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Black-Scholes Model and Its Application
Abstract
The purpose of the paper is to research on the Black-Scholes (BS) Model as a popular pricing model in the stock market. In addition, the paper discusses the definitions of key stochastic terms that are in important on the derivation and developing of the BS formula and the partial differential of the derived equation. In addition the paper examines in detail the assumptions, applications with examples and the limitation of the Black-Scholes Model.
Key terms: Black-Scholes, Volatility, Geometric Brownian motion, European Call Option, Portfolio, Option Price. Hedge
Introduction
Fischer Black and Myron Scholes developed the Black—Scholes in the year 1973 and it marked a huge impact in the financial industry. The impact made the two mathematicians to be awarded a Nobel Prize in the field of economics. The two researchers achieved to change the option pricing problem into solving the parabolic and partial differential equation with a financial condition. The primary conceptual idea of the Black and Scholes based in the development of riskless portfolio taking positions in cash option as well as the underlying stock. Deriving a Black Scholes equation from a closed-form solution highly depends on the concept of the heat equation. The transformation of the Black-Scholes to heat equation is essential as it leads to change of variables. In other words, once the closed solution is transmuted to heat equation, it becomes possible to transform it back to obtain the corresponding answer of the Black-Scholes. Today, this concept has become applicable in many pricing models and techniques in finance which are rooted to the finance model. The model is typically used in the ito calculus, which derived from financial mathematics is the pricing of options. Among all the applications, black schools formula for pricing become the most popular model in the European market.
Definition of Key Terms
Black Scholes: It is a mathematical stochastic formula that is used in the calculations of options ‘values;
Volatility: Refers to a tool of measuring the fluctuations of stock prices as well as other financial instruments.
Hedge: A transaction that is applied to eliminate or mitigate the risk of investment
Portfolio: Refers to a collection of financial assets such as bonds, cash equivalents and stocks held by some investment institution
Geometric Brownian motion: Refers to a continuous period and stochastic process, whereby the randomly varying quantity relates to the principle of Brownian motion.
European Option. Refers to an option which cannot be traded until the option expires.
Background of Black Scholes Model
Back in 1985, two men, John Gutfreund and Craig Coats Jr came into a conclusion that the Great Depression was imminent and an urgent and reliable solution was necessary. Gutfreund was the head of Salomon Brother (Klebaner, 2012). A strong players business in the Wall Street while Coats Jr has also led Salomon Brothers’ government-bond trading group. In 1987, another concerned investors known as Michael Lewis, also experienced serious scramble to investors when they attemp...