MSF Derivatives Homework 1

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Florida State University *

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5537

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Finance

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Feb 20, 2024

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FIN 5537-01: Financial Derivatives - Homework 1 You may complete this homework individually or as a group assignment. 1. Trader A enters into futures contracts to buy 4.0 million euros for 4.4 million dollars in three months. Trader B enters into a forward contract to do the same thing. The exchange (dollars per euro) declines sharply during the first two months and then increases in the third month to close at 1.1120. Ignoring daily settlement, what is the total profit of each trader? When the impact of daily settlement at the exchange is considered, which trader does better? 2. A trader owns gold as part of a long-term investment portfolio. The trader can buy gold for $2,029 per ounce and sell gold for $2,028 per ounce. The trader can borrow funds at 5.5% per year and invest funds at 5.3% per year. (Both interest rates are expressed with annual compounding.) For what range of one-year forward prices of gold will the trader have no arbitrage opportunities? Assume there is no bid offer spread for forward prices. 3. A company wishes to hedge its exposure to a new fuel whose price changes have a 0.94 correlation with crude oil futures price changes. The company will lose $2,350,000 for each $0.01 increase in the price per gallon of the new fuel over the next three months. The new fuel's price changes have an estimated standard deviation 12% greater than the standard deviation of crude oil futures prices. a. What is the company's total exposure measured in gallons of the new fuel? b. If crude oil futures are to be used to hedge the fuel exposure, what should the hedge ratio be? c. What position measured - in barrels - should the company take in crude oil futures? Assume there are 42 gallons to a barrel. d. How many crude oil futures contracts should be traded? Assume each crude oil contract represents 1,000 barrels. 4. A trader owns 28,000 troy oz. of silver and decides to hedge price exposure with 3-month silver futures contracts. Each futures contract is for 5,000 troy oz. The standard deviation of the change in the spot price of silver is 0.25. The standard deviation of the change in silver futures prices is 0.27. The coefficient of correlation between the two is 0.96. a. What is the minimum variance hedge ratio? b. What is the optimal number of futures contracts? 5. A stock provides a dividend yield of 1.75% paid semi-annually (equivalent to 1.74% continuously compounded). The spot price of the stock is currently $64, and the risk-free rate is 4.60% with continuous compounding. a. What is the two-year forward price for a stock? b. What is the continuously compounded cost of carry for the stock?
6. A US investor sees an arbitrage opportunity in the currency markets. The spot exchange rate between the Swiss Franc and US Dollar is 0.9624 ($ per CHF). Assume the continuously compounded interest rates in the US and Switzerland are 4.35% and 3.85%, respectively. The 3-month currency forward price is 0.9585 ($ per CHF). a. What is the theoretically correct forward price? b. What is the investor’s total profit (in $), assuming she begins by borrowing 4,000,000 CHF? 7. The following table gives data on 12 monthly changes in the spot price and the futures price for a certain commodity. Use this data to calculate a minimum variance hedge ratio. Spot Price Change -0.3734 -0.4511 1.1818 -0.1499 -0.3560 -0.0676 Futures Price Change -0.4221 -0.6031 1.4676 -0.2218 -0.3207 -0.1388 Spot Price Change 0.0048 -0.0837 0.3906 1.9142 -0.0958 0.3503 Futures Price Change -0.1023 -0.2634 -0.0766 2.1223 0.4797 0.3637 8. The current price of a stock is $57, and three-month call options with a strike price of $60 currently sell for $2.85. An investor who feels that the price of the stock will increase is trying to decide between buying 100 shares and buying 2,000 call options (20 contracts). Both strategies involve an investment of $5,700. What advice would you give? How high does the stock price have to rise for the all-option strategy to be profitable if the options are held until expiration? 9. An interest rate is quoted as 5.2% per annum with semiannual compounding. What is the equivalent rate with (a) annual compounding, (b) monthly compounding, and (c) continuous compounding? 10. Suppose that LIBOR rates for maturities of one month, two months, three months, four months, five months and six months are 4.30%, 4.65%, 4.90%, 5.15%, 5.35%, and 5.50% with continuous compounding? What are the forward rates for the five future one-month periods? 11. The 6-month, 12-month, 18-month, and 24-month zero rates are 4.60%, 4.90%, 5.20%, and 5.45%, with semiannual compounding. a. What are the rates with continuous compounding? b. What is the forward rate for the 6-month period beginning in 18 months? 12. Assume the risk-free rates are as described in Question 11. What is the value of an FRA where the holder pays LIBOR and receives 5.5% (semiannually compounded) for a six- month period beginning in 18 months? The current forward rate for this period is 4.0% (semiannually compounded) and the principal is $75 million.
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