American Institute of Mathematical Sciences

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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 & 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.   Google Scholar

[2]

F. Black and M. Scholes, The pricing of options and corporate liabilities,, Journal of Political Economy, 81 (1973), 637.   Google Scholar

[3]

P. P. Boyle and K. S. Tan, Lure of the linear,, Risk, 7 (1994), 43.   Google Scholar

[4]

P. P Boyle and T. Vorst, Option replication in discrete time with transaction costs,, The Journal of Finance, 47 (1992), 271.   Google Scholar

[5]

L. Clewlow and S. Hodge, Optimal delta-hedging under transaction costs. Computational financial modelling,, Journal of Economic Dynamics and Control, 21 (1997), 1353.  doi: 10.1016/S0165-1889(97)00030-4.  Google Scholar

[6]

M. G. Crandall and P.-L. Lions, Viscosity solution of Hamilton-Jacobi equations,, Trans. Am. Math. Soc., 277 (1983), 1.  doi: 10.2307/1999343.  Google Scholar

[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.  doi: 10.1090/S0273-0979-1992-00266-5.  Google Scholar

[8]

A. Damgaard, Utility based option evaluation with proportional transaction costs,, Journal of Economic Dynamics and Control, 27 (2003), 667.  doi: 10.1016/S0165-1889(01)00068-9.  Google Scholar

[9]

A. Damgaard, Computation of reservation prices of options with proportional transaction costs,, Journal of Economic Dynamics and Control, 30 (2006), 415.  doi: 10.1016/j.jedc.2005.03.001.  Google Scholar

[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.  doi: 10.1137/0331022.  Google Scholar

[11]

M. H. A. Davis and T. Zariphopoulou, American options and transaction fees,, in, (1995).   Google Scholar

[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.   Google Scholar

[13]

S. Figlewski, Options arbitrage in imperfect markets,, The Journal of Finance, 44 (1989), 1289.   Google Scholar

[14]

W. H. Fleming and H. M. Soner, "Controlled Markov Processes and Viscosity Solutions,'', Applications of Mathematics (New York), 25 (1993).   Google Scholar

[15]

S. D. Hodges and A. Neuberger, Optimal replication of contingent claims under transaction costs,, Review of Futures Markets, 8 (1989), 222.   Google Scholar

[16]

C. C. Huang and S. Wang, A power penalty approach to a nonlinear complementarity problem,, Operations Research Letters, 38 (2010), 72.  doi: 10.1016/j.orl.2009.09.009.  Google Scholar

[17]

C. C. Huang and S. Wang, A penalty method for a mixed nonlinear complementarity problem,, Nonlinear Analysis, 75 (2012), 588.  doi: 10.1016/j.na.2011.08.061.  Google Scholar

[18]

M. A. Katsoulakis, Viscosity solutions of second order fully nonlinear elliptic equations with state constrains,, Indiana Univ. Math. J., 43 (1994), 493.  doi: 10.1512/iumj.1994.43.43020.  Google Scholar

[19]

H. E. Leland, Option pricing and replication with transaction costs,, The Journal of Finance, 40 (1985), 1283.   Google Scholar

[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.  doi: 10.1007/s10957-009-9559-7.  Google Scholar

[21]

W. Li and S. Wang, A numerical method for pricing European option with proportional transaction costs,, submitted., ().   Google Scholar

[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.  doi: 10.1016/S0165-1889(03)00059-9.  Google Scholar

[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.  doi: 10.3934/jimo.2010.6.161.  Google Scholar

[24]

H. M. Soner, Optimal control with state-space constraint. I,, SIAM J. Control Optimization., 24 (1986), 552.  doi: 10.1137/0324032.  Google Scholar

[25]

K. B. Toft, On the mean-variance tradeoff in option replication with transaction costs,, Journal of Financial and Quantitative Analysis, 31 (1996), 233.   Google Scholar

[26]

R. S. Varga, "Matrix Iterative Analysis,", Prentice-Hall, (1962).   Google Scholar

[27]

C. Vázquez, An upwind numerical approach for an American and European option pricing model,, Appl. Math. Comput., 97 (1998), 273.  doi: 10.1016/S0096-3003(97)10122-9.  Google Scholar

[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.  doi: 10.1023/A:1024980623095.  Google Scholar

[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.  doi: 10.1093/imanum/24.4.699.  Google Scholar

[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.  doi: 10.1007/s10957-006-9062-3.  Google Scholar

[31]

S. Wang and X. Yang, A power penalty method for linear complementarity problems,, Operations Research Letters, 36 (2008), 211.  doi: 10.1016/j.orl.2007.06.006.  Google Scholar

[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.  doi: 10.1016/j.jedc.2004.11.002.  Google Scholar

[33]

V. I. Zakamouline, American option pricing and exercising with transaction costs,, Journal of Computational Finance, 8 (2005), 81.   Google Scholar

[34]

K. Zhang and S. Wang, Convergence property of an interior penalty approach to pricing American option,, J. Ind. Manag. Optim., 7 (2011), 435.  doi: 10.3934/jimo.2011.7.435.  Google Scholar

[35]

K. Zhang and S. Wang, Pricing American bond options using a penalty method,, Automatica, 48 (2012), 472.  doi: 10.1016/j.automatica.2012.01.009.  Google Scholar

