The stock price is currently at $50 per share. A European call option with 1 year to maturity and a strike price $60 is trading on this stock. Risk-free asset pays 5% interest rate per year. In one year, it is estimated that the stock price will either go up to $80 per share or go down to $31.25 per share.

Find the terminal payoffs of the call option in the up state and in the down state; that is, find cu and cd.

Find u, d, and R to plug into the binomial tree formula.

Use the binomial tree formula to find the risk-neutral probabilities and the call option premium today c.

Find the option delta

If you have written 100 call option contracts, what risk are you facing?

To hedge the risk, should you buy or short the stock? How many shares?

If you buy stock, how much do you borrow? If you sell, how much of the proceeds should you invest?

Now suppose the option expires in two years, keeping everything else the same as in Problem 2.

Based on u, d, and R calculated in the previous problem build the binomial tree for the stock.

Find the terminal option prices c_uu, c_ud, and c_dd.

Using the binomial formula, find c_u, c_d, and c.

Compare c that you calculated in this problem to c that you calculated in the previous problem. What is the difference? How do you explain it?

Find option delta in every node of the tree.

You write 100 call contracts today. How should you hedge this position?

In the up state one year from now, how should you adjust your hedge? Show that the value of your hedge after the adjustment will be exactly equal to the option value in that state.

In the down state one year from now, how should you adjust your hedge? Show that the value of your hedge after the adjustment will be exactly equal to the option value in that state.