Optimization of the Black-Scholes Equation with the Numerical Method of Local Expansion to Minimize Risk Coverage
Subject Areas : Numerical Methods in Mathematical FinanceAmirreza 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.
Keywords:
Abstract :
[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