ISSN 1749-3889 (print), 1749-3897 (online) International Journal of Nonlinear Science Vol.17(2014) No.2,pp.105-110 European Option Pricing of Fractional Version of the Black-Scholes Model: Approach Via Expansion in Series M. A. M. Ghandehari , M. Ranjbar ∗ Department of Applied Mathematics, Azarbaijan Shahid Madani University, Tabriz, Iran (Received 31 December 2013, accepted 10 April 2014) Abstract: This paper presents the decomposition method for solution of the fractional Black-Scholes equa- tion with boundary condition for a European option pricing problem. Undoubtedly this model is the most well known model for pricing financial derivatives. This method finds the analytical solution without any dis- cretization or additive assumption. The numerical method has been applied in the form of convergent power series with easily computable components, to solve the fractional Black-Scholes equations. Keywords: Fractional Black-Scholes equations; European option pricing problem; Analytical solution 1 Introduction There is an immense amount of interest and literature on the pricing of financial derivatives. A financial derivative is an instrument whose price depends on, or is derived from, the value of another asset [14]. Often, this underlying asset is a stock. The concept of financial derivatives is not new. While there remains some historical debate as to the exact date of the creation of financial derivatives, it is well accepted that the first attempt at modern derivative pricing began with the work of Charles Castelly [6] published in 1877. Castell’s book was a general introduction to concepts such as hedging and speculative trading, but it laked mathematical rigor. In 1969, Fisher Black and Myron Scholes got an idea that would change the world of finance forever. The central idea of their paper revolved around the discovery that one did not need to estimate the expected return of a stock in order to price an option written on that stock. The Black-Scholes model (BS) for pricing stock options has been applied to many different commodities and payoff structures. The Black-Scholes model for value of an option is described by the equation ∂V ∂t + 1 2 σ 2 x 2 ∂ 2 V ∂x 2 + r(t)x ∂V ∂x - r(t)V =0, (x, t) ∈ R + × (0,T ) (1) where V (x, t) is the European option price at asset price x and at time t, T is the maturity, r(t) is the risk free interest rate and σ(x, t) represents the volatility function of underlying asset. It is well-known that problem (1) has a closed-form solution obtained for the price of a European call or European put option after several changes of variables and solving certain related diffusion equations [5, 14]. We denoted the payoff functions c(x, t) and p(x, t) for the European call and put options, respectively. Thus c(x, t)= max(x - E, 0) , p(x, t)= max(E - x, 0), where E is the exercise price. During the last decades, several numerical and analytical methods have been proposed in the literature to solve the Black-Scholes model by Ankudinova and Ehrhardt [2], Gulkac [13], Jodar and et al. [16], Cen and Le [7] and Company and et al. in [8]. The fractional calculus is used in many fields of science and engineering [3, 4, 20, 25]. In the area of financial markets, fractional order models have been recently used to described the probability distributions of log-price in the log-time limit, which is useful to characterise the natural variability in prices in the log term. Meerschaert and Scalas [19] introduced a time-space fractional diffusion equation to model the CTRW scaling limit process densities when the waiting times * Corresponding author. E-mail address:m ranjbar@azaruniv.edu Copyright c ⃝World Academic Press, World Academic Union IJNS.2014.04.15/792