Optimization of the Black-Scholes Equation with the Numerical Method of Local Expansion to Minimize Risk Coverage

محورهای موضوعی : Numerical Methods in Mathematical Finance

Amirreza Keyghobadi ¹ , Shadan Behzadi ² , Fatemeh Gervei ³

1 - Department of Accounting, Central Tehran Branch, Islamic Azad University, Tehran, Iran.
2 - Department of Mathematics, Qazvin Branch, Islamic Azad University, Qazvin, Iran.
3 - Department of Mathematics, Qazvin Branch, Islamic Azad University, Qazvin, Iran.

تاریخ دریافت : 1397/08/27 تاریخ پذیرش : 1398/12/21 تاریخ انتشار : 1400/07/09

کلید واژه: Fractional equations, Market power Exchange, Jacobi polynomial, Black-Scholes equation, Airfoil polynomial,

چکیده مقاله :

In this paper, we present an efficient and accurate method for calculating the Black-Scholes differential equations and solve the Black-Scholes equations using Jacoby and Airfoil orthogonal bases, with the collocation method. The Black-Scholes equation is a partial differential equation, which describes the price of choice in terms of time and the collocation method is a method of deter-mining coefficients. Then we show the computational results and examine the performance of the method for the two options, the price of basic assets and its issues. These results show that the Jacoby method is more efficient in solving the Black Scholes equation, and the method error is less and the convergence rate is higher. In this paper, by applying numerical methods to the desired equation, nonlinear devices can be solved by nonlinear solution methods, such as Newton's iterative method. The existence, uniqueness of the solution, and convergence of the methods are examined, and we will show in an example that by repeating then |u n+1-u n |/|u n | <ε can be reached and this indicates the accuracy of the response to these methods.

منابع و مأخذ:

[1] Behzadi, Sh.S., Yildirim, A., Application of Quintic B-Spline Collocation Method for Solving the Coupled-BBM System, Middle-East Journal of Scientific Research, 2013, 15(11), P.1478-1486. Doi: 10.5829/idosi.mejsr.2013.15.11.2147

 

[2] Behzadi, Sh.S., Allahviranloo, T., Abbasbandy, S., Fuzzy Collocation Methods for Second-Order Fuzzy Abel-Volterra Integro-Differential Equations, Iranian Journal of Fuzzy Systems, 2014, 11(2), P. 71-88. Doi:10.22111/IJFS.2014.1503

 

[3] Black, F., and Scholes, M., The Pricing of Options and Corporate Liabilities, Journal of Political Economy, 1973, 81(3), P.637-659. Doi: org/10.1086/260062

 

[4] Cen, Z., Le, A., Xu, A., A Second-Order Difference Scheme for the Penalized Black-Scholes Equation Governing American Put Option Pricing, Computational Economics, 2012, 40(1), P. 49–62. Doi:org/10.1155/2013/651573

 

[5] Chen, Sh., Shen, J., Wang, L., Generalized Jacobi Functions and Their Applications to Fractional Differential Equations, Mathematics of Computation, 2016, 85(300), P. 1603-1638 Doi: org/10.1090/mcom3035

 

[6] Dalvand, KH., Tabatabaie, M., Studying the Role of Marketing Intensity on the Relation of Finance Leverage and Firm Function, Advances in Mathematical Finance, 2018, 3(3), P. 27-39. Doi:10.22034/AMFA.2018.544947

 

[7] Dibachi, H., Behzadi, M.H., Izadikhah, M., Stochastic Modified MAJ Model for Measuring the Efficiency and Ranking of DMUs, Indian Journal of Science and Technology, 2015, 8(8), P. 1-7, Doi: 10.17485/ijst/2015/v8iS8/71505

 

 [8] Edeki, S.O., Owoloko, E.A., and Ugbebor, O.O., The Modified Black–Scholes Model Via Constant Elasticity of Variance for Stock Options Valuation, AIP Conference Proceedings, 2016, 1705(1) P. 1705-1716. Doi:org/10.1063/1.4940289

 

[9]  Edeki, S.O., Ugbebor, O.O., and Owoloko, E.A., Analytical Solutions of a Time-Fractional Nonlinear Transaction-Cost Model for Stock Option Valuation in an Illiquid Market Setting Driven by a Relaxed Black–Scholes Assumption, Edeki et al., Cogent Mathematics, 2017, 4(13),P. 52-81. Doi:.org/10.1080/23311835.2017.1352118

 

[10] Fei, Z., Goto, Y., Kita, E., Solution of Black-Scholes Equation by Using RBF Approximation, Frontiers of Computational Science, 2007, P. 339-343. Doi:org/10.1007/978-3-540-46375-7_53.

