American Institute of Mathematical Sciences

Advanced
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

Wen Li 1, and  Song Wang 1,

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.
Keywords: American option valuation, upwind finite difference method., penalty methods, HJB equations.
Mathematics Subject Classification: Primary: 49L20, 65M06; Secondary: 91G6.
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.

show all references

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.

[1]

Junkee Jeon, Jehan Oh. Valuation of American strangle option: Variational inequality approach. Discrete and Continuous Dynamical Systems - B, 2019, 24 (2) : 755-781. doi: 10.3934/dcdsb.2018206
[2]

Kai Zhang, Song Wang. Convergence property of an interior penalty approach to pricing American option. Journal of Industrial and Management Optimization, 2011, 7 (2) : 435-447. doi: 10.3934/jimo.2011.7.435
[3]

Kai Zhang, Xiaoqi Yang, Kok Lay Teo. A power penalty approach to american option pricing with jump diffusion processes. Journal of Industrial and Management Optimization, 2008, 4 (4) : 783-799. doi: 10.3934/jimo.2008.4.783
[4]

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
[5]

Xiaohai Wan, Zhilin Li. Some new finite difference methods for Helmholtz equations on irregular domains or with interfaces. Discrete and Continuous Dynamical Systems - B, 2012, 17 (4) : 1155-1174. doi: 10.3934/dcdsb.2012.17.1155
[6]

Lunji Song, Wenya Qi, Kaifang Liu, Qingxian Gu. A new over-penalized weak galerkin finite element method. Part Ⅱ: Elliptic interface problems. Discrete and Continuous Dynamical Systems - B, 2021, 26 (5) : 2581-2598. doi: 10.3934/dcdsb.2020196
[7]

Kai Zhang, Kok Lay Teo. A penalty-based method from reconstructing smooth local volatility surface from American options. Journal of Industrial and Management Optimization, 2015, 11 (2) : 631-644. doi: 10.3934/jimo.2015.11.631
[8]

Na Song, Yue Xie, Wai-Ki Ching, Tak-Kuen Siu. A real option approach for investment opportunity valuation. Journal of Industrial and Management Optimization, 2017, 13 (3) : 1213-1235. doi: 10.3934/jimo.2016069
[9]

Qing-Qing Yang, Wai-Ki Ching, Wan-Hua He, Na Song. Effect of institutional deleveraging on option valuation problems. Journal of Industrial and Management Optimization, 2021, 17 (4) : 2097-2118. doi: 10.3934/jimo.2020060
[10]

Xinfu Chen, Huibin Cheng. Regularity of the free boundary for the American put option. Discrete and Continuous Dynamical Systems - B, 2012, 17 (6) : 1751-1759. doi: 10.3934/dcdsb.2012.17.1751
[11]

Zalman Balanov, Carlos García-Azpeitia, Wieslaw Krawcewicz. On variational and topological methods in nonlinear difference equations. Communications on Pure and Applied Analysis, 2018, 17 (6) : 2813-2844. doi: 10.3934/cpaa.2018133
[12]

Xuefei He, Kun Wang, Liwei Xu. Efficient finite difference methods for the nonlinear Helmholtz equation in Kerr medium. Electronic Research Archive, 2020, 28 (4) : 1503-1528. doi: 10.3934/era.2020079
[13]

Baojun Bian, Shuntai Hu, Quan Yuan, Harry Zheng. Constrained viscosity solution to the HJB equation arising in perpetual American employee stock options pricing. Discrete and Continuous Dynamical Systems, 2015, 35 (11) : 5413-5433. doi: 10.3934/dcds.2015.35.5413
[14]

Xinfu Chen, Bei Hu, Jin Liang, Yajing Zhang. Convergence rate of free boundary of numerical scheme for American option. Discrete and Continuous Dynamical Systems - B, 2016, 21 (5) : 1435-1444. doi: 10.3934/dcdsb.2016004
[15]

Junxiang Li, Yan Gao, Tao Dai, Chunming Ye, Qiang Su, Jiazhen Huo. Substitution secant/finite difference method to large sparse minimax problems. Journal of Industrial and Management Optimization, 2014, 10 (2) : 637-663. doi: 10.3934/jimo.2014.10.637
[16]

Hongsong Feng, Shan Zhao. A multigrid based finite difference method for solving parabolic interface problem. Electronic Research Archive, 2021, 29 (5) : 3141-3170. doi: 10.3934/era.2021031
[17]

Brittany Froese Hamfeldt, Jacob Lesniewski. A convergent finite difference method for computing minimal Lagrangian graphs. Communications on Pure and Applied Analysis, 2022, 21 (2) : 393-418. doi: 10.3934/cpaa.2021182
[18]

Jinye Shen, Xian-Ming Gu. Two finite difference methods based on an H2N2 interpolation for two-dimensional time fractional mixed diffusion and diffusion-wave equations. Discrete and Continuous Dynamical Systems - B, 2022, 27 (2) : 1179-1207. doi: 10.3934/dcdsb.2021086
[19]

Nan Li, Song Wang, Shuhua Zhang. Pricing options on investment project contraction and ownership transfer using a finite volume scheme and an interior penalty method. Journal of Industrial and Management Optimization, 2020, 16 (3) : 1349-1368. doi: 10.3934/jimo.2019006
[20]

Xiaozhong Yang, Xinlong Liu. Numerical analysis of two new finite difference methods for time-fractional telegraph equation. Discrete and Continuous Dynamical Systems - B, 2021, 26 (7) : 3921-3942. doi: 10.3934/dcdsb.2020269

2021 Impact Factor: 1.411

Tools

Metrics

  • PDF downloads (188)
  • HTML views (0)
  • Cited by (18)

Other articles
by authors

[Back to Top]

Copyright © 2022 American Institute of Mathematical Sciences | Privacy Policy