Non-linear partial differential equations have been increasingly used to model the price of options in the realistic market setting when transaction costs arising in the hedging of portfolios are taken into account. This paper focuses on finding the numerical solution of the non-linear partial differential equation corresponding to a Bermudan call option price with variable transaction costs for an asset under the information-based framework. The finite difference method is implemented to approximate the option price and its Greeks. Numerical examples are presented and the option prices compared to the closed-form solution of the information-based model and the Black Scholes model with zero transaction costs. The results show that the approximated option prices correspond to the analytical solution of the information-based model but are slightly higher than the prices under Black-Scholes model. These findings validate the finite difference method as an efficient way of approximating the information-based non-linear partial differential equation.
References
[1]
Black, F. and Scholes, M. (1973) The Pricing of Options and Corporate Liabilities. Journal of Political Economy, 81, 637-654. https://doi.org/10.1086/260062
[2]
Brody, D.C., Hughston, L.P. and Macrina, A. (2008) Information-Based Asset Pricing. International Journal of Theoretical and Applied Finance, 11, 107-142. https://doi.org/10.1142/S0219024908004749
[3]
Hughston, L.P. and Sanchez-Betancourt, L. (2020) Pricing with Variance Gamma Information. Risks, 8, Article 105. https://doi.org/10.3390/risks8040105
[4]
Rutkowski, M. and Yu, N. (2007) An Extension of the Brody-Hughston-Macrina Approach to Modeling of Defaultable Bonds. International Journal of Theoretical and Applied Finance, 10, 557-589. https://doi.org/10.1142/S0219024907004263
[5]
Brody, D.C., Davis, M.H.A., Friedman, R.L. and Hughston, L.P. (2009) Informed traders. Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, 465, 1103-1122. https://doi.org/10.1098/rspa.2008.0465
[6]
Macrina, A. (2008) An Information-Based Framework for Asset Pricing: X-Factor Theory and Its Applications. arXiv preprint arXiv:0807.2124
[7]
Hoyle, E., Hughston, L.P. and Macrina, A. (2011) Levy Random Bridges and the Modelling of Financial Information. Stochastic Processes and Their Applications, 121, 856-884. https://doi.org/10.1016/j.spa.2010.12.003
[8]
Odin, M., Aduda, J.A. and Omari, C.O. (2022) Pricing Bermudan Option with Variable Transaction Costs under the Information-Based Model. Open Journal of Statistics, 12, 549-562. https://doi.org/10.4236/ojs.2022.125033
[9]
He, T., Coolen, F.P.A. and Coolen-Maturi, T. (2021) Nonparametric Predictive Inference for American Option Pricing Based on the Binomial Tree Model. Communications in Statistics—Theory and Methods, 50, 4657-4684. https://doi.org/10.1080/03610926.2020.1764040
[10]
El Filali Ech-Chafiq, Z., Henry-Labordere, P. and Lelong, J. (2021) Pricing Bermudan Options Using Regression Trees/Random Forests. Available at SSRN 3984200. https://doi.org/10.2139/ssrn.3984200
[11]
Vellekoop, M. and Nieuwenhuis, H. (2009) A Tree-Based Method to Price American Options in the Heston Model. The Journal of Computational Finance, 13, 1-21. https://doi.org/10.21314/JCF.2009.197
[12]
Sahar, S., Zouhir, M., et al. (2022) Pricing American Put Option Using RBF-NN: New Simulation of Black-Scholes. Moroccan Journal of Pure and Applied Analysis, 8, 78-91. https://doi.org/10.2478/mjpaa-2022-0007
[13]
Xiang, J. and Wang, X. (2022) Quasi-Monte Carlo Simulation for American Option Sensitivities. Journal of Computational and Applied Mathematics, 413, Article ID: 114268. https://doi.org/10.1016/j.cam.2022.114268
[14]
Lapeyre, B. and Lelong, J. (2021) Neural Network Regression for Bermudan Option Pricing. Monte Carlo Methods and Applications, 27, 227-247. https://doi.org/10.1515/mcma-2021-2091
[15]
Abdelkawy, M.A. and Lopes, A.M. (2022) Spectral Solutions for Fractional Black-Scholes Equations. Mathematical Problems in Engineering, 2022, Article ID: 9365292. https://doi.org/10.1155/2022/9365292
