Basic Terminology. As the purpose of this paper is not entirely related to the derivation or assumptions necessary for Black-Scholes we will shortly present the fundamental components that came together to spawn this equation. Black-Scholes equation was derived back in the early 70’s by Xxxxxxx Xxxxx and Xxxxx Xxxxxxx [12]. The equation solves the value of a stock option. It is a partial differential equation that relies on simplifying assumptions of financial structures and the mathematical consequences of a perfectly hedged portfolio strategy. Let us quickly recap the terminology necessary for a basic understanding of Black-Scholes and what will be used in this paper. An option is a derivative that one can buy (sell); it gives the buyer the choice to buy (sell) a stock for a specific price, at a future date. The option to buy the stock is called a call option, where the option to sell is called a put option. The price that the underlying can be bought at is called the exercise or strike price and the date by which the transaction must occur is called the date of maturity. In addition, one can buy or sell (short) each option. In the case of selling, one would be paid to give someone the option to buy or sell from them at a future date at the exercise price. The payoff diagrams below help in understanding each option. In addition, there are American options and European options. American options can be exercised any time up until the date of maturity where European options can only be exercised on the date of maturity. The value of an option, V , is a combination of the options intrinsic value and its time value. Intrinsic value is how much an option is in-the- money, that is, max(S − E, 0) or max(E − S, 0) depending on if one has a call or put option. The time value relates to the likelihood that the option will become in-the-money and by how much by the time of expiration. Figure 2.1: Pay-off diagram.
