correct answers

Uncertainty

Johnny maximizes his expected (or von Neumann Morgenstern-) utility on his personal wealth. That is, Johnny's expected utility from a random gamble X is

E[u( + X)],

where is the money in his wallet, and the expectation is taken according to the distribution of X. Also, Johnny's Bernoulli utility u is CRRA.

Johnny goes to the beach to play cards with the local kids. His winnings are proportional to the money he decides to gamble. Denoting the amount of money that he decides to gamble by a, his stochastic wealth (after playing cards) is w + R a, where is his initial wealth, and R is a random variable. You can assume that R can take 3 possible values R {RW , RT , RL}, corresponding to win, tie, or loss, with associated probabilities {pW , pT , pL}.

(a) Write down Johnny's maximization problem for choosing a (no need to solve for a, just write down Johnny's objective).

(b) Johnny's mother gives him a birthday gift of w, so now Johnny's initial wealth before going to the beach is 2 w. What will be the effect of this gift on Johnny's behavior, is he going to gamble more or less? Provide an brief explanation why.

4. For a two-period binomial model, you are given: (i) Each period is one year. (ii) The current price for a nondividend-paying stock is 20. (iii) u = 1.2840, where u is one plus the rate of capital gain on the stock per period if the stock price goes up. (iv) d = 0.8607, where d is one plus the rate of capital loss on the stock per period if the stock price goes down. (v) The continuously compounded risk-free interest rate is 5%. Calculate the price of an American call option on the stock with a strike price of 22. (A) 0 (B) 1 (C) 2 (D) 3 (E) 4

5. Consider a 9-month dollar-denominated American put option on British pounds. You are given that: (i) The current exchange rate is 1.43 US dollars per pound. (ii) The strike price of the put is 1.56 US dollars per pound. (iii) The volatility of the exchange rate is = 0.3. (iv) The US dollar continuously compounded risk-free interest rate is 8%. (v) The British pound continuously compounded risk-free interest rate is 9%. Using a three-period binomial model, calculate the price of the put. (A) 0.23 (B) 0.25 (C) 0.27 (D) 0.29 (E) 0.31

6. You are considering the purchase of 100 units of a 3-month 25-strike European call option on a stock. You are given: (i) The Black-Scholes framework holds. (ii) The stock is currently selling for 20. (iii) The stock's volatility is 24%. (iv) The stock pays dividends continuously at a rate proportional to its price. The dividend yield is 3%. (v) The continuously compounded risk-free interest rate is 5%. Calculate the price of the block of 100 options. (A) 0.04 (B) 1.93 (C) 3.63 (D) 4.22 (E) 5.09

7. Company A is a U.S. international company, and Company B is a Japanese local company. Company A is negotiating with Company B to sell its operation in Tokyo to Company B. The deal will be settled in Japanese yen. To avoid a loss at the time when the deal is closed due to a sudden devaluation of yen relative to dollar, Company A has decided to buy at-the-money dollar-denominated yen put of the European type to hedge this risk. You are given the following information: (i) The deal will be closed 3 months from now. (ii) The sale price of the Tokyo operation has been settled at 120 billion Japanese yen. (iii) The continuously compounded risk-free interest rate in the U.S. is 3.5%. (iv) The continuously compounded risk-free interest rate in Japan is 1.5%. (v) The current exchange rate is 1 U.S. dollar = 120 Japanese yen. (vi) The daily volatility of the yen per dollar exchange rate is 0.261712%. (vii) 1 year = 365 days; 3 months = year. Calculate Company A's option cost.