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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 |
References:
[1] |
G. Barles and P. E. Souganidis, Convergence of approximation schemes for fully nonlinear second order equations,, Asymptotic Analysis, 4 (1991), 271.
|
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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).
|
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[24] |
H. M. Soner, Optimal control with state-space constraint. I,, SIAM J. Control Optimization., 24 (1986), 552.
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. Google Scholar |
[26] |
R. S. Varga, "Matrix Iterative Analysis,", Prentice-Hall, (1962).
|
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
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.
|
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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).
|
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[24] |
H. M. Soner, Optimal control with state-space constraint. I,, SIAM J. Control Optimization., 24 (1986), 552.
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. Google Scholar |
[26] |
R. S. Varga, "Matrix Iterative Analysis,", Prentice-Hall, (1962).
|
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
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