# American Institute of Mathematical Sciences

April  2013, 9(2): 365-389. doi: 10.3934/jimo.2013.9.365

## Pricing American options under proportional transaction costs using a penalty approach and a finite difference scheme

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

Received  January 2012 Revised  May 2012 Published  February 2013

In this paper we propose a penalty method combined with a finite difference scheme for the Hamilton-Jacobi-Bellman (HJB) equation arising in pricing American options under proportional transaction costs. In this method, the HJB equation is approximated by a nonlinear partial differential equation with penalty terms. We prove that the viscosity solution to the penalty equation converges to that of the original HJB equation when the penalty parameter tends to positive infinity. We then present an upwind finite difference scheme for solving the penalty equation and show that the approximate solution from the scheme converges to the viscosity solution of the penalty equation. A numerical algorithm for solving the discretized nonlinear system is proposed and analyzed. Numerical results are presented to demonstrate the accuracy of the method.
Citation: Wen Li, Song Wang. Pricing American options under proportional transaction costs using a penalty approach and a finite difference scheme. Journal of Industrial and Management Optimization, 2013, 9 (2) : 365-389. doi: 10.3934/jimo.2013.9.365
##### References:
 [1] G. Barles and P. E. Souganidis, Convergence of approximation schemes for fully nonlinear second order equations, Asymptotic Analysis, 4 (1991), 271-283. [2] F. Black and M. Scholes, The pricing of options and corporate liabilities, Journal of Political Economy, 81 (1973), 637-654. [3] P. P. Boyle and K. S. Tan, Lure of the linear, Risk, 7 (1994), 43-46. [4] P. P Boyle and T. Vorst, Option replication in discrete time with transaction costs, The Journal of Finance, 47 (1992), 271-293. [5] L. Clewlow and S. Hodge, Optimal delta-hedging under transaction costs. Computational financial modelling, Journal of Economic Dynamics and Control, 21 (1997), 1353-1376. doi: 10.1016/S0165-1889(97)00030-4. [6] M. G. Crandall and P.-L. Lions, Viscosity solution of Hamilton-Jacobi equations, Trans. Am. Math. Soc., 277 (1983), 1-42. doi: 10.2307/1999343. [7] M. G. Crandall, H. Ishii and P.-L. Lions, User's guide to viscosity solutions of second order partial differential equations, Bull. Amer. Math. Soc. (N.S.), 27 (1992), 1-67. doi: 10.1090/S0273-0979-1992-00266-5. [8] A. Damgaard, Utility based option evaluation with proportional transaction costs, Journal of Economic Dynamics and Control, 27 (2003), 667-700. doi: 10.1016/S0165-1889(01)00068-9. [9] A. Damgaard, Computation of reservation prices of options with proportional transaction costs, Journal of Economic Dynamics and Control, 30 (2006), 415-444. doi: 10.1016/j.jedc.2005.03.001. [10] M. H. A. Davis, V. G. Panas and T. Zariphopoulou, European option pricing with transaction costs, SIAM J. Control and Optimization, 31 (1993), 470-493. doi: 10.1137/0331022. [11] M. H. A. Davis and T. Zariphopoulou, American options and transaction fees, in "Mathemtical Finance" (eds. M. H. A. Davis, et al.), Springer-Verlag, 1995. [12] C. Edirisinghe, V. Naik and R. Uppal, Optimal replication of options with transaction costs and trading restrictions, Journal of Financial and Quantitative Analysis, 28 (1993), 