IOSR Journal of Mathematics (IOSRJM) ISSN: 2278-5728 Volume 2, Issue 2 (July-Aug 2012), PP 38-42 www.iosrjournals.org www.iosrjournals.org 38 | Page Mathematical Finance: Applications of Stochastic Process 1 S. K. Sahoo, 2 M. N. Mishra 1 Asst. Professor, Department of Mathematics, VITS Engineering College, Damanabhuin, NH-5, Khurda-20, Odisha, India 2 Professor, Mathematical Finance, Institute of Mathematics & Applications, Bhubaneswar, Odisha, India Abstract: One of the momentous equations in financial mathematics is the Black-Scholes equation, a partial differential equation that governs the value of financial derivatives, such as options. In this paper, we attempt to show the application of Stochastic Process. We have shown how geometric Brownian motion & Ito’s Lemma overlaps on Option Pricing. Key Words: - Geometric Brownian motion, Ito’s Lemma, Black-Scholes Equation I. Introduction Investors pay for stocks and bonds in the monetary market, putting their funds at risk for the chance to receive a return. As the time of Phoenicians, they have sought to reduce this risk value for each level of expected return. In order to do so, a whole range financial tool have been developed, known as derivatives, assets who derive assets as of another financial asset. The scenery of derivative assets provides an interesting means of expression for the analysis and application of Brownian motion and solving partial derivative equations, while maintaining its real world applications. Several articles have been written on modeling movements in financial markets with stochastic calculus. Possibly the most eminent of these described the Nobel Prize winning Black-Scholes option pricing model [4]. In several articles, mathematicians, specifically Robert Almgren's[5] and Anastasios Malliaris[1], have attempted to more rigorously bridge the gap between random motion and option pricing. II. Terminology 2.1 Financial  Asset: An object that provides a claim to future cash flows.  Efficient Market Hypothesis: There is no opportunity for arbitrage in the market.  Derivative: A financial asset that derives its value from another asset.  Option: A derivative that provides the opportunity, but not obligation to buy or sell an asset at a predetermined price in the future.  Strike Price: The predetermined price for executing an option. For a call option, if the market price rises above the strike price, the investor will be willing to buy. For a put option, if the market price falls below the strike price, the investor will want to sell the underlying asset. 2.2 Stochastics  Probability Space: A construct of three components, Ω , ܨ, , where 1. Ω is the set of all possible outcomes. 2. ܨis the set of all events, where each event has zero or more outcomes. 3.  is the assignment of probabilities to each event.  With Probability 1: Also known as almost surely. The probability of an event occurring tends to 1 given some limit. Note that this differs from surely in that surely indicates that no other event is possible, while almost surely indicates that other events become less and less likely.  A collection of sets F is called σ-algebra if for a sequence of sets   ∈۴ ,   ∈۴ ∞  and is closed under complementation. The sets ∈۴ are F-measurable.  [0 , T] denotes the set of functions (ݐ) such that (ݐ) is defined on 0 , T, measurable with respect to the σ-algebra ۴ ݐ for all ݐ, and  (ݐ) 2  ݐ  0 is finite with probability 1.