1 © 2020 Conscientia Beam. All Rights Reserved. NUMERICAL SOLUTIONS OF BLACK-SCHOLES MODEL BY DU FORT-FRANKEL FDM AND GALERKIN WRM Md. Shorif Hossan 1 A B M Shahadat Hossain 2 Md. Shafiqul Islam 3+ 1,2,3 Department of Applied Mathematics, University of Dhaka, Bangladesh. (+ Corresponding author) ABSTRACT Article History Received: 27 November 2019 Revised: 30 December 2019 Accepted: 3 February 2020 Published: 25 February 2020 Keywords Black-Scholes equation European call option European put option Du Fort-Frankel finite difference method (DF3DM) Galerkin weighted residual method (GWRM) Modified legendre polynomials. The main objective of this paper is to find the approximate solutions of the Black- Scholes (BS) model by two numerical techniques, namely, Du Fort-Frankel finite difference method (DF3DM), and Galerkin weighted residual method (GWRM) for both (call and put) type of European options. Since both DF3DM and GWRM are the most familiar numerical techniques for solving partial differential equations (PDE) of parabolic type, we estimate options prices by using these techniques. For this, we first convert the Black-Scholes model into a modified parabolic PDE, more specifically, in DF3DM, the first temporal vector is calculated by the Crank-Nicolson method using the initial boundary conditions and then the option price is evaluated. On the other hand, in GWRM, we use piecewise modified Legendre polynomials as the basis functions of GWRM which satisfy the homogeneous form of the boundary conditions. We may observe that the results obtained by the present methods converge fast to the exact solutions. In some cases, the present methods give more accurate results than the earlier results obtained by the adomian decomposition method [14]. Finally, all approximate solutions are shown by the graphical and tabular representations. Contribution/Originality: The paper’s primary contribution is finding that the approximate results of Black- Scholes model by DF3DM, and GWRM with modified Legendre polynomials as basis functions. 1. INTRODUCTION Options are treated as the most important part of the security markets from the beginning of the Chicago Board Options Exchange (CBOE) in 1973, which is the largest options market in the world [1]. During last decades, the valuation of options has become important problem for both financial and mathematical point of view. Details about options are available in Hull [1]; Privault [2]. There are many models for calculating the value of options but among all of those models, the Black-Scholes model is a suitable way to find the European options price. The discovery of the Black-Scholes model took long time. Fishers Black took the first step to make a model for valuation of stock. Afterward Myron Scholes added with Black and today we use their result for finding the value of different kinds of stocks. In 1973, the concept of the Black-Scholes model was first disclosed in the paper entitled, “The pricing of options and corporate liabilities” in the Journal of political economy by Black and Scholes [3] and then advanced in “Theory of rational option pricing" by Robert Merton. In 2003, Chawla, et al. [4] approximate European put option value by using Generalized trapezoidal formula and found better approximation than Crank- Nicolson method especially near the strike price. In Hackmann [5] Crank-Nicolson method has been used for evaluation of European options price with accuracy up to 3 decimal places. In 2012, Shinde and Takale [6] have International Journal of Mathematical Research 2020 Vol. 9, No. 1, pp. 1-10 ISSN(e): 2306-2223 ISSN(p): 2311-7427 DOI: 10.18488/journal.24.2020.91.1.10 © 2020 Conscientia Beam. All Rights Reserved.