editor@tjprc.org www.tjprc.org International Journal of Mathematics and Computer Applications Research (IJMCAR) ISSN(P): 2249-6955; ISSN(E): 2249-8060 Vol. 4, Issue 3, Jun 2014, 33-46 © TJPRC Pvt. Ltd. ACCELERATED GENETIC ALGORITHM SOLUTION OF LINEAR BLACK – SCHOLES EQUATION EMAN ALI HUSSAIN & YASEEN MERZAH ALRAJHI Al-Mustansiriyah University, College of Science, Department of Mathematics, Baghdad, Iraq ABSTRACT In this research, the development of a fast numerical method was performed in order to solve the Option Pricing problems governed by the Black-Scholes equation using an accelerated genetic algorithm method. Where the Black-Scholes equation is a well known partial differential equation in financial mathematics. A discussion of the solutions was introduced for the linear Black-Scholes model with the European options (Call and Put) analytically and numerically ,with and without transformation to heat equation .Comparisons of the presented approximation solutions of these models with the exact solution were achieved. KEYWORDS: Linear Black-Scholes Equation, American and European Options, Genetic Algorithm 1. INTRODUCTION Many mathematical models have been proposed to describe the evolution of the value of financial derivatives. Financial models were generally formulated in terms of stochastic differential equations, [1]. Linear Black-Scholes equation the model was developed by Fischer Black and Myron Scholes [2], as an important equation in the financial mathematics. 0=  +       +   −  (1) Where S:= S(t) > 0 and t є (0, T ), provides both the price for a European option and a hedging portfolio that replicates the option assuming that, [4]. • The price of the asset price or underlying derivative S(t) follows a Geometric Brownian motion W(t), meaning that S satisfies the following stochastic differential equation (SDE). dS(t) = µS(t)dt + σS(t)dW(t). • The trend or drift µ (measures the average rate of growth of the asset price), the volatility σ (measures the standard deviation of the returns) and the riskless interest rate r are constant for 0 ≤ t ≤ T and no dividends are paid in that time period. Equation (1) can be transformed into the heat equation to find the price of the option analytically, [1]. In this research, the application of the accelerated Genetic Algorithm method was taken into account to solve the linear Black–Scholes model for European options. Studying (1) for an American options would be redundant, since the value of an American options equals the value of a European options if no dividends are paid and the volatility is constant, [6], [7], [8], [9], [10].