007.00328.x
[30]	E. Ekstrom, “Convexity of the Optimal Stopping Boundary for the American Put Option,” Journal of Mathematical Analysis and Applications, Vol. 299, No. 1, 2004, pp. 147-156.
[31]	F. Black and M. Scholes, “The Pricing of Options and Corporate Liabilities,” Journal of Politics and Economics, Vol. 81, No. 3, 1973, pp. 637-654.
[32]	R. C. Merton, “Theory of Rational Option Pricing,” Bell Journal of Economics and Management Science, Vol. 4, No. 1, 1973, pp. 141-183.
[33]	E. Ekstrom, “Properties of American Option Prices,” Stochastic Processes and Their Applications, Vol. 114, 2004, pp. 265-278.
[34]	O. A. Ladyzhenskaya, “The Boundary Value Problems of Mathematical Physics,” Springer-Verlag, New York, 1985.