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 can’t purchase more in April or May at the guaranteed price. The current price of the stock is $50. Each month, assume that the stock price either goes up by a dollar (with probability 0.55) or goes down by a dollar (with probability 0.45). 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. On the other hand, if the stock price goes down, he doesn’t have to exercise his option; he can just throw the contract away.
a. Use a decision tree to find the investor’s optimal strategy (that is, when he should exercise the option), assuming he purchases the contract.
b. How much should he be willing to pay for such a contract?