# American Institute of Mathematical Sciences

January  2015, 11(1): 241-264. doi: 10.3934/jimo.2015.11.241

## A finite difference method for pricing European and American options under a geometric Lévy process

 1 School of Mathematics & Statistics, The University of Western Australia, 35 Stirling Highway, Crawley, WA 6009, Australia

Received  September 2013 Revised  January 2014 Published  May 2014

In this paper we develop a numerical approach to a fractional-order differential Linear Complementarity Problem (LCP) arising in pricing European and American options under a geometric Lévy process. The LCP is first approximated by a nonlinear penalty fractional Black-Scholes (fBS) equation. We then propose a finite difference scheme for the penalty fBS equation. We show that both the continuous and the discretized fBS equations are uniquely solvable and establish the convergence of the numerical solution to the viscosity solution of the penalty fBS equation by proving the consistency, stability and monotonicity of the numerical scheme. We also show that the discretization has the 2nd-order truncation error in both the spatial and time mesh sizes. Numerical results are presented to demonstrate the accuracy and usefulness of the numerical method for pricing both European and American options under the geometric Lévy process.
Citation: Wen Chen, Song Wang. A finite difference method for pricing European and American options under a geometric Lévy process. Journal of Industrial and Management Optimization, 2015, 11 (1) : 241-264. doi: 10.3934/jimo.2015.11.241
##### References:
 [1] L. Angermann and S. Wang, Convergence of a fitted finite volume method for the penalized BlackScholes equation governing European and American Option pricing, Numerische Mathematik, 106 (2007), 1-40. doi: 10.1007/s00211-006-0057-7. [2] F. Black and M. Scholes, The pricing of options and corporate liabilities, The Journal of Political Economy, 81 (1973), 637-654. [3] G. Barles, Convergence of numerical schemes for degenerate parabolic equations arising in finance theory, Numerical methods in finance, Cambridge University Press, (1997), 1-21. [4] S. I. Boyarchenko and S. Levendorskii, Non-Gaussian Merton-Black-Scholes Theory, World Scientific Singapore, 2002. doi: 10.1142/9789812777485. [5] P. Carr, H. Geman, D. B. Madan and M. Yor, The fine structure of asset returns: An empirical investigation, The Journal of Business, 75 (2002), 305-332. doi: 10.1086/338705. [6] A. Cartea and D. del-Castillo-Negrete, Fractional diffusion models of option prices in markets with jumps, Physica A: Statistical Mechanics and its Applications, 374 (2007), 749-763. doi: 10.1016/j.physa.2006.08.071. [7] C. M. Chen, F. Liu, I. Turner and V. Anh, A Fourier method for the fractional diffusion equation describing sub-diffusion, Journal of Computational Physics, 227 (2007), 886-897. doi: 10.1016/j.jcp.2007.05.012. [8] M. Chen and C. Huang, A power penalty method for the general traffic assignment problem with elastic demand, Journal of Industrial and Management Optimization, 10 (2014), 1019-1030. doi: 10.3934/jimo.2014.10.1019. [9] W. Chen and S. Wang, A penalty method for a fractional order parabolic variational inequality governing American put option valuation, Computers and Mathematics with Applications, 67 (2014), 77-90. doi: 10.1016/j.camwa.2013.10.007. [10] R. Cont and P. Tankov, Financial Modelling With Jump Processes, Chapman & Hall, 2004. [11] R. Cont and E. Voltchkova, A finite difference scheme for option pricing in jump-diffusion and exponential Lévy models, SIAM Journal on Numerical Analysis, 43 (2005), 1596-1626. doi: 10.1137/S0036142903436186. [12] V. J. Ervin and J. P. Roop, Variational formulation for the stationary fractional advection dispersion equation, Numerical