Problem 4 (Hedging using Simulation, 20 pts) Background Info: When a US. company does business around the world, it has to deal with foreign exchange risk, that is, the risk that a foreign currency loses value against the dollar. For example, a company whose operations are in the U.S. (and thus pays for R&D in USD) would be hurt if it held German marks (DM) and the German mark loses value. Suppose the current exchange rate is $0.6513/DM and the company has 100 million DM. This translates to $65.13 million. It the DM loses value (i.e., the USD strengthens), then the exchange rate may be $06413 and the company has lost $1 million. A put option is a special contract designed to mitigate (hedge) such risk. This is how it works: you pay a cost of c for a "put option\" with strike k. The put option gives you the option, hence the name, to convert your DM next year at an exchange rate of I: (this is called exercising the op- tion). In other words, it next year '5 exchange rate D falls below it, then your option is valuable because it allows you to \"bring up\" the exchange rate to k. We have to subtract the cost, so the option gives you the net value of k D c. If D 2 k, then our option is worthless and it has a net value of c. Putting this together, we have Net Payoff = max(k D, 0) c (make sure you see why). Note: D is a random variable (next year's rate), but 1: and c are known (characteristics of the contract known this year). Suppose k = 0.62 and c = 0.01327. 0 Suppose D = 0.5 (i.e., rate falls alot). Then Net Payoff = 0.12 00137 = 0.1063. 0 Suppose D = 0.61 (i.e., rate falls at little). Then Net Payoff = 0.01 00137 = 0.0037. In both cases, we exercised the option, but in the second case, the value of the option was not enough to offset c, the cost of the contract. Let Ec be the revenue generated next year in Germany. Here are two formulas: Unhedged USD Revenue = EG . D. Hedged USD Revenue = Unhedged USD Revenue + Number of Options . Net Payoff. Problem: We'll consider a company that generates revenue EG = 643 million DM next year in Germany and EB = 272 million pounds next year in Great Britain. Using the current exchange rates of 0.6531 for DM and 1.234 for the British pound, the total revenue is around 756 million USD. We will denote next year's exchange rates by D for the DM and B for the British pound. The are determined by the formulas D = 0.6531 . (1+ RD RB 100 and B = 1.234 . 1 + 100 where RD ~ N(0, 92) and RB ~ N(0, 112). They are correlated with p = 0.675. RD and RB are the percentage change from 0.6531 and 1.234 (this year's rates). We consider 9 possible put options on the German mark: k = 0.66, 0.65, 0.64, 0.63, 0.62, 0.61, 0.60, 0.59, 0.55, with respective costs c = 0.0858, 0.0322, 0.0208, 0.0170, 0.0137, 0.0108, 0.0084, 0.0063, 0.0014. We consider also consider 9 possible put options on the British pound: k = 1.30, 1.25, 1.20, 1.15, 1.10, 1.05, 1.00, 0.95, 0.90, with respective costsQuestion: The company will buy 500 million put options on DM and 500 million put options on the British pound. The question now is, which contracts should they buy (i.e., which k?) Assume that they will choose one type of put option for each currency and buy 500 million of it (they won't split the 500 million over different types of options). For each of the 31 possible combinations, estimate the probability that the total hedged revenue will be greater than 706 million USD (make a table). Wlch combination maximizes the estimated probability that the hedged revenue will be greater than 706 million USD? What is this probability? (Estimate the probability in the same way we did in class.) Question: Rather than buying 500 million of each type of option, we could try other combi- nations. Try 100-100 (100 million DM options, 100 million pound options), 100-300, 300-100, 300-300, 300-500, 500-300. What is the maximum estimated probability for each? Which com- bination do you suggest (use the 706 million USD as the benchmark again)? Other Directions, Hints: Generate 10,000 samples for your simulations. The file hedging_skeleton . 111 gives you a basic skeleton