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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 attempted to tackle the stock market crash, which is currently known as Black Monday (Klebaner, 2012). In particular, Gutfreund and Coats Jr feared the repeat of the Great Crash of 1929 and for these reason, they decided to purchase a $2 billion worth of 30-year U.S. Treasury bonds. The strategy was to move in front of investors’ flight to safety.
Similarly, various leaders from group of Salomon traders, settled to differing conclusion, but over time, the academic groups of John Gutfreund and Craig Coats Jr proved to be right after they lost approximately $75 million. Therefore, there was need for a transition by transferring power on Wall Street from cursing and veterans traders with formulas (Klebaner, 2012). Later, the nerd traders opted to develop their own hedge fund, long term capital management, and this marked a rejuvenation of the Wall Street, as aforementioned, Fischer Black and Myron Scholes boosted the industry by inventing the Black-Scholes formula, which was being applied by many call traders (Klebaner, 2012). The formula featured among the best financial models that spawned the Wall Street Market and won the two professors a Nobel Prize in economics. 
The idea behind Black-Scholes model became more famous and important than the formula itself. As much as the two researcher did not explain why stock prices rise and falls unpredictably, the assumption that rise and fall of stock prices unpredictability and predictability is similar to the bombardment of atoms and air which are in constant random motion covered up the explanation (Klebaner, 2012). This makes sense as the stock market, especially at Wall Street are random and they can be controlled and modelled as well. This is actually the role of the Black-Scholes formula. In other words, the model does not offer predictions on the changes in the stock prices but instead, the price will randomly change but in a normal distribution curve (Klebaner, 2012). This introduces the element of probability that the stock prices are likely to increase or decrease based on the rate of its volatility.
Despite the facts that the formula majorly described a single type of derivative, which is a stock option, still it had many application that is being used up to today. Understanding the formula was worth for all traders in the Wall Street. The concept behind the formula was that, when traders purchase a stock option, they obtain it with a right to trade the stock option at a particular time in future (Hirsa & Neftci, 2013). This aspect was essential as it prevented chances of making losses in case a market of a particular stock crushes. In other words, the Black-Scholes Formula made options to be flexible, hence making them more valuable. Traders were pleased with Black-Scholes formula as it answered the question as to how the value of a stock changes the date when it has to be exercised approaches (Hirsa & Neftci, 2013). In addition, the formula dwells on the concept that free lunch no longer exist. For example, if Facebook stock price drops to $25, its selling value should increase to $ 33
In addition, many traders focused mostly at the option prices using the Black-Scholes formula as it employs the factor of price volatility to determine the actual prices of the stock option (Klebaner, 2012). This revealed that, when the prices rises, the stock option became very volatile. Also, in circumstances of higher than expected volatility, Long-Term Capital have the likelihood of reverting the volatility created by the current prices to normal historic prices.
Derivation of Black-Scholes Model
To derive the Black-Scholes model, the ito lemma concept is applied which employs the basic assumption of geometric Brownian motion that is used in asset pricing. The geometric Brownian motion, under it lemma, is donated as; Xt = St, a= μSt and b= σSt. Subsequently, the subscript t is dropped to make the formula simple and understandable (Klebaner, 2012). The basic assumption in is that the prices of assets S takes the form of geometric Brownian motion as; ds = uS dt + σS dW. In this case, the inputs u and σ are constant while W represents the Wiener Process. Hereafter, Let V= V(S,t) represent or deliver the value of a stock or any smooth and an uncertain claim, whose order second derivative is differentiated with respect to S, while the first derivative with respect to t. This is because S and t are continuous in the domain. Dv = {(S, t): S ≥ t ≤ T} (Klebaner, 2012). Thereafter, it takes the form from ito lemma that; dv = (∂V/∂S*μS + ∂V/∂t + ½ ∂2V2/∂S2* ∂2 S2) dt + ∂V/∂S*σS dW. This is equation is simply a rephrasing of ito lemma, and plays little in the derivation of the Black-Scholes equation after the applying the no-arbitrage maxim
Due to the fact that the Wienie process W drives both the two stochastic process S and V, it becomes possible to drop the stochastic term σS*av/as * dW by developing a portfolio that is made up of the underlying asset or a stock option; a typical practice in finance. For this reason, Let II be the portfolios’ asset that comprise a single short position with the price S. the value ∆ and V units of the underlying asset. In assumption, the initial value of the wealth/asset is II0, therefore, the value of the portfolio at a time t can be determined from (Klebaner, 2012).
II= -V + ∆S.
Therefore, the significant and influential change in the portfolio in the the portfolio becomes; which is the Black-Scholes equation.
dII = -dV + ∆ds
= - (μS [∆ -∂V/ ∂S] + ∂V/∂t + ½ σ2 S2* S2 ∂2 V/∂S2 ) dt + (- ∂v/ ∂S + ∆) σS dW)
Notably, the increment of the Wiener process has a coefficient (-∂V/∂S + ∆), that relays on ∆, the number of shares of the underlying asset, which causes the fluctuations in the Black-Scholes equation. ∆ = ∂V/∂S (Katz & McCormick, 2012). Therefore shares of the underlying asset, the insignificant change, dII of the portfolio within the time interval dt is precisely deterministic.
dII = - (∂V/ ∂t + ½ σ2S2 * ∂2 V/ ∂S2 ) dt,
Moreover, he drift rate μ has been eliminated to signify the gain in II, the initial wealth invested in the ri...