The Black-Scholes Model

Financial analysts have reached the point where they are able to calculate, with high accuracy, the value of a stock option or derivative. The models and techniques employed by today's analysts are based on the Nobel-prize-winning model developed by R. Merton and F. Black and M. Scholes. An example of a derivative is a call option, which is defined as the right, but not obligation, to buy a fixed amount of stock for a certain price on a future specified date. The goal is to buy an option and make money if the stock value rises to cover the price of the option. The basic idea behind the Black-Scholes model is that the option price and stock price depend on the same underlying source of uncertainty, and it is possible to create a portfolio consisting of stocks and options that eliminates this source of uncertainty. The desired portfolio is then instantaneously risk free and enjoys the risk-free interest rate.

To understand the basis of the model, consider three assets:

• Bonds, which accrue at a fixed rate of interest, r, cashable on a date in the future.
• Stocks, which are risky but, on average, appreciate and can't go negative.
• Derivatives, which have a value that depends on the underlying stock (a call option, for instance).

Bonds, stock, and derivatives are modeled respectively by Example 4(a), where the option value c is written as some unknown function of the stock S at time t. To derive the Black-Scholes equation, you construct a portfolio of stock and derivatives such that its value tracks that of a bond. The portfolio may be written as Example 4(b). Assuming no money is added or taken from the portfolio, you can write this as Example 4(c). Next, insert dS and dc from Example 4(a) into Example 4(c). To make the portfolio risk free, eliminate the dW term and make dP=rPdt so that the portfolio behaves like the bond. This yields Example 4(d), the Black-Scholes equation whose solution gives c(S,t) as the price of the option at a given time t.

— D.B.