Introduction to Options
An "option" is a type of futures contract that gives the buyer the right to buy or sell something of value at a designated date. Options are used to manage risk. They are used when the value of the item being traded fluctuates a great deal. A "call option" can be bought to hedge the risk of an item's price going up. Similarly, a "put option" can hedge against falling prices.

History of Options Trading
No one really knows the specifics regarding how options trading began. However, it is believed that the history is similar to that of the forward contract which began in ancient times and was mainly used in farming. A forward contract is an agreement to buy something in the future. For example, a farmer may be growing corn and wishes to guarantee that his investment will be profitable. To do this, he would find a buyer of the corn and write an agreement stating that the corn will be sold in six months at an agreed upon price. Apparently, at some point in the past, it occurred to someone to create an option rather than an obligation to engage in a future transaction.

Options trading formally began in the United States in 1848 with the creation of the Chicago Board of Trade. Soon after, other exchanges opened in the U.S. including the Kansas City Board of Trade, the Minneapolis Grain Exchange, and the New York Cotton Exchange. "Historically the pricing of options was entirely ad hoc. Traders with good intuition about how other traders would price options made money and those without it lost money," (Wikipedia.org). Throughout the 1900's, there were many formal attempts at creating an equation to value options. However, all of these equations made unreasonable assumptions that could not be applied in the real world.

Black-Scholes Model
Fischer Black was an economist with a PhD in applied math that he earned from Harvard University in 1964. A contemporary of his named Myron Scholes, received an MBA from the University of Chicago in 1964 and a PhD in 1969. Scholes left Chicago to become an assistant professor of finance at the MIT Sloan School of Finance. There he worked closely with Black. Together they developed an equation to value stock options. They named their formula, the "Black-Scholes Model." It is depicted below:

and

In their formula, S = current stock price, K = strike price, T = time in years, r = risk-free interest rate, ? = the stock's volatility

, and N = the cumulative normal distribution function. Basically this equation uses probability theory to predict the chance of a stock (or other item) reaching a certain price. This equation, which was introduced in 1973, is used to value a European option. The two indefinite variables in the equation are the risk-free interest rate and the volatility because both of these variables are constantly changing. In general, the risk-free rate is much easier to estimate because it is basically the rate treasuries are yielding (which is very predictable) during the options' period. Volatility measures how much a stock price will change between now and the expiration date. People continue to argue about the best way to measure this variable and those traders who can estimate volatility the best that are the best options traders.

As noted by Wilmott, Howison & Dewynne, (1995, p. 43), "It is hard to overemphasize the fact that, under the assumptions stated earlier, any derivative security whose price depends only on the current value of S and on t, and which is paid for up-front, must satisfy the Black-Scholes equation (or a variant incorporating dividends or time-dependent parameters). In other words, the Black-Scholes Model can be very effective when used correctly to value options for any type of security.

The Options Market after Black-Scholes
The greatest benefit of being able to accurately price options is the greater ability to hedge risk. Still another important benefit is that companies are able to more accurately account for employee stock options. Another benefit of accurate options' valuation is that more people are willing to enter the options' markets. However, a detrimental effect is that more people are trading rather than actually producing something for the economy.

"Before 1973, all option contracts were what is now called 'over-the-counter' (OTC). That is, they were individually negotiated by a broker on behalf of two clients, one being the buyer and the other the seller. Trading on an official exchange began in 1973 on the Chicago Board Options Exchange (CBOE)" (Wilmott, et al., 1995, p. 10). Once options began to be traded on the CBOE, they began to be traded on other exchanges as well. This resulted in a dramatic increase in options trading.

Originally, the Black-Scholes model was used for stocks. As others have continued to work on the model, they have adapted it to apply to more than just European call options and it is now used with futures contracts, options on futures contracts, American options, bonds, commodities, and currencies. The Black-Scholes model has also been expanded to account for more variables such as transaction costs, dividends, and taxes.

Conclusion
The Black-Scholes Model spearheaded the growth in popularity of options trading. As a result of the introduction of this model, options markets have soared. During the introduction of Black-Scholes and the second year (1974) of operations at the Chicago Board Options Exchange, trading volume was about 200,000 contracts. By the year 2000, trading volume had exploded beyond 200,000,000 contracts per year.

At this time and in the foreseeable future, the Black-Scholes Model is the most trusted and utilized method for options valuation. Whether the model continues to be the primary means of valuing stock options or some new method becomes more popular, it will be essential for those employed in the world of finance to understand the math involved in their valuations. There is a "rapidly growing impetus for new mathematical models and modern mathematical methods; the area is an expanding source of novel and relevant 'real-world' mathematics. The demand from financial institutions for well-qualified mathematicians is substantial," (Wilmott, et al., 1995, p. ix). Therefore, it is critical for those in the investment world to understand the mathematical concepts applicable to investment. Since there is no reason to suggest we will see a decrease in options trading, it will continue to be important for investors and those who counsel them to fully understand the options market and the relevant models.

References:

"Black-Scholes." Wikipedia. 15 December 2005. http://en.wikipedia.org/wiki/Black-Scholes

Walker, Joseph A. (1991) How the Options Markets Work, NYIF Corp: New York.

Wilmott, P, Howison, S. & Dewynne. (1995) The Mathematics of Financial Derivatives: A student Introduction, Press Syndicate of the University of Cambridge.