Suppose an investor has the opportunity to buy the following contract (a stock call option) on March 1. The contract allows him to buy 100 shares of ABC stock at the end of March, April, or May at a guaranteed price of $50 per share. He can exercise this option at most once. For example, if he purchases the stock at the end of March, he cannot purchase more in April or May at the guaranteed price. If the investor buys the contract, he is hoping that the stock price will go up. The reasoning is that if he buys the contract, the price goes up to $51, and he buys the stock (that is, he exercises his option) for $50, he can then sell the stock for $51 and make a profit of $1 per share. Of course, if the stock price goes down, he doesn’t have to exercise his option; he can just throw the contract away
Assume that the stock price change each month is normally distributed with mean 0 and standard deviation 2. The investor uses the following strategy. At the end of March, he exercises the option only if the stock price is above $51.50. At the end of April, he exercises the option (assuming he hasn’t exercised it yet) only if the price is above $50.75. At the end of May, he exercises the option (assuming he hasn’t exercised it yet) only if the price is above $50.00. (This isn’t necessarily his best strategy, but it’s a reasonable one.) Simulate 250 replications of this strategy and answer the following:
a. Estimate the probability that he will exercise his option.
b. Estimate his net profit with this strategy. (This doesn’t include the price of the contract.)
c. Estimate the probability that he will net over $300.
d. Estimate the worth of this contract to him.