Document Type : Research Article
Authors
Department of Applied Mathematics, Faculty of Mathematics and Computer, Shahid Bahonar University of Kerman, Kerman, Iran
Abstract
The European option can be exercised only at the expiration date while an American option can be exercised on or at any time before the expiration date.
In this paper, we will study the numerical solutions of a class of complex partial differential equations (PDE) systems with free boundary conditions. This kind of problems arise naturally in pricing (finite-maturity) American options, which is applies to a wide variety of asset price models including the constant elasticity of variance (CEV), hyper-exponential jump-diffusion (HEJD) and the finite moment log stable (FMLS) models. Developing efficient numerical schemes will have significant applications in finance computation. These equations have already been solve by the Hybrid Laplace transformfinite difference methods and the Laplace transform method(LTM). In this paper we will introduce a method to solve these equations by Tau method. Also, we will show that using this method will end up to a faster convergence. Numerical examples demonstrate the accuracy and velocity of the method in CEV models.
Keywords
[2] Carr, P. and Wu, L. (2003). The nite moment log stable process and option pricing. The journal of nance, 58(2):753{777.
[3] Cox, J. C. (1996). The constant elasticity of variance option pricing model. The Journal of Portfolio Management, 23(5):15{17.
[4] Cox, J. C. and Ross, S. A. (1976). The valuation of options for alternative stochas-tic processes. Journal of nancial economics, 3(1-2):145{166.
[5] Davydov, D. and Linetsky, V. (2001). The valuation and hedging of barrier and lookback options under the cev process. Management Science, 47(7):949{965.
[6] Detemple, J. (2005). American-style derivatives: Valuation and computation. Chapman and Hall/CRC.
[7] Hamilton, J. D. (1989). A new approach to the economic analysis of nonstationary time series and the business cycle. Econometrica: Journal of the Econometric Society, pages 357{384.
[8] Khaliq, A., Kleefeld, B., and Liu, R. (2013). Solving complex pde systems for pricing american options with regime-switching by efficient exponential time differencing schemes. Numerical Methods for Partial Differential Equations, 29(1):320{336.
[9] Khaliq, A. Q. and Liu, R. (2009). New numerical scheme for pricing american option with regime-switching. International Journal of Theoretical and Applied Finance, 12(03):319{340.
[10] Khaliq, A. Q., Voss, D. A., and Kazmi, K. (2008). Adaptive -methods for pricing american options. Journal of Computational and Applied Mathematics, 222(1):210{227.
Operational Research, 189(2):363{374.
[13] Lipton, A. (2002). Assets with jumps. Risk-London-Risk Magazine Limited,15(9):149{153.
[14] Longstaff, F. A. and Schwartz, E. S. (2001). Valuing american options by simulation: a simple least-squares approach. The review of nancial studies, 14(1):113{147.
[15] Ma, J., Zhou, Z., and Cui, Z. (2017). Hybrid laplace transform and nite difference methods for pricing american options under complex models. Computers & Mathematics with Applications, 74(3):369{384.
[16] Nielsen, B. F., Skavhaug, O., and Tveito, A. (2008). Penalty methods for the numerical solution of american multi-asset option problems. Journal of Computational and Applied Mathematics, 222(1):3{16.
[17] Ortiz, E. L. (1974). Canonical polynomials in the lanczos tau method. Studies in Numerical Analysis, pages 73{93.
[18] Rivaz, A., Mohdeni Moghadam, M., and Bani Asadi, S. (2019). Numerical solutions of black-scholes integro-differential equations with convergence analysis. Turkish Journal of Mathematics, 43(3):1080{1094.
[19] Rivaz, A., Mohseni Moghadam, M., Bani Asadi, S., and Zangoei Zadeh, S. (2017). Stochastic delay differential equation with the price jump in nance and its numerical solution. Stochastic Analysis and Applications, page submitted.
[20] Rogers, L. C. (2002). Monte carlo valuation of american options. Mathematical Finance, 12(3):271{286.
[21] Wong, H. Y. and Zhao, J. (2010). Valuing american options under the cev model by laplace{carson transforms. Operations Research Letters, 38(5):474{481.
[22] Zhou, Z. and Gao, X. (2016). Numerical methods for pricing american options with time-fractional pde models. Mathematical Problems in Engineering, 2016.
[23] Zhu, S.-P. (2006). A new analytical approximation formula for the optimal exercise boundary of american put options. International Journal of Theoretical and Applied Finance, 9(07):1141{1177.
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