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 and

April 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 at the

guaranteed price. The current price of the stock is $$50.$ Each month, assume that the stock

price goes up by a dollar $($with a probability of $0\mathrm{.}55)$ or goes down $($with a probability of $0\mathrm{.}45).$

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, and the price goes up to $$51,$ considering that 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 that he already owns the contract.

B$)$ Now suppose he does not own the contract and is considering buying the contract.

How much should he be willing to pay for such a contract?