117-138. [13] S. Figlewski, Options arbitrage in imperfect markets, The Journal of Finance, 44 (1989), 1289-1311. [14] W. H. Fleming and H. M. Soner, "Controlled Markov Processes and Viscosity Solutions,'' Applications of Mathematics (New York), 25, Springer-Verlag, New York, 1993. [15] S. D. Hodges and A. Neuberger, Optimal replication of contingent claims under transaction costs, Review of Futures Markets, 8 (1989), 222-239. [16] C. C. Huang and S. Wang, A power penalty approach to a nonlinear complementarity problem, Operations Research Letters, 38 (2010), 72-76. doi: 10.1016/j.orl.2009.09.009. [17] C. 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. [18] M. A. Katsoulakis, Viscosity solutions of second order fully nonlinear elliptic equations with state constrains, Indiana Univ. Math. J., 43 (1994), 493-518. doi: 10.1512/iumj.1994.43.43020. [19] H. E. Leland, Option pricing and replication with transaction costs, The Journal of Finance, 40 (1985), 1283-1301. [20] 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. [21] W. Li and S. Wang, A numerical method for pricing European option with proportional transaction costs, submitted. [22] M. Monoyios, Option pricing with transaction costs using a Markov chain approximation. Financial decision models in a dynamical setting, Journal of Economic Dynamics and Control, 28 (2004), 889-913. doi: 10.1016/S0165-1889(03)00059-9. [23] S. Richardson and S. Wang, The viscosity approximation to the Hamilton-Jacobi-Bellman equation in optimal feedback control: Upper bounds for extended domains, J. Ind. Manag. Optim., 6 (2010), 161-175. doi: 10.3934/jimo.2010.6.161. [24] H. M. Soner, Optimal control with state-space constraint. I, SIAM J. Control Optimization., 24 (1986), 552-561. doi: 10.1137/0324032. [25] K. B. Toft, On the mean-variance tradeoff in option replication with transaction costs, Journal of Financial and Quantitative Analysis, 31 (1996), 233-262. [26] R. S. Varga, "Matrix Iterative Analysis," Prentice-Hall, Inc., Englewood Cliffs, N.J., 1962. [27] C. Vázquez, An upwind numerical approach for an American and European option pricing model, Appl. Math. Comput., 97 (1998), 273-286. doi: 10.1016/S0096-3003(97)10122-9. [28] S. Wang, L. S. Jennings and K. L. Teo, Numerical solution of Hamilton-Jacobi-Bellman equations by an upwind finite volume method, Journal of Global Optimization, 27 (2003), 177-192. doi: 10.1023/A:1024980623095. [29] S. Wang, A novel fitted finite volume method for the Black-Scholes equations governing option pricing, IMA Journal of Numerical Analysis, 24 (2004), 699-720. doi: 10.1093/imanum/24.4.699. [30] 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. [31] S. Wang and X. Yang, A power penalty method for linear complementarity problems, Operations Research Letters, 36 (2008), 211-217. doi: 10.1016/j.orl.2007.06.006. [32] V. I. Zakamouline, European option pricing and hedging with both fixed and proportional transaction costs, Journal of Economic Dynamics and Control, 30 (2006), 1-25. doi: 10.1016/j.jedc.2004.11.002. [33] V. I. Zakamouline, American option pricing and exercising with transaction costs, Journal of Computational Finance, 8 (2005), 81-115. [34] K. Zhang and S. Wang, Convergence property of an interior penalty approach to pricing American option, J. Ind. Manag. Optim., 7 (2011), 435-447. doi: 10.3934/jimo.2011.7.435. [35] 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.

