In the following problems, we consider a binomial tree where only one step is taken. The risk-free interest rate is r = 0.05, the continuous dividend rate is = 0.04, and the time interval corresponding to the step we take is h = 1/12. The options that we consider expire at time t = h. The current stock price is S = 50. We construct a portfolio consisting of shares of the stock and the amount B in a risk-free bond with interest rate r, where each of and B can be any real number. We assume that uS = 60 and dS = 40. We consider a put option with strike price K = 50. All problems refer to this same situation. 1. Determine the numerical values of u and d. Also determine the numerical values of the payoffs Cu and Cd at time h. Show your work. 2.Write down the formulas that give , B, and the risk-neutral probability p . Then find the numerical value of each of , B, and p , using 5 decimals in each case (so that the numbers are sufficiently accurate to be used in problem 3). 3. Find the numerical value of the option price at the present time. You may have a choice of more than one formula to do this, and any correct formula can be used. Use two decimal places for the option price. Use sufficiently many decimals in intermediate calculations so that the two decimals in the final answer are certainly correct.

Write down the formulas that give , B, and the risk-neutral probability p Then find the numerical value of each of , B, and p , using 5 decimals in each case (so that the numbers are sufficiently accurate to be used in problem 3).