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Preface: New trends on numerical analysis and analytical methods with their applications to real world problems
June  2019, 12(3): 435-445. doi: 10.3934/dcdss.2019028

## Generalised class of Time Fractional Black Scholes equation and numerical analysis

 1 Department of Mathematics and Applied Mathematics, Faculty of Natural and Agricultural Sciences, University of the Free State, P.O. Box 339 Bloemfontein 9300, South Africa 2 Department of Mathematics University of Namibia, Private bag 13301 Windhoek, Namibia 3 Institute of Groundwater Studies, University of the Free State, IB 56 UFS P.O. Box 339 Bloemfontein 9300, South Africa

* Corresponding author: Rodrigue Gnitchogna Batogna

Received  May 2017 Revised  October 2017 Published  September 2018

It is well known now, that a Time Fractional Black Scholes Equation (TFBSE) with a time derivative of real order $\alpha$ can be obtained to describe the price of an option, when for example the change in the underlying asset is assumed to follow a fractal transmission system. Fractional derivatives as they are called were introduced in option pricing in a bid to take advantage of their memory properties to capture both major jumps over small time periods and long range dependencies in markets. Recently new derivatives of Fractional Calculus with non local and/or non singular Kernel, have been introduced and have had substantial changes in modelling of some diffusion processes. Based on consistency and heuristic arguments, this work generalises previously obtained Time Fractional Black Scholes Equations to a new class of Time Fractional Black Scholes Equations. A numerical scheme solution is also derived. The stability of the numerical scheme is discussed, graphical simulations are produced to price a double barriers knock out call option.

Citation: Rodrigue Gnitchogna Batogna, Abdon Atangana. Generalised class of Time Fractional Black Scholes equation and numerical analysis. Discrete & Continuous Dynamical Systems - S, 2019, 12 (3) : 435-445. doi: 10.3934/dcdss.2019028
##### References:
 [1] E. Alos and Y. Yang, A fractional Heston model with $H>1/2.$, Stochastics, 89 (2017), 384-399.  doi: 10.1080/17442508.2016.1218496.  Google Scholar [2] F. Black and M. Scholes, The pricing of options and corporate liabilities, J. Polit. Econ., 81 (1973), 637-654.  doi: 10.1086/260062.  Google Scholar [3] S. I. Boyarchenko and S. Levendorskii, Non-Gaussian Merton-Black-Scholes Theory, World Scientific, Singapore, 2002. doi: 10.1142/9789812777485.  Google Scholar [4] M. Caputo and F. Fabrizio, A New Definition of Fractional Derivative without singular Kernel, Progr. Frac. differ. Appl., 1 (2015), 1-13.   Google Scholar [5] P. Carr, H. Geman, D. B. Madan and M. Yor, the finite structure of asset returns: An empirical investigation, The journal of Business, 75 (2002), 305-332.   Google Scholar [6] P. Carr and L. Wu, The finite moment log stable process and option pricing, J. Finance, 58 (2003), 753-777.  doi: 10.1111/1540-6261.00544.  Google Scholar [7] A. Cartea and D. del-castillo-Negrete, Fractional diffusion models of option prices in markets with jumps, Physica A-Statistical Mechanics and its Applications, 374 (2) (2007), 749–763. Google Scholar [8] W. Chen and S. Wang, A penalty method for for a fractional order parabolic variational inequality governing American put option valuation, Computers & Mathematics with Apllications, 67 (2014), 77-90.  doi: 10.1016/j.camwa.2013.10.007.  Google Scholar [9] W. Chen, X. Xu and S.-P. Zhu, Analytically pricing double barrier options based on a time-fractional Black Scholes equation, Computers & Mathematics with Applications, 69 (2015), 1407-1419.  doi: 10.1016/j.camwa.2015.03.025.  Google Scholar [10] J.-R. Liang, J. Wang, W.-J. Zhang, W.-Y. Qiu and F.-Y. Ren, The solution to a bi-fractional Black-Scholes-Merton differential equation, Int. J. Pure Appl. Math., 58 (2010), 99-112.   Google Scholar [11] R. C. Merton, The theory of rational option pricing, Bell Journal of Economics and Management Science, 4 (1973), 141-183.  doi: 10.2307/3003143.  Google Scholar [12] A. A. Tateishi, H. V. Ribeiro and E. K. Lenzi, The role of fractional time-derivative operators on anomalous diffusion. Front. Phys., 5 (2017), p52. Google Scholar [13] W. Wyss, the fractional Black-Scholes equation, Fractional Calculus and Applied Analysis Theory Applications, 3 (2000), 51-61.   Google Scholar [14] H. Zhang, F. Liu, I. Turner and Q. Yang, Numerical solution of the time fractional Black-Scholes model governing European options, Comput. Math. Appl., 71 (2016), 1772-1783.  doi: 10.1016/j.camwa.2016.02.007.  Google Scholar

