G6505-001-S16: Homework set 3 Due Wednesday March 2 Submit your solutions as a hardcopy but the spreadsheet must be submitted on CourseWorks 1. For this problem you need to find a source for financial data. Please write down which one you use. Make sure that the market prices that you use are read off at the same time and that you answer the questions to this problem (1) during the same day. (a) Find the current value of the S&P500 index. (b) On what date in April, 2016 do call and put options on the S&P500 index expire? (c) Are these options European or American? (d) How many trading days are there until the expiration date? How many trading days are there in a year? (e) Find the market value of a European call option on the S&P500 index expiring in April, 2016. The strike price should be slightly above the current value of the S&P500 index. (f) Find the market value of a European put option expiring in April, 2016 and the strike price slightly below the current value of the S&P500 index. 2. For this problem please use Excel or similar softwares. Consider an N -period binomial model for the S&P500 index, where N is the number of trading days in (1d). (a) Suppose the interest rate is 0.5% per year. What value of r should you use in your model? (b) Assume that ud = 1 and u, d are constant throughout the tree. Calibrate your model to the market price of the option in (1c). Do this by choosing u and d so that the option price computed by the binomial model is within 1% of the market price found in (1e). Write down: (i) the calibrated values of u and d; (ii) the corresponding conditional risk-neutral probabilities p and q. (c) Using the calibrated model in part (b), compute the price of the following options: (a) the European put option in (1f); (b) a (hypothetical?) American put with the same strike price and expiration date as in (3a). 3. Consider the 1-period trinomial model so that = {H, E, T }. The stock price process S satisfies S0 = 18, S1 (H) = 27, S1 (E) = 21 and S1 (T ) = 15. The interest rate is r = 1/6. Consider a call option expiring at time 1 and with strike price 18. Also consider a put option expiring at time 1 and with strike price 21. (a) Find a portfolio using stocks, call options and the money market account that replicates the put option. (b) Suppose the market price of the call option at time 0 is 2.5. Construct an arbitrage portfolio using the call option, the stock market and the money market. (c) Suppose instead that the market price of the call option at time 0 is 3.9. Construct an arbitrage portfolio as in (b). (d) Find the range of no-arbitrage prices for the call option. Do this by an arbitrage argument (and not using risk-neutral measures). (e) Suppose the market price of the call option is 3. Find the price of the put option using your answer to (a). You should write down P() (f) Find all the risk-neutral measures P. for all . (g) Find the range of no-arbitrage prices for the call option using risk-neutral pricing. In other words, compute the discounted expected payoff under all risk-neutral measures in (f). Verify that you get the same answer as in (d). (h) Suppose the market price of the call option is 3. Given this information, compute the risk-neutral measure P. (i) Using the risk neutral measure in (h), compute the price of the put option. Verify that you get the same answer as in (e). 4. Consider a probability space = {HH, HE, HT, EH, ET, T H, T T }. The interest rate is zero and the adapted stock price process S satisfies S0 = 18, S1 (H) = 27, S1 (E) = 18, S1 (T ) = 9, S2 (HH) = 33, S2 (HE) = 27, S2 (HT ) = 21, S2 (EH) = 27, S2 (ET ) = 15, S2 (T H) = 12, S2 (T T ) = 6. The actual probability measure P is given by P(HH) = P(HE) = P(HT ) = 2/9, P(EH) = P(ET ) = P(T H) = P(T T ) = 1/12. You should write down P() (a) Find all risk-neutral measures P. for all outcomes . (b) Suppose the market prices of a call option with strike 30 and a put option with strike 9 are 1/4 and 3/8, respectively. Given this information, find the risk neutral measure P. 5. Let B be a Brownian motion and Xt = et Be2t . Show that X is a Gaussian process. Compute the mean and covariance functions EXt and Cov(Xt , Xs ). 6. Assume the stock price follows a Bachelier model St = 80 + 10Bt where B is a Brownian motion under the risk-neutral measure. Compute the price of a down-and-out call with strike K = 100, maturity T = 5 and knockout barrier L = 70