# American Institute of Mathematical Sciences

2017, 13(4): 1793-1813. doi: 10.3934/jimo.2017019

## A numerical scheme for pricing American options with transaction costs under a jump diffusion process

 1 Department of Mathematics, Bogor Agricultural University, Kampus IPB Darmaga, Bogor, Jawa Barat 16680, Indonesia 2 Department of of Mathematics & Statistics, Curtin University, GPO Box U1987, WA 6845, Australia

Received  April 2016 Revised  July 2016 Published  December 2016

In this paper we develop a numerical method for a nonlinear partial integro-differential complementarity problem arising from pricing American options with transaction costs when the underlying assets follow a jump diffusion process. We first approximate the complementarity problem by a nonlinear partial integro-differential equation (PIDE) using a penalty approach. The PIDE is then discretized by a combination of a spatial upwind finite differencing and a fully implicit time stepping scheme. We prove that the coefficient matrix of the system from this scheme is an M-matrix and that the approximate solution converges to the viscosity solution to the PIDE by showing that the scheme is consistent, monotone, and unconditionally stable. We also propose a Newton's iterative method coupled with a Fast Fourier Transform for the computation of the discretized integral term for solving the fully discretized system. Numerical results will be presented to demonstrate the convergence rates and usefulness of this method.

Citation: Donny Citra Lesmana, Song Wang. A numerical scheme for pricing American options with transaction costs under a jump diffusion process. Journal of Industrial & Management Optimization, 2017, 13 (4) : 1793-1813. doi: 10.3934/jimo.2017019
##### References:
 [1] A. Almendral and C. W. Oosterlee, Numerical valuation of options with jumps in the underlying, Appl. Math. Comput., 53 (2005), 1-18. doi: 10.1016/j.apnum.2004.08.037. [2] A. Anderson and J. Andresen, Jump diffusion process: volatility smile fitting and numerical methods for option pricing, Rev. Derivat. Res., 4 (2000), 231-262. [3] J. Ankudinova and M. Ehrhardt, On the numerical solution of nonlinear Black-Scholes equations, Computers and Mathematics with Applications, 56 (2008), 799-812. doi: 10.1016/j.camwa.2008.02.005. [4] C. G. Averbuj, Nonlinear differential evolution equation arising in option pricing when including transaction costs: a viscosity solution approach, R. Bras. Eco. de Emp., 12 (2012), 81-90. [5] G. Barles, Convergence of numerical schemes for degenerate parabolic equations arising in finance theory, in: L. C. G. Rogers, D. Talay (Eds), Numerical Methods in Finance, Cambridge University Press, Cambridge, (1997), 1-21. [6] F. Black and M. Scholes, The pricing of options and corporate liabilities, The Journal of Political Economy, 81 (1973), 637-654. doi: 10.1086/260062. [7] W. Chen and S. Wang, A penalty method for a fractional order parabolic variational inequality governing American put option valuation, Computers & Mathematics with Applications, 67 (2014), 77-90. doi: 10.1016/j.camwa.2013.10.007. [8] W. Chen and S. Wang, A finite difference method for pricing European and American options under a geometric Levy process, Journal of Industrial and Management Optimization, 11 (2015), 241-264. doi: 10.3934/jimo.2015.11.241. [9] R. Company, L. Jodar and J. R. Pintos, A numerical method for European option pricing with transaction costs nonlinear equation, Mathematics and Computer Modelling, 50 (2009), 910-920. doi: 10.1016/j.mcm.2009.05.019. [10] R. Cont and P. Tankov, Financial Modelling with Jump Processes, Chapman and Hall/CRC, Boca Raton, FL, 2004. [11] R. Cont and E. Voltchkova, A finite difference scheme for option pricing in jump diffusion and exponential lěvy models, SIAM J. Numer. Anal., 43 (2005), 1596-1626. doi: 10.1137/S0036142903436186. [12] B. Dupire, Pricing with a smile, Risk, 7 (1994), 18-20. [13] B. D. üring, M. Fournier and A. J. üngel, High order compact finite difference schemes for a nonlinear Black-Scholes equation, International Journal of Theoretical and Applied Finance, 6 (2003), 767-789. [14] Y. d'Halluin, P. A. Forsyth and K. R. Vetzal, Robust numerical methods for contingent claims under jump diffusion processes, IMA J. Numer. Anal., 25 (2005), 