Suppose that in each period of length T the cost of a security either goes up by a factor of u = 3/2, or down by a factor of d = 2/3. Suppose that the initial price of the security is S(0) = e1296, and that the interest rate r is such that erT = 21/20. Consider the situation after 4 periods. (a) Determine the possible final prices S(4) for this security, and the probabilities with which they occur. (b) Determine the no-arbitrage cost of a European call option to purchase the security at the end of the 4 periods for a price of e2000. (c) A European asset-or-nothing call option pays its holder a fixed amount F if the price of the security at the exercise time is larger that the strike price K, and pays zero otherwise. Find the fair price of such a call option for an exercise time of 4T, when F = e3000 and K = e1000.

2. An investor pays a random amount of money into his bank account at the start of each month for a total of 5 years. Let Xi denote the amount paid in at the start of month i, and assume that is normally distributed with mean m and variance s2; here m and s do not depend on the month. Assume also that these monthly investments are independent of one another. The annual interest of 5% is compounded monthly. Let Yi denote the present-value of the payment made at the beginning of month i, and Y the present-value of the total investment. (a) Calculate E[Yi] and Var(Yi) in terms of m and s. (b) Suppose that m = e250 and s = e10. Compute E[Y ] and Var(Y ). (c) Suppose that the investor wants to withdraw e400 at the beginning of each month for the three years following the period of investment. Assuming that s = e10 still, how large should m be to ensure that the probability of being able to make these withdrawals is 0.85?