Document Type : Original Paper


1 Faculty of Mathematical Sciences, Department of Actuarial Science, Shahid Beheshti University, Tehran , Iran

2 Faculty of Mathematical Sciences, Department of Applied Mathematics, Shahid Beheshti University, Tehran, Iran


In this article, we examine the issue of pricing perpetual American option with a differential equation approach with free boundary properties. To describe the underlying asset dynamics in these options, we use the feature of jump-diffusion models under regime switching. In pricing these perpetual options, due to the possibility of early application, we need to solve the ordinary integro-differential equation with a free boundary. For this purpose, we write the equation created from this model first as a linear complementarity problems and then discretize by using the finite difference method. We use linear interpolation to approximate the

integral term. The discrete maximum principle is applied to the linear complementarity

problems to obtain the error estimates. We also illustrate some numerical results in order

to demonstrate and compare the accuracy of the method for our problem.


Main Subjects

[1] S. Asmussen, F. Avram, and M. R Pistorius. Russian and american put options under exponential phase–type lévy models. Stochastic Processes and their Applications, 109(1):79–111, 2004.
[2] S. I. Boyarchenko and S. Z Levendorskii. Option pricing for truncated lévy processes. International Journal of Theoretical and Applied Finance, 3(03):549–552, 2000.
[3] S. I. Boyarchenko and S. Z Levendorskii. Perpetual american options under lévy processes. SIAM Journal on Control and Optimization, 40(6):1663–1696, 2002.
[4] T. Chan. Pricing perpetual american options driven by spectrally one–sided lévy processes. preprint, 2000.
[5] X. liang Cheng and L. Xue. On the error estimate of finite difference method for the obstacle problem. Applied mathematics and computation, 183(1):416–422, 2006.
[6] A. Holmes, H. Yang, and S. Zhang. A front–fixing finite element method for the valuation of American options with regime–switching. International Journal of Computer Mathematics, 89(9):1094–1111, 2012.
[7] Z. Jiang and M. R. Pistorius. On perpetual American put valuation and first–passage in a regime–switching model with jumps. Finance and Stochastics, 12(3):331–355, 2008.
[8] A. Jobert and L. CG Rogers. Option pricing with markov-modulated dynamics. SIAM Journal on Control and Optimization, 44(6):2063–2078, 2006.
[9] S. G. Kou and H. Wang. Option pricing under a double exponential jump diffusion model. Management science, 50(9):1178–1192, 2004.
[10] A. Le, Z. Cen, and A. Xu. A robust upwind difference scheme for pricing perpetual American put options under stochastic volatility. International Journal of Computer Mathematics, 89(9):1135–1144,2012.
[11] H. P. McKean. Appendix: A free boundary problem for the heat equation arising from a problem in mathematical economics. Sloan Management Review, 6(2):32, 1965.
[12] E. Mordecki. Optimal stopping and perpetual options for lévy processes. Finance and Stochastics, 6(4):473–493, 2002.
[13] S. Salmi and J. Toivanen. An iterative method for pricing American options under jump–diffusion models. Applied Numerical Mathematics, 61:821–831, 2011.
[14] T. Jinying and Z. Zhenzhong . An explicit solution for perpetual American put options in a Markov–modulated jump–diffusion model. Progress in Applied Mathematics, 4(2):65–77, 2012.
[15] Q. Zhang and X. Guo. Closed–form solutions for perpetual American put options with regime–switching. SIAM Journal on Applied Mathematics, 64(6):2034–2049, 2004.