THE BLACK – SCHOLES FORMULA
In the previous chapter the definition of an option was given. Furthermore, some examples showed what the payoff and profit of an option look like. In these examples the price of the option was always given. It is however possible to identify what the fair price of a European option should be. In this perspective ‘fair’ means that the expected profit for both sides of the option contract is 0. The Black – Scholes formula is a good tool for determining the fair price of an option. From the definition of an option it is clear that the price should depend on the strike price, the price of the underlying stock and the time to maturity. It appears that the price of an option also depends on less obvious variables. These other variables are interest rates, the volatility of the underlying stock (the way the stock moves) and the dividends on the stock. By some simple examples it can be clarified that the option price should also depend on the last mentioned variables:
• Interest rate. Suppose that the interest rate given on a savings account is 5% per year. Consider a put option with a time to maturity of 1 year, and, given an interest rate of 5%, the price of this option is $10. Since the holder of the short position in this option gets this $10, he can put this money in a savings account, getting a 5% interest rate. By doing so he will have $10.5 (10 1.05) at the expiration date of the option. Since the price of the option is fair, the expected ...