87-112. doi: 10.1093/imanum/drh011. [15] P. Heider, Numerical methods for nonlinear Black-Scholes equations, Applied Mathematical Finance, 17 (2010), 59-81. doi: 10.1080/13504860903075670. [16] S. L. Heston, A closed-form solution for options with stochastic volatility with application to bond and currency options, Rev. Financial Stud., 6 (1993), 327-343. doi: 10.1093/rfs/6.2.327. [17] 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. [18] C. Huang and S. Wang, A penalty method for a mixed nonlinear complementarity problem, Nonlinear Analysis TMA, 75 (2012), 588-597. doi: 10.1016/j.na.2011.08.061. [19] C. S. Huang, C. H. Hung and S. Wang, A fitted finite volume method for the valuation of options on assets with stochastic volatilities, Computing, 77 (2006), 297-320. doi: 10.1007/s00607-006-0164-4. [20] J. Hull, Options, Futures, and Other Derivatives, Prentice-Hall, Englewood Cliffs, 2005. [21] J. Hull and A. White, The pricing of options on assets with stochastic volatilities, J. Finance, 42 (1987), 281-300. doi: 10.1111/j.1540-6261.1987.tb02568.x. [22] H. E. Leland, Option pricing and replication with transaction costs, Journal of Finance, 40 (1985), 1283-1301. [23] 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. [24] D. C. Lesmana and S. Wang, Penalty approach to a nonlinear obstacle problem governing American put option valuation under transaction costs, Applied Mathematics and Computation, 251 (2015), 318-330. doi: 10.1016/j.amc.2014.11.060. [25] 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. [26] C. Van Loan, Computational Frameworks for the Fast Fourier Transform, Frontier in applied mathematics, Vol. 10.SIAM, Philadelphia, PA, 1992. doi: 10.1137/1.9781611970999. [27] R. C. Merton, Option pricing when underlying stock returns are discontinuous, J. Financial Econ., 3 (1976), 125-144. doi: 10.1016/0304-405X(76)90022-2. [28] A. Mocioalca, Jump diffusion options with transaction costs, Rev. Roumaine Math. Pures Appl., 52 (2007), 349-366. [29] R. S. Varga, Matrix Iterative Analysis, Prentice-Hall, Engelwood Cliffs, NJ, 1962. [30] S. Wang, A penalty method for a finite-dimensional obstacle problem with derivative constraints, Optimization Letters, 8 (2014), 1799-1811. doi: 10.1007/s11590-013-0651-4. [31] S. Wang, A penalty approach to a discretized double obstacle problem with derivative constraints, Journal of Global Optimization, 62 (2015), 775-790. doi: 10.1007/s10898-014-0262-3. [32] 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. [33] S. Wang and X. Q. Yang, A power penalty method for a bounded nonlinear complementarity problem, Optimization, 64 (2015), 2377-2394. doi: 10.1080/02331934.2014.967236. [34] 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 & Applications, 129 (2006), 227-254. doi: 10.1007/s10957-006-9062-3. [35] S. Wang, S. Zhang and Z. Fang, A superconvergent fitted finite volume method for Black-Scholes equations governing European and American option valuation, Numerical Methods for Partial Differential Equations, 31 (2015), 1190-1208. doi: 10.1002/num.21941. [36] Y.-P. Wang and S.-L. Tao, Application of regularization technique to variational adjoint method: A case for nonlinear convection-diffusion problem, Applied Mathematics and Computation, 218 (2011), 4475-4482. doi: 10.1016/j.amc.2011.10.028. [37] P. Wilmott, J. Dewynne and S. Howison, Option Pricing: Mathematical Models and Computation, Oxford Financial Press, Oxford, 1993. [38] D. M. Young, Iterative Solution of Large Linear Systems, Academic Press, 1971. [39] K. Zhang and S. Wang, Pricing options under jump diffusion processes with fitted finite volume method, Applied Mathematics & Computation, 201 (2008), 398-413. doi: 10.1016/j.amc.2007.12.043. [40] K. Zhang and S. Wang, A computational scheme for options under jump diffusion processes, International Journals of Numerical Analysis and Modeling, 6 (2009), 110-123. [41] 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. [42] X. L. Zhang, Numerical analysis of American option pricing in a jump diffusion model, Math. Oper. Res., 22 (1997), 668-690. doi: 10.1287/moor.22.3.668. [43] Y. Y. Zhou, S. Wang and X. Q. Yang, A penalty approximation method for a semilinear parabolic double obstacle problem, Journal of Global Optimization, 60 (2014), 531-550. doi: 10.1007/s10898-013-0122-6.

