International Journal of Partial Differential Equations and Applications, 2016, Vol. 4, No. 2, 20-24 Available online at http://pubs.sciepub.com/ijpdea/4/2/1 ©Science and Education Publishing DOI:10.12691/ijpdea-4-2-1 Fractional Black Scholes Option Pricing with Stochastic Arbitrage Return Bright O. Osu 1,* , Chukwunezu A. Ifeoma 2 1 Department of Mathematics, Michael Okpara University of Agriculture, Umudike 2 Department of Mathematics/Statistics, Federal Polytechnic, Nekede, Owerri *Corresponding author: megaobrai@hotmail.com Abstract Option price and random arbitrage returns change on different time scales allow the development of an asymptotic pricing theory involving the options rather than exact prices. The role that random arbitrage opportunities play in pricing financial derivatives can be determined. In this paper, we construct Green’s functions for terminal boundary value problems of the fractional Black-Scholes equation. We follow further an approach suggested in literature and focus on the pricing bands for options that account for random arbitrage opportunities and got similar result for the fractional Black- Scholes option pricing. Keywords: arbitrage returns, option pricing, green function, FBS equation Cite This Article: Bright O. Osu, and Chukwunezu A. Ifeoma, “Fractional Black Scholes Option Pricing with Stochastic Arbitrage Return.” International Journal of Partial Differential Equations and Applications, vol. 4, no. 2 (2016): 20-24. doi: 10.12691/ijpdea-4-2-1. 1. Introduction Fractional calculus has become of increasing use for analyzing not only stochastic processes driven by fractional Brownian processes [16], but also non -random fractional phenomena in physics [8], like the study of porous systems, for instance, and quantum mechanics [14]. Whichever the framework is, we believe that the very reason for introducing and using fractional derivative is to deal with non-differentiable functions. In financial literature for example, stochastic volatility models the Merton jump-diffusion model [9], non-Gaussian option pricing models [4,5], amongst others have been proposed. Each of these is based on the assumption of the absence of arbitrage. However, it is well-known that arbitrage opportunities always exist in the real world (see Refs. [6,15]). Of course, arbitragers ensure that the prices of securities do not get out of line with their equilibrium values, and therefore virtual arbitrage is always short-lived. An arbitrage possibility is essentially equivalent to the possibility of making a positive amount of money out of nothing without taking any risk. It is thus essentially a riskless money making machine. An arbitrage possibility is a serious case of mispricing in the market. It is well- known that arbitrage opportunities always exist in the real world [10]. Of course, arbitragers ensure that the prices of securities do not get out of line with their equilibrium values, and therefore virtual arbitrage is always short-lived. The first attempt to take into account virtual arbitrage in option pricing was made by Physicists Refs [1,7,13]. The authors assume that arbitrage returns exist, appearing and disappearing over a short time scale. Asma et al [2] applied the homotopy perturbation method for fractional Black-Scholes equation by using He’s polynomials and Sumudu transform to obtain the solution of fractional Black-Scholes equation. At this point, Belgacem et al. [3,9] had applied the Laplace transform and extended the theory and the applications of the Sumudu transform to the solution of fractional differential equations. In this work, a technique is proposed for the construction of Green’s functions for terminal boundary value problems of the fractional Black-Scholes equation. The technique is based on the method of integral Laplace transform and the method of variation of parameters. It provides closed form analytic representations for the constructed Green’s functions [12]. We follow further an approach suggested in Ref. [15] and focus on the pricing bands for options that account for random arbitrage opportunities and got similar result for the fractional Black- Scholes option pricing. 2. Derivation of the Black-Scholes Equation We base our derivation on replicating portfolio that ensures that no arbitrage opportunities are allowed. As in the discrete case, consider a portfolio ⋀ ={⋀  }  >0 , which is   - measurable ( we can choose as we go, but at any point in time the choice is deterministic), ⋀  denotes the proportion of shares invested at time  , the rest of the money is invested in the money market account, giving risk-free rate of return, , say. In what follows, we state: Theorem 2.1 Let a generic payoff function ()= (, ), the PDE associated with the price of derivative on the stock price is 2 2 22 1 2 0. H V V V rS H St rV t S S σ − ∂ ∂ ∂ + + − = ∂ ∂ ∂ (2.1)