 

[11] Fey, R., & and Polte, U., Nonlinear Black-Scholes Equations in Finance: Associated Control Problems and Properties of Solutions, SIAM Journal on Control and Optimization, 2011, 49(1), Doi: org/10.1137/090773647

 

[12] Izadikhah, M., Azadi, M., Shokri Kahi, V., Farzipoor Saen, R., Developing a new chance constrained NDEA model to measure the performance of humanitarian supply chains, International Journal of Production Research, 2019, 57(3), P. 662-682, Doi: 10.1080/00207543.2018.1480840

 

[13] Jeong, D., Kim, J., and Wee, I.S., An Accurate and Efficient Numerical Method for Black-Scholes Equations, Commun. Korean Math. Soc, 2009, 4(24), P. 617–628. Doi:10.4134/CKMS.2009.24.4.617

 

[14] Ksendal, B., Mathematics and Finance, The Black-Scholes Option Pricing Formula and Beyond, Dept. of Math./CMA University of Oslo "Matilde", Danish Mathematical Society Pure Mathematics, Denmark ISSN 0806, 2011, 11.

 

[15] Ksendal, B., Sulem, A., Maximum Principles for Optimal Control of Forward-Backward Stochastic Differential Equations with Jumps, E-Print, University of Oslo, 2008, 48(5), P. 2945–2976. Doi:10.1137/080739781

 

[16] Kumar, S., Kumar, D., Singh, J., Numerical Computation of Fractional Blacke-Scholes Equation Arising in Financial Market, Egyptian Journal of Basic and Applied Sciences 1, 2014, 1(3-4), P. 177-183.  Doi: org/10.1016/j.ejbas.2014.10.003

 

[17] Izadikhah, M., Using goal programming method to solve DEA problems with value judgments, Yugoslav Journal of Operations Research, 2016, 24 (2), P. 267–282. Doi: 10.2298/YJOR121221015I

 

[18] Rodrigue C., Moutsinga B., Pindza E., Maré E., Homotopy Perturbation Transform Method for Pricing Under Pure Diffusion Models with Affine Coefficients, Journal of King Saud University – Science, 2018, 30(1) P. 1-13. Doi: org/10.1016/j.jksus.2016.09.004

 

[19] Malekian, E., Fakhari, H., Gasemi, J., Farzadi, F., Predict the Stock Price of Crash Risk by Using Firefly Algorithm and Comparison with Regression, Advances in Mathematical Finance, 2018, 3(2) P. 43-58. Doi:10.22034/AMFA.2018.540830

 

[20] Michael Steele, J., Stochastic Calculus and Financial Applications, Springer-Verlag, 2001, 274-280.

 

[21] Ouafoudi, M. Gao, F., Exact Solution of Fractional Black-Scholes European Option Pricing Equations, Applied Mathematics, 2018, 9(1), P. 86-100. Doi: org/10.4236/am.2018.91006

 

[22] Slavova A., Kyurkchiev, N., Programme Packages for Implementation of Modifications of Black–Scholes Model and Bulgar, Institute of Mathematics and Informatics (Bulgarian Academy of Sciences), 2014, 23, P. 141-158.

 

[23] Slavova A., Kyurkchiev, N., Numerical Implementation of Generalizations of Black–Scholes Model for Estimation of Call- and Put-Option, Institute of Mathematics and Informatics (Bulgarian Academy of Sciences), 2014, 67(8), 1053–1060.

 

[24] Sakthivel, K. Kim, J.H., Controllability and Hedgibility of Black-Scholes Equations with N Stocks, Acta Applicandae Mathematicae, 2010, 111(3), P. 339-363. Doi: 10.1007/s10440-009-9548-8

 

[25] Saadat R., Sheykhimehrabadi M., Masoudian A., A Model for the Mechanism of Monetary Policy and Inflation Control in the Framework of the Interest-Free Banking Act, Advances in Mathematical Finance, 2016,1(2), P. 29-41. Doi: 10.22034/AMFA.2016.527814

 

[26] Wilmott, P., Howison, S., Dewynne, J., The Mathematics of Financial Derivatives, Cambridge University Press, 1995. Doi: org/10.1017/CBO9780511812545

 

[247] Xiaozhong, Y., Lifei, Wu., A Universal Difference Method for Time-Space Fractional Black-Scholes Equation, Advances Difference Equations, 2016, 71(1), P. 1-14. Doi: 10.1186/s13662-016-0792-8