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.   Google Scholar

[2]

F. Black and M. Scholes, The pricing of options and corporate liabilities,, Journal of Political Economy, 81 (1973), 637.   Google Scholar

[3]

P. P. Boyle and K. S. Tan, Lure of the linear,, Risk, 7 (1994), 43.   Google Scholar

[4]

P. P Boyle and T. Vorst, Option replication in discrete time with transaction costs,, The Journal of Finance, 47 (1992), 271.   Google Scholar

[5]

L. Clewlow and S. Hodge, Optimal delta-hedging under transaction costs. Computational financial modelling,, Journal of Economic Dynamics and Control, 21 (1997), 1353.  doi: 10.1016/S0165-1889(97)00030-4.  Google Scholar

[6]

M. G. Crandall and P.-L. Lions, Viscosity solution of Hamilton-Jacobi equations,, Trans. Am. Math. Soc., 277 (1983), 1.  doi: 10.2307/1999343.  Google Scholar

[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.  doi: 10.1090/S0273-0979-1992-00266-5.  Google Scholar

[8]

A. Damgaard, Utility based option evaluation with proportional transaction costs,, Journal of Economic Dynamics and Control, 27 (2003), 667.  doi: 10.1016/S0165-1889(01)00068-9.  Google Scholar

[9]

A. Damgaard, Computation of reservation prices of options with proportional transaction costs,, Journal of Economic Dynamics and Control, 30 (2006), 415.  doi: 10.1016/j.jedc.2005.03.001.  Google Scholar

[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.  doi: 10.1137/0331022.  Google Scholar

[11]

M. H. A. Davis and T. Zariphopoulou, American options and transaction fees,, in, (1995).   Google Scholar

[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.   Google Scholar

[13]

S. Figlewski, Options arbitrage in imperfect markets,, The Journal of Finance, 44 (1989), 1289.   Google Scholar

[14]

W. H. Fleming and H. M. Soner, "Controlled Markov Processes and Viscosity Solutions,'', Applications of Mathematics (New York), 25 (1993).   Google Scholar

[15]

S. D. Hodges and A. Neuberger, Optimal replication of contingent claims under transaction costs,, Review of Futures Markets, 8 (1989), 222.   Google Scholar

[16]

C. C. Huang and S. Wang, A power penalty approach to a nonlinear complementarity problem,, Operations Research Letters, 38 (2010), 72.  doi: 10.1016/j.orl.2009.09.009.  Google Scholar

[17]

C. C. Huang and S. Wang, A penalty method for a mixed nonlinear complementarity problem,, Nonlinear Analysis, 75 (2012), 588.  doi: 10.1016/j.na.2011.08.061.  Google Scholar

[18]

M. A. Katsoulakis, Viscosity solutions of second order fully nonlinear elliptic equations with state constrains,, Indiana Univ. Math. J., 43 (1994), 493.  doi: 10.1512/iumj.1994.43.43020.  Google Scholar

[19]

H. E. Leland, Option pricing and replication with transaction costs,, The Journal of Finance, 40 (1985), 1283.   Google Scholar

[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.  doi: 10.1007/s10957-009-9559-7.  Google Scholar

[21]

W. Li and S. Wang, A numerical method for pricing European option with proportional transaction costs,, submitted., ().   Google Scholar

[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.  doi: 10.1016/S0165-1889(03)00059-9.  Google Scholar

[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.  doi: 10.3934/jimo.2010.6.161.  Google Scholar

[24]

H. M. Soner, Optimal control with state-space constraint. I,, SIAM J. Control Optimization., 24 (1986), 552.  doi: 10.1137/0324032.  Google Scholar

[25]

K. B. Toft, On the mean-variance tradeoff in option replication with transaction costs,, Journal of Financial and Quantitative Analysis, 31 (1996), 233.   Google Scholar

[26]

R. S. Varga, "Matrix Iterative Analysis,", Prentice-Hall, (1962).   Google Scholar

[27]

C. Vázquez, An upwind numerical approach for an American and European option pricing model,, Appl. Math. Comput., 97 (1998), 273.  doi: 10.1016/S0096-3003(97)10122-9.  Google Scholar

[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.  doi: 10.1023/A:1024980623095.  Google Scholar

[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.  doi: 10.1093/imanum/24.4.699.  Google Scholar

[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.  doi: 10.1007/s10957-006-9062-3.  Google Scholar

[31]

S. Wang and X. Yang, A power penalty method for linear complementarity problems,, Operations Research Letters, 36 (2008), 211.  doi: 10.1016/j.orl.2007.06.006.  Google Scholar

[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.  doi: 10.1016/j.jedc.2004.11.002.  Google Scholar

[33]

V. I. Zakamouline, American option pricing and exercising with transaction costs,, Journal of Computational Finance, 8 (2005), 81.   Google Scholar

[34]

K. Zhang and S. Wang, Convergence property of an interior penalty approach to pricing American option,, J. Ind. Manag. Optim., 7 (2011), 435.  doi: 10.3934/jimo.2011.7.435.  Google Scholar

[35]

K. Zhang and S. Wang, Pricing American bond options using a penalty method,, Automatica, 48 (2012), 472.  doi: 10.1016/j.automatica.2012.01.009.  Google Scholar

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