INTERNATIONAL JOURNAL OF SCIENTIFIC & TECHNOLOGY RESEARCH VOLUME 3, ISSUE 12, December 2014 ISSN 2277-8616 200 IJSTR©2014 www.ijstr.org Black-Scholes Partial Differential Equation In The Mellin Transform Domain Fadugba Sunday Emmanuel, Ogunrinde Roseline Bosede Abstract: This paper presents Black-Scholes partial differential equation in the Mellin transform domain. The Mellin transform method is one of the most popular methods for solving diffusion equations in many areas of science and technology. This method is a powerful tool used in the valuation of options. We extend the Mellin transform method proposed by Panini and Srivastav [7] to derive the price of European power put options with dividend yield. We also derive the fundamental valuation formula known as the Black-Scholes model using the convolution property of the Mellin transform method. 2010 Mathematics Subject Classification: 44A15, 60H30, 91G99 Keywords: Black-Scholes Model, Black-Scholes Partial Differential Equation, Dividend Yield, European Option, Mellin Transform Method, Option ———————————————————— 1 INTRODUCTION An option is an instrument whose value derives from that of another asset; hence it is called a “derivative”. In other words an option on an underlying asset is an asymmetric contract that is negotiated today with the following conditions in the future. The holder has either the right, but not the obligation to buy, as it is the case with the European call option, or the possibility to sell, as in the case of the European put option, an asset for a certain price at a prescribed date in the future. The American type of option can be exercised at any time up to and including the date of expiry. The distinctive features of American option are its early exercise privilege. The pricing of American options has been the subject of extensive research in the last decades. There is no known closed form solution and many numerical and analytic approximations have been proposed. Black and Scholes [1] published their seminal work on option pricing in which they described a mathematical frame work for finding the fair price of a European option. They used a no-arbitrage argument to describe a partial differential equation which governs the evolution of the option price with respect to the maturity time and the price of the underlying asset. The Black- Scholes model for pricing options has been applied to many different commodities and payoff structures. In spite of the market crash of 1987, in practice simple Black-Scholes models are widely used because they are very easy to use [4]. We present an overview of the Mellin transform method for the valuation of options in the context of Black and Scholes [1]. The Mellin transform is an integral transform named after the Finnish mathematician Hjalmar Mellin (1854-1933). The Mellin transforms in option theory were introduced by Panini and Srivastav [6]. They derived the expression for the free boundary and price of an American perpetual put as the limit of a finite-lived option. For mathematical backgrounds, the Mellin transform method in derivatives pricing and some numerical methods for options valuation see [3], [5], [8], [9], [10] just to mention a few. 2 BLACK-SCHOLES MODEL Let us consider a market where the risk neutral asset price ,0 t S t T   , which is governed by the stochastic differential equation of the form ( ) t t t t ds S S dW       (1) Where  is the dividend yield, r is the riskless interest rate,  is called the volatility, T is the maturity date and t W is called the Wiener process or Brownian motion. 2.1 Derivation of Black-Scholes Partial Differential Equation Black and Scholes derived the famous Black-Scholes partial differential equation that must be satisfied by the price of any derivative dependent on a non-dividend paying stock. The Black-Scholes model can also be extended to deal with European call and put options on dividend-paying stocks. In the sequel, we derive here the Black-Scholes partial differential equation with a dividend paying stock using portfolio approach. We recall from (1) that; ( ) t t t t dS S dt S dW       where t W follows a Wiener process on a filtered probability space   , , , ( ) B FB   in which ____________________________  Fadugba Sunday Emmanuel is currently a Lecturer in the Department of Mathematical Sciences, Ekiti State University, Ado Ekiti, Nigeria, PH-+2348067032044. E-mail: emmasfad2006@yahoo.com , sunday.fadugba@eksu.edu.ng  Ogunrinde Roseline Bosede is currently a Senior Lecturer in the Department of Mathematical Sciences, Ekiti State University, Ado Ekiti, Nigeria, PH-+2347031343029. E-mail: rowbose@yahoo.com , roseline.ogunrinde@eksu.edu.ng