A European put option allows an investor to sell a share of stock at the exercise price on the exercise data. For example, if the exercise price is $48, and the stock price is $45 on the exercise date, the investor can sell the stock for $48 and then immediately buy it back (that is, cover his position) for $45, making $3 profit. But if the stock price on the exercise date is greater than the exercise price, the option is worthless at that date. So for a put, the investor is hoping that the price of the stock decreases. Using the same parameters as in Example 11.8, find a fair price for a European put option. (Note: As discussed in the text, an actual put option is usually for 100 shares.)
EXAMPLE 11.8 PRICING A EUROPEAN CALL OPTION
Ashare of AnTech stock currently sells for $42. A European call option with an expiration date of six months and an exercise price of $40 is available. The stock has an annual standard deviation of 20%. The stock price has tended to increase at a rate of 15% per year. The risk-free rate is 10% per year. What is a fair price for this option?
Objective To use simulation to find the price of a European call option.
WHERE DO THE NUMBERS COME FROM? All of this information is publicly available. The mean and standard deviation would probably be found as in Figure 11.28 from historical stock price data. Interestingly, however, financial analysts often infer the standard deviation of the stock’s returns from the known price of an option on it. They call this standard deviation the implied volatility. Essentially, they “back into” the standard deviation that must have caused the option price to be what it is.