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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 and 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.

[2]

F. Black and M. Scholes, The pricing of options and corporate liabilities, J. Polit. Econ., 81 (1973), 637-654.  doi: 10.1086/260062.

[3]

S. I. Boyarchenko and S. Levendorskii, Non-Gaussian Merton-Black-Scholes Theory, World Scientific, Singapore, 2002. doi: 10.1142/9789812777485.

[4]

M. Caputo and F. Fabrizio, A New Definition of Fractional Derivative without singular Kernel, Progr. Frac. differ. Appl., 1 (2015), 1-13. 

[5]

P. CarrH. GemanD. B. Madan and M. Yor, the finite structure of asset returns: An empirical investigation, The journal of Business, 75 (2002), 305-332. 

[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.

[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.

[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.

[9]

W. ChenX. 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.

[10]

J.-R. LiangJ. WangW.-J. ZhangW.-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. 

[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.

[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.

[13]

W. Wyss, the fractional Black-Scholes equation, Fractional Calculus and Applied Analysis Theory Applications, 3 (2000), 51-61. 

[14]

H. ZhangF. LiuI. 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.

show all references

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.

[2]

F. Black and M. Scholes, The pricing of options and corporate liabilities, J. Polit. Econ., 81 (1973), 637-654.  doi: 10.1086/260062.

[3]

S. I. Boyarchenko and S. Levendorskii, Non-Gaussian Merton-Black-Scholes Theory, World Scientific, Singapore, 2002. doi: 10.1142/9789812777485.

[4]

M. Caputo and F. Fabrizio, A New Definition of Fractional Derivative without singular Kernel, Progr. Frac. differ. Appl., 1 (2015), 1-13. 

[5]

P. CarrH. GemanD. B. Madan and M. Yor, the finite structure of asset returns: An empirical investigation, The journal of Business, 75 (2002), 305-332. 

[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.

[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.

[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.

[9]

W. ChenX. 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.

[10]

J.-R. LiangJ. WangW.-J. ZhangW.-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. 

[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.

[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.

[13]

W. Wyss, the fractional Black-Scholes equation, Fractional Calculus and Applied Analysis Theory Applications, 3 (2000), 51-61. 

[14]

H. ZhangF. LiuI. 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.

Figure 1.  Double barrier option price solutions. Model parameters are $\sigma = 0.45, r = 0.03, T = 1, K = 10, DO = 3, UO = 15$
Figure 2.  Approximate solutions from equation (15) Double barrier option prices approximate solutions. Model parameters are $\sigma = 0.45, r = 0.03, T = 1, K = 10, DO = 3 ,UO = 15$
Figure 3.  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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