Methods for Partial Differential Equations, 22 (2006), 558-576. doi: 10.1002/num.20112. [13] C. Huang and S. Wang, A power penalty approach to a nonlinear complementary problem, Operations Research Letters, 38 (2010), 72-76. doi: 10.1016/j.orl.2009.09.009. [14] C. Huang and S. Wang, A penalty method for a mixed nonlinear complementarity problem, Nonlinear Analysis, 75 (2012), 588-597. doi: 10.1016/j.na.2011.08.061. [15] D. C. Lesmana and S. Wang, An upwind finite difference method for a nonlinear Black-Scholes equation governing European option valuation, Applied Mathematics and Computation, 219 (2013), 8818-8828. doi: 10.1016/j.amc.2012.12.077. [16] W. Li and S. Wang, Penalty approach to the HJB equation arising in European stock option pricing with proportional transaction costs, Journal of Optimization Theory and Applications, 143 (2009), 279-293. doi: 10.1007/s10957-009-9559-7. [17] W. Li and S. Wang, Pricing American options under proportional transaction costs using a penalty approach and a finite difference scheme, Journal of Industrial and Management Optimization, 9 (2013), 365-389. doi: 10.3934/jimo.2013.9.365. [18] V. E. Lynch, B. A. Carreras, D. del-Castillo-Negrete, K. M. Ferreira-Mejias and H. R. Hicks, Numerical methods for the solution of partial differential equations of fractional order, Journal of Computational Physics, 192 (2003), 406-421. doi: 10.1016/j.jcp.2003.07.008. [19] K. S. Miller and B. Ross, An Introduction to the Fractional Calculus and Fractional Differential Equations, A Wiley-Interscience Publication. John Wiley & Sons, Inc., New York, 1993. [20] K. B. Oldham and J. Spanier, The fractional calculus, Theory and applications of differentiation and integration to arbitrary order. With an annotated chronological bibliography by Bertram Ross. Mathematics in Science and Engineering, 111 (1974), 76-81. [21] C. Tadjeran, M. M. Meerschaert and H. P. Scheffler, A second-order accurate numerical approximation for the fractional diffusion equation, Journal of Computational Physics, 213 (2006), 205-213. doi: 10.1016/j.jcp.2005.08.008. [22] C. Tadjeran and M. M. Meerschaert, A second-order accurate numerical method for the two-dimensional fractional diffusion equation, Journal of Computational Physics, 220 (2007), 813-823. doi: 10.1016/j.jcp.2006.05.030. [23] S. Wang, A novel fitted finite volume method for the Black-Scholes equation governing option pricing, IMA Journal of Numerical Analysis, 24 (2004), 699-720. doi: 10.1093/imanum/24.4.699. [24] S. Wang, X. Q. Yang and K. L. Teo, Power penalty method for a linear complementarity problem arising from American option valuation, Journal of Optimization Theory and Applications, 129 (2006), 227-254. doi: 10.1007/s10957-006-9062-3. [25] S. Wang and X. Q. Yang, A power penalty method for linear complementarity problems, Operations Research Letters, 36 (2008), 211-214. doi: 10.1016/j.orl.2007.06.006. [26] S. B. Yuste, Weighted average finite difference methods for fractional diffusion equations, Journal of Computational Physics, 216 (2006), 264-274. doi: 10.1016/j.jcp.2005.12.006. [27] P. Wilmott, J. Dewynne and S. Howison, Option Pricing: Mathematical Models and Computation. Oxford Financial Press, Oxford, 1993. [28] K. Zhang and S. Wang, Convergence property of an interior penalty approach to pricing American option, Journal of Industrial and Management Optimization, 7 (2011), 435-447. doi: 10.3934/jimo.2011.7.435. [29] K. Zhang and S. Wang, Pricing American bond options using a penalty method, Automatica, 48 (2012), 472-479. doi: 10.1016/j.automatica.2012.01.009. [30] R. Zvan, P. A. Forsyth P. A. and K. R. Vetzal, Penalty methods for American options with stochastic volatility, Comput. Appl. Math., 91 (1998), 199-218. doi: 10.1016/S0377-0427(98)00037-5.