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##### References:
 [1] G. Barles and P. E. Souganidis, Convergence of approximation schemes for fully nonlinear second order equations, Asymptotic Analysis, 4 (1991), 271-283. [2] F. Black and M. Scholes, The pricing of options and corporate liabilities, Journal of Political Economy, 81 (1973), 637-654. [3] P. P. Boyle and K. S. Tan, Lure of the linear, Risk, 7 (1994), 43-46. [4] P. P Boyle and T. Vorst, Option replication in discrete time with transaction costs, The Journal of Finance, 47 (1992), 271-293. [5] L. Clewlow and S. Hodge, Optimal delta-hedging under transaction costs. Computational financial modelling, Journal of Economic Dynamics and Control, 21 (1997), 1353-1376. doi: 10.1016/S0165-1889(97)00030-4. [6] M. G. Crandall and P.-L. Lions, Viscosity solution of Hamilton-Jacobi equations, Trans. Am. Math. Soc., 277 (1983), 1-42. doi: 10.2307/1999343. [7] M. G. Crandall, H. Ishii and P.-L. Lions, User's guide to viscosity solutions of second order partial differential equations, Bull. Amer. Math. Soc. (N.S.), 27 (1992), 1-67. doi: 10.1090/S0273-0979-1992-00266-5. [8] A. Damgaard, Utility based option evaluation with proportional transaction costs, Journal of Economic Dynamics and Control, 27 (2003), 667-700. doi: 10.1016/S0165-1889(01)00068-9. [9] A. Damgaard, Computation of reservation prices of options with proportional transaction costs, Journal of Economic Dynamics and Control, 30 (2006), 415-444. doi: 10.1016/j.jedc.2005.03.001. [10] M. H. A. Davis, V. G. Panas and T. Zariphopoulou, European option pricing with transaction costs, SIAM J. Control and Optimization, 31 (1993), 470-493. doi: 10.1137/0331022. [11] M. H. A. Davis and T. Zariphopoulou, American options and transaction fees, in "Mathemtical Finance" (eds. M. H. A. Davis, et al.), Springer-Verlag, 1995. [12] C. Edirisinghe, V. Naik and R. Uppal, Optimal replication of options with transaction costs and trading restrictions, Journal of Financial and Quantitative Analysis, 28 (1993), 117-138. [13] S. Figlewski, Options arbitrage in imperfect markets, The Journal of Finance, 44 (1989), 1289-1311. [14] W. H. Fleming and H. M. Soner, "Controlled Markov Processes and Viscosity Solutions,'' Applications of Mathematics (New York), 25, Springer-Verlag, New York, 1993. [15] S. D. Hodges and A. Neuberger, Optimal replication of contingent claims under transaction costs, Review of Futures Markets, 8 (1989), 222-239. [16] C. C. Huang and S. Wang, A power penalty approach to a nonlinear complementarity problem, Operations Research Letters, 38 (2010), 72-76. doi: 10.1016/j.orl.2009.09.009. [17] C. 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. [18] M. A. Katsoulakis, Viscosity solutions of second order fully nonlinear elliptic equations with state constrains, Indiana Univ. Math. J., 43 (1994), 493-518. doi: 10.1512/iumj.1994.43.43020. [19] H. E. Leland, Option pricing and replication with transaction costs, The Journal of Finance, 40 (1985), 1283-1301. [20] 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. [21] W. Li and S. Wang, A numerical method for pricing European option with proportional transaction costs, submitted. [22] M. Monoyios, Option pricing with transaction costs using a Markov chain approximation. Financial decision models in a dynamical setting, Journal of Economic Dynamics and Control, 28 (2004), 889-913. doi: 10.1016/S0165-1889(03)00059-9. [23] S. Richardson and S. Wang, The viscosity approximation to the Hamilton-Jacobi-Bellman equation in optimal feedback control: Upper bounds for extended domains, J. Ind. Manag. Optim., 6 (2010), 161-175. doi: 10.3934/jimo.2010.6.161. [24] H. M. Soner, Optimal control with state-space constraint. I, SIAM J. Control Optimization., 24 (1986), 552-561. doi: 10.1137/0324032. [25] K. B. Toft, On the mean-variance tradeoff in option replication with transaction costs, Journal of Financial and Quantitative Analysis, 31 (1996), 233-262. [26] R. S. Varga, "Matrix Iterative Analysis," Prentice-Hall, Inc., Englewood Cliffs, N.J., 1962. [27] C. Vázquez, An upwind numerical approach for an American and European option pricing model, Appl. Math. Comput., 97 (1998), 273-286. doi: 10.1016/S0096-3003(97)10122-9. [28] S. Wang, L. S. Jennings and K. L. Teo, Numerical solution of Hamilton-Jacobi-Bellman equations by an upwind finite volume method, Journal of Global Optimization, 27 (2003), 177-192. doi: 10.1023/A:1024980623095. [29] S. Wang, A novel fitted finite volume method for the Black-Scholes equations governing option pricing, IMA Journal of Numerical Analysis, 24 (2004), 699-720. doi: 10.1093/imanum/24.4.699. [30] 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. [31] S. Wang and X. Yang, A power penalty method for linear complementarity problems, Operations Research Letters, 36 (2008), 211-217. doi: 10.1016/j.orl.2007.06.006. [32] V. I. Zakamouline, European option pricing and hedging with both fixed and proportional transaction costs, Journal of Economic Dynamics and Control, 30 (2006), 1-25. doi: 10.1016/j.jedc.2004.11.002. [33] V. I. Zakamouline, American option pricing and exercising with transaction costs, Journal of Computational Finance, 8 (2005), 81-115. [34] K. Zhang and S. Wang, Convergence property of an interior penalty approach to pricing American option, J. Ind. Manag. Optim., 7 (2011), 435-447. doi: 10.3934/jimo.2011.7.435. [35] 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.
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