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##### References:
 [1] E. Alos and Y. Yang, A fractional Heston model with $H>1/2.$, Stochastics, 89 (2017), 384-399.  doi: 10.1080/17442508.2016.1218496.  Google Scholar [2] F. Black and M. Scholes, The pricing of options and corporate liabilities, J. Polit. Econ., 81 (1973), 637-654.  doi: 10.1086/260062.  Google Scholar [3] S. I. Boyarchenko and S. Levendorskii, Non-Gaussian Merton-Black-Scholes Theory, World Scientific, Singapore, 2002. doi: 10.1142/9789812777485.  Google Scholar [4] M. Caputo and F. Fabrizio, A New Definition of Fractional Derivative without singular Kernel, Progr. Frac. differ. Appl., 1 (2015), 1-13.   Google Scholar [5] P. Carr, H. Geman, D. B. Madan and M. Yor, the finite structure of asset returns: An empirical investigation, The journal of Business, 75 (2002), 305-332.   Google Scholar [6] P. Carr and L. Wu, The finite moment log stable process and option pricing, J. Finance, 58 (2003), 753-777.  doi: 10.1111/1540-6261.00544.  Google Scholar [7] A. Cartea and D. del-castillo-Negrete, Fractional diffusion models of option prices in markets with jumps, Physica A-Statistical Mechanics and its Applications, 374 (2) (2007), 749–763. Google Scholar [8] W. Chen and S. Wang, A penalty method for for a fractional order parabolic variational inequality governing American put option valuation, Computers & Mathematics with Apllications, 67 (2014), 77-90.  doi: 10.1016/j.camwa.2013.10.007.  Google Scholar [9] W. Chen, X. Xu and S.-P. Zhu, Analytically pricing double barrier options based on a time-fractional Black Scholes equation, Computers & Mathematics with Applications, 69 (2015), 1407-1419.  doi: 10.1016/j.camwa.2015.03.025.  Google Scholar [10] J.-R. Liang, J. Wang, W.-J. Zhang, W.-Y. Qiu and F.-Y. Ren, The solution to a bi-fractional Black-Scholes-Merton differential equation, Int. J. Pure Appl. Math., 58 (2010), 99-112.   Google Scholar [11] R. C. Merton, The theory of rational option pricing, Bell Journal of Economics and Management Science, 4 (1973), 141-183.  doi: 10.2307/3003143.  Google Scholar [12] A. A. Tateishi, H. V. Ribeiro and E. K. Lenzi, The role of fractional time-derivative operators on anomalous diffusion. Front. Phys., 5 (2017), p52. Google Scholar [13] W. Wyss, the fractional Black-Scholes equation, Fractional Calculus and Applied Analysis Theory Applications, 3 (2000), 51-61.   Google Scholar [14] H. Zhang, F. Liu, I. Turner and Q. Yang, Numerical solution of the time fractional Black-Scholes model governing European options, Comput. Math. Appl., 71 (2016), 1772-1783.  doi: 10.1016/j.camwa.2016.02.007.  Google Scholar
Double barrier option price solutions. Model parameters are $\sigma = 0.45, r = 0.03, T = 1, K = 10, DO = 3, UO = 15$
Approximate solutions from equation (15) Double barrier option prices approximate solutions. Model parameters are $\sigma = 0.45, r = 0.03, T = 1, K = 10, DO = 3 ,UO = 15$
Approximate solutions from equation (15) Double barrier option prices approximate solutions. Model parameters are $\sigma = 0.45, r = 0.03, T = 1, K = 10, DO = 3 ,UO = 15$

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