show all references

##### References:
 [1] L. Angermann and S. Wang, Convergence of a fitted finite volume method for the penalized BlackScholes equation governing European and American Option pricing, Numerische Mathematik, 106 (2007), 1-40. doi: 10.1007/s00211-006-0057-7. [2] F. Black and M. Scholes, The pricing of options and corporate liabilities, The Journal of Political Economy, 81 (1973), 637-654. [3] G. Barles, Convergence of numerical schemes for degenerate parabolic equations arising in finance theory, Numerical methods in finance, Cambridge University Press, (1997), 1-21. [4] S. I. Boyarchenko and S. Levendorskii, Non-Gaussian Merton-Black-Scholes Theory, World Scientific Singapore, 2002. doi: 10.1142/9789812777485. [5] P. Carr, H. Geman, D. B. Madan and M. Yor, The fine structure of asset returns: An empirical investigation, The Journal of Business, 75 (2002), 305-332. doi: 10.1086/338705. [6] A. Cartea and D. del-Castillo-Negrete, Fractional diffusion models of option prices in markets with jumps, Physica A: Statistical Mechanics and its Applications, 374 (2007), 749-763. doi: 10.1016/j.physa.2006.08.071. [7] C. M. Chen, F. Liu, I. Turner and V. Anh, A Fourier method for the fractional diffusion equation describing sub-diffusion, Journal of Computational Physics, 227 (2007), 886-897. doi: 10.1016/j.jcp.2007.05.012. [8] M. Chen and C. Huang, A power penalty method for the general traffic assignment problem with elastic demand, Journal of Industrial and Management Optimization, 10 (2014), 1019-1030. doi: 10.3934/jimo.2014.10.1019. [9] W. Chen and S. Wang, A penalty method for a fractional order parabolic variational inequality governing American put option valuation, Computers and Mathematics with Applications, 67 (2014), 77-90. doi: 10.1016/j.camwa.2013.10.007. [10] R. Cont and P. Tankov, Financial Modelling With Jump Processes, Chapman & Hall, 2004. [11] R. Cont and E. Voltchkova, A finite difference scheme for option pricing in jump-diffusion and exponential Lévy models, SIAM Journal on Numerical Analysis, 43 (2005), 1596-1626. doi: 10.1137/S0036142903436186. [12] V. J. Ervin and J. P. Roop, Variational formulation for the stationary fractional advection dispersion equation, Numerical Methods for Partial Differential Equations, 22 (2006), 558-576. doi: 10.1002/num.20112. [13] C. Huang and S. Wang, A power penalty approach to a nonlinear complementary problem, Operations Research Letters, 38 (2010), 72-76. doi: 10.1016/j.orl.2009.09.009. [14] C. Huang and S. Wang, A penalty method for a mixed nonlinear complementarity problem, Nonlinear Analysis, 75 (2012), 588-597. doi: 10.1016/j.na.2011.08.061. [15] D. C. Lesmana and S. Wang, An upwind finite difference method for a nonlinear Black-Scholes equation governing European option valuation, Applied Mathematics and Computation, 219 (2013), 8818-8828. doi: 10.1016/j.amc.2012.12.077. [16] W. Li and S. Wang, Penalty approach to the HJB equation arising in European stock option pricing with proportional transaction costs, Journal of Optimization Theory and Applications, 143 (2009), 279-293. doi: 10.1007/s10957-009-9559-7. [17] W. Li and S. Wang, Pricing American options under proportional transaction costs using a penalty approach and a finite difference scheme, Journal of Industrial and Management Optimization, 9 (2013), 365-389. doi: 10.3934/jimo.2013.9.365. [18] V. E. Lynch, B. A. Carreras, D. del-Castillo-Negrete, K. M. Ferreira-Mejias and H. R. Hicks, Numerical methods for the solution of partial differential equations of fractional order, Journal of Computational Physics, 192 (2003), 406-421. doi: 10.1016/j.jcp.2003.07.008. [19] K. S. Miller and B. Ross, An Introduction to the Fractional Calculus and Fractional Differential Equations, A Wiley-Interscience Publication. John Wiley & Sons, Inc., New York, 1993. [20] K. B. Oldham and J. Spanier, The fractional calculus, Theory and applications of differentiation and integration to arbitrary order. With an annotated chronological bibliography by Bertram Ross. Mathematics in Science and Engineering, 111 (1974), 76-81. [21] C. Tadjeran, M. M. Meerschaert and H. P. Scheffler, A second-order accurate numerical approximation for the fractional diffusion equation, Journal of Computational Physics, 213 (2006), 205-213. doi: 10.1016/j.jcp.2005.08.008. [22] C. Tadjeran and M. M. Meerschaert, A second-order accurate numerical method for the two-dimensional fractional diffusion equation, Journal of Computational Physics, 220 (2007), 813-823. doi: 10.1016/j.jcp.2006.05.030. [23] S. Wang, A novel fitted finite volume method for the Black-Scholes equation governing option pricing, IMA Journal of Numerical Analysis, 24 (2004), 699-720. doi: 10.1093/imanum/24.4.699. [24] S. Wang, X. Q. Yang and K. L. Teo, Power penalty method for a linear complementarity problem arising from American option valuation, Journal of Optimization Theory and Applications, 129 (2006), 227-254. doi: 10.1007/s10957-006-9062-3. [25] S. Wang and X. Q. Yang, A power penalty method for linear complementarity problems, Operations Research Letters, 36 (2008), 211-214. doi: 10.1016/j.orl.2007.06.006. [26] S. B. Yuste, Weighted average finite difference methods for fractional diffusion equations, Journal of Computational Physics, 216 (2006), 264-274. doi: 10.1016/j.jcp.2005.12.006. [27] P. Wilmott, J. Dewynne and S. Howison, Option Pricing: Mathematical Models and Computation. Oxford Financial Press, Oxford, 1993. [28] K. Zhang and S. Wang, Convergence property of an interior penalty approach to pricing American option, Journal of Industrial and Management Optimization, 7 (2011), 435-447. doi: 10.3934/jimo.2011.7.435. [29] K. Zhang and S. Wang, Pricing American bond options using a penalty method, Automatica, 48 (2012), 472-479. doi: 10.1016/j.automatica.2012.01.009. [30] R. Zvan, P. A. Forsyth P. A. and K. R. Vetzal, Penalty methods for American options with stochastic volatility, Comput. Appl. Math., 91 (1998), 199-218. doi: 10.1016/S0377-0427(98)00037-5.
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