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##### References:
 [1] A. Almendral and C. W. Oosterlee, Numerical valuation of options with jumps in the underlying, Appl. Math. Comput., 53 (2005), 1-18. doi: 10.1016/j.apnum.2004.08.037. [2] A. Anderson and J. Andresen, Jump diffusion process: volatility smile fitting and numerical methods for option pricing, Rev. Derivat. Res., 4 (2000), 231-262. [3] J. Ankudinova and M. Ehrhardt, On the numerical solution of nonlinear Black-Scholes equations, Computers and Mathematics with Applications, 56 (2008), 799-812. doi: 10.1016/j.camwa.2008.02.005. [4] C. G. Averbuj, Nonlinear differential evolution equation arising in option pricing when including transaction costs: a viscosity solution approach, R. Bras. Eco. de Emp., 12 (2012), 81-90. [5] G. Barles, Convergence of numerical schemes for degenerate parabolic equations arising in finance theory, in: L. C. G. Rogers, D. Talay (Eds), Numerical Methods in Finance, Cambridge University Press, Cambridge, (1997), 1-21. [6] F. Black and M. Scholes, The pricing of options and corporate liabilities, The Journal of Political Economy, 81 (1973), 637-654. doi: 10.1086/260062. [7] W. Chen and S. Wang, A penalty method for a fractional order parabolic variational inequality governing American put option valuation, Computers & Mathematics with Applications, 67 (2014), 77-90. doi: 10.1016/j.camwa.2013.10.007. [8] W. Chen and S. Wang, A finite difference method for pricing European and American options under a geometric Levy process, Journal of Industrial and Management Optimization, 11 (2015), 241-264. doi: 10.3934/jimo.2015.11.241. [9] R. Company, L. Jodar and J. R. Pintos, A numerical method for European option pricing with transaction costs nonlinear equation, Mathematics and Computer Modelling, 50 (2009), 910-920. doi: 10.1016/j.mcm.2009.05.019. [10] R. Cont and P. Tankov, Financial Modelling with Jump Processes, Chapman and Hall/CRC, Boca Raton, FL, 2004. [11] R. Cont and E. Voltchkova, A finite difference scheme for option pricing in jump diffusion and exponential lěvy models, SIAM J. Numer. Anal., 43 (2005), 1596-1626. doi: 10.1137/S0036142903436186. [12] B. Dupire, Pricing with a smile, Risk, 7 (1994), 18-20. [13] B. D. üring, M. Fournier and A. J. üngel, High order compact finite difference schemes for a nonlinear Black-Scholes equation, International Journal of Theoretical and Applied Finance, 6 (2003), 767-789. [14] Y. d'Halluin, P. A. Forsyth and K. R. Vetzal, Robust numerical methods for contingent claims under jump diffusion processes, IMA J. Numer. Anal., 25 (2005), 87-112. doi: 10.1093/imanum/drh011. [15] P. Heider, Numerical methods for nonlinear Black-Scholes equations, Applied Mathematical Finance, 17 (2010), 59-81. doi: 10.1080/13504860903075670. [16] S. L. Heston, A closed-form solution for options with stochastic volatility with application to bond and currency options, Rev. Financial Stud., 6 (1993), 327-343. doi: 10.1093/rfs/6.2.327. [17] 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. [18] C. Huang and S. Wang, A penalty method for a mixed nonlinear complementarity problem, Nonlinear Analysis TMA, 75 (2012), 588-597. doi: 10.1016/j.na.2011.08.061. [19] C. S. Huang, C. H. Hung and S. Wang, A fitted finite volume method for the valuation of options on assets with stochastic volatilities, Computing, 77 (2006), 297-320. doi: 10.1007/s00607-006-0164-4. [20] J. Hull, Options, Futures, and Other Derivatives, Prentice-Hall, Englewood Cliffs, 2005. [21] J. Hull and A. White, The pricing of options on assets with stochastic volatilities, J. Finance, 42 (1987), 281-300. doi: 10.1111/j.1540-6261.1987.tb02568.x. [22] H. E. Leland, Option pricing and replication with transaction costs, Journal of Finance, 40 (1985), 1283-1301. [23] 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. [24] D. C. Lesmana and S. Wang, Penalty approach to a nonlinear obstacle problem governing American put option valuation under transaction costs, Applied Mathematics and Computation, 251 (2015), 318-330. doi: 10.1016/j.amc.2014.11.060. [25] 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. [26] C. Van