[16]
Jain, S. and Oosterlee, C.W. (2015) The Stochastic Grid Bundling Method: Efficient Pricing of Bermudan Options and Their Greeks. Applied Mathematics and Computation, 269, 412-431. https://doi.org/10.1016/j.amc.2015.07.085
[17]
Tour, G., Thakoor, N., Khaliq, A.Q.M. and Tangman, D.Y. (2018) Cos Method for Option Pricing under a Regime-Switching Model with Time-Changed Levy Processes. Quantitative Finance, 18, 673-692. https://doi.org/10.1080/14697688.2017.1412494
[18]
Fang, F. and Oosterlee, C.W. (2011) A Fourier-Based Valuation Method for Bermudan and Barrier Options under Heston’s Model. SIAM Journal on Financial Mathematics, 2, 439-463. https://doi.org/10.1137/100794158
[19]
Zhang, R., Zhang, Q. and Song, H. (2015) An Efficient Finite Element Method for Pricing American Multi-Asset Put Options. Communications in Nonlinear Science and Numerical Simulation, 29, 25-36. https://doi.org/10.1016/j.cnsns.2015.03.022
[20]
Izem, N., Malek, M., Saoud, S., et al. (2021) A Partition of Unity Finite Element Method for Valuation American Option under Black-Scholes Model. Moroccan Journal of Pure and Applied Analysis, 7, 324-336. https://doi.org/10.2478/mjpaa-2021-0021
[21]
Khalsaraei, M.M., Shokri, A., Mohammadnia, Z. and Sedighi, H.M. (2021) Qualitatively Stable Nonstandard Finite Difference Scheme for Numerical Solution of the Nonlinear Black-Scholes Equation. Journal of Mathematics, 2021, Article ID: 6679484. https://doi.org/10.1155/2021/6679484
[22]
Khalsaraei, M.M., Rashidi, M.M., Shokri, A., Ramos, H. and Khakzad, P. (2022) A Nonstandard Finite Difference Method for a Generalized Black-Scholes Equation. Symmetry, 14, Article 141. https://doi.org/10.3390/sym14010141
[23]
Uprety, K.N. and Panday, G.P. (2020) Numerical Solution of European and American Option with Dividends Using Finite Difference Methods. Scientific World, 13, 55-61. https://doi.org/10.3126/sw.v13i13.30540
[24]
Yousuf, M., Khaliq, A.Q.M. and Kleefeld, B. (2012) The Numerical Approximation of Nonlinear Black-Scholes Model for Exotic Path-Dependent American Options with Transaction Cost. International Journal of Computer Mathematics, 89, 1239-1254. https://doi.org/10.1080/00207160.2012.688115
[25]
Company, R., Egorova, V., Jodar, L. and Vazquez, C. (2016) Finite Difference Methods for Pricing American Put Option with Rationality Parameter: Numerical Analysis and Computing. Journal of Computational and Applied Mathematics, 304, 1-17. https://doi.org/10.1016/j.cam.2016.03.001
[26]
Ševčovič, D. and Žitňanská, M. (2016) Analysis of the Nonlinear Option Pricing Model under Variable Transaction Costs. Asia-Pacific Financial Markets, 23, 153-174. https://doi.org/10.1007/s10690-016-9213-y
[27]
Fakharany, M., Company, R. and Jodar, L. (2014) Positive Finite Difference Schemes for a Partial Integro-Differential Option Pricing Model. Applied Mathematics and Computation, 249, 320-332. https://doi.org/10.1016/j.amc.2014.10.064
[28]
Amster, P., Averbuj, C.G., Mariani, M.C. and Rial, D. (2005) A Black-Scholes Option Pricing Model with Transaction Costs. Journal of Mathematical Analysis and Applications, 303, 688-695. https://doi.org/10.1016/j.jmaa.2004.08.067
[29]
Ikamari, C., Ngare, P. and Weke, P. (2020) Multi-Asset Option Pricing Using an Information-Based Model. Scientific African, 10, e00564. https://doi.org/10.1016/j.sciaf.2020.e00564
[30]
Bjork, T. (2009) Arbitrage Theory in Continuous Times. Oxford University Press, Oxford.
[31]
Jalghaf, H.K., Kovacs, E., Majar, J., Nagy, A. and Askar, A.H. (2021) Explicit Stable finite Difference Methods for Diffusion-Reaction Type Equations. Mathematics, 9, Article 3308. https://doi.org/10.3390/math9243308
[32]
Lee, W.T. (2011) Tridiagonal Matrices: Thomas Algorithm. MS6021, Scientific Computation, University of Limerick, Limerick.
[33]
Kociński, M. (2014) Transaction Costs and Market Impact in Investment Management. e-Finanse: Financial Internet Quarterly, 10, 28-35.