Loan, Computational Frameworks for the Fast Fourier Transform, Frontier in applied mathematics, Vol. 10.SIAM, Philadelphia, PA, 1992. doi: 10.1137/1.9781611970999. [27] R. C. Merton, Option pricing when underlying stock returns are discontinuous, J. Financial Econ., 3 (1976), 125-144. doi: 10.1016/0304-405X(76)90022-2. [28] A. Mocioalca, Jump diffusion options with transaction costs, Rev. Roumaine Math. Pures Appl., 52 (2007), 349-366. [29] R. S. Varga, Matrix Iterative Analysis, Prentice-Hall, Engelwood Cliffs, NJ, 1962. [30] S. Wang, A penalty method for a finite-dimensional obstacle problem with derivative constraints, Optimization Letters, 8 (2014), 1799-1811. doi: 10.1007/s11590-013-0651-4. [31] S. Wang, A penalty approach to a discretized double obstacle problem with derivative constraints, Journal of Global Optimization, 62 (2015), 775-790. doi: 10.1007/s10898-014-0262-3. [32] 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. [33] S. Wang and X. Q. Yang, A power penalty method for a bounded nonlinear complementarity problem, Optimization, 64 (2015), 2377-2394. doi: 10.1080/02331934.2014.967236. [34] 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 & Applications, 129 (2006), 227-254. doi: 10.1007/s10957-006-9062-3. [35] S. Wang, S. Zhang and Z. Fang, A superconvergent fitted finite volume method for Black-Scholes equations governing European and American option valuation, Numerical Methods for Partial Differential Equations, 31 (2015), 1190-1208. doi: 10.1002/num.21941. [36] Y.-P. Wang and S.-L. Tao, Application of regularization technique to variational adjoint method: A case for nonlinear convection-diffusion problem, Applied Mathematics and Computation, 218 (2011), 4475-4482. doi: 10.1016/j.amc.2011.10.028. [37] P. Wilmott, J. Dewynne and S. Howison, Option Pricing: Mathematical Models and Computation, Oxford Financial Press, Oxford, 1993. [38] D. M. Young, Iterative Solution of Large Linear Systems, Academic Press, 1971. [39] K. Zhang and S. Wang, Pricing options under jump diffusion processes with fitted finite volume method, Applied Mathematics & Computation, 201 (2008), 398-413. doi: 10.1016/j.amc.2007.12.043. [40] K. Zhang and S. Wang, A computational scheme for options under jump diffusion processes, International Journals of Numerical Analysis and Modeling, 6 (2009), 110-123. [41] 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. [42] X. L. Zhang, Numerical analysis of American option pricing in a jump diffusion model, Math. Oper. Res., 22 (1997), 668-690. doi: 10.1287/moor.22.3.668. [43] Y. Y. Zhou, S. Wang and X. Q. Yang, A penalty approximation method for a semilinear parabolic double obstacle problem, Journal of Global Optimization, 60 (2014), 531-550. doi: 10.1007/s10898-013-0122-6.
Prices of the European call and put options with $a=0.01$ and $b=0.07$
Prices of the European call and put option for different values of the transaction cost parameter
Computed American and European put option prices
Computed American put option prices for different values of the transaction cost parameter
Computed rates of convergence for the call option with $a = 0.01$ and $b=0.07$
 $M$ $N$ $\|\cdot\|_{h,2}$ Ratio$(\|\cdot\|_{h,2})$ 21 11 0.215680 41 21 0.116543 1.85 81 41 0.061550 1.89 161 81 0.031986 1.92 321 161 0.016228 1.97 641 321 0.007861 2.06 1281 641 0.003457 2.27 2561 1281 0.001170 2.96
 $M$ $N$ $\|\cdot\|_{h,2}$ Ratio$(\|\cdot\|_{h,2})$ 21 11 0.215680 41 21 0.116543 1.85 81 41 0.061550 1.89 161 81 0.031986 1.92 321 161 0.016228 1.97 641 321 0.007861 2.06 1281 641 0.003457 2.27 2561 1281 0.001170 2.96
Computed rates of convergence for the put option with $a = 0.01$ and $b=0.07$
 $M$ $N$ $\|\cdot\|_{h,2}$ Ratio$(\|\cdot\|_{h,2})$ 21 11 0.454596 41 21 0.438884 1.04 81 41 0.390547 1.12 161 81 0.327934 1.19 321 161 0.259319 1.26 641 321 0.189478 1.37 1281 641 0.121703 1.56 2561 1281 0.058168 2.09
 $M$ $N$ $\|\cdot\|_{h,2}$ Ratio$(\|\cdot\|_{h,2})$ 21 11 0.454596 41 21 0.438884 1.04 81 41 0.390547 1.12 161 81 0.327934 1.19 321 161 0.259319 1.26 641 321 0.189478 1.37 1281 641 0.121703 1.56 2561 1281 0.058168 2.09
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