# Practice Set #1: Forward pricing & hedging.

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1 Derivatives (3 credits) Professor Michel Robe What to do with this practice set? Practice Set #1: Forward pricing & hedging To help students with the material, eight practice sets with solutions shall be handed out These sets contain mostly problems of my own design as well as a few carefully chosen, workedout end-of-chapter problems from Hull None of these Practice Sets will be graded: the number of "points" for a question solely indicates its difficulty in terms of the number of minutes needed to provide an answer Students are strongly encouraged to try hard to solve the practice sets and to use office hours to discuss any problems they may have doing so The best self-test for a student of her/his command of the material is whether s/he can handle the questions of the relevant practice sets The questions on the mid-term and final exams will cover the material covered in class Their format, in particular, shall in large part reflect questions such as the numerical exercises solved in class and/or the questions in the practice sets Question 1 Forward Spot Parity (5 points) In Feb09, gold was trading at \$950 per ounce for spot delivery The lease rate was 0125% 1

4 Derivatives (3 credits) Professor Michel Robe Practice Set #1: Solutions Question 1 (5 points) In Feb 2009, gold was trading at \$950 per ounce for spot delivery The lease rate was about 0% a If the 1-month interest rate was 36% (LIBOR, annualized) and gold storage costs were 1325% per year (annualized), what was the 1-month (net) cost of carry? Assume no convenience yield, and treat all rates as being given under the APR convention Given the lease rate is 0125% and the convenience yield are 0, the (annualized) cost of carry was (36%+1325%-0125%) = 48% b If a gold brokerage was contemporaneously selling gold at \$960 for 1-month forward delivery, was there an arb opportunity? Explain briefly Assume commissions of \$15 per ounce to buy or sell gold, for both forward and spot purchases/sales (Hint: What should have been the 1-month forward gold price?) Gold should have been trading at \$950 (1+36%+12%) 1/12 using the APR convention to annualize/scale periodic rates of return or approximately \$95375 The price of \$960 therefore looks like it offers an arbitrage opportunity, until you take brokerage fees into account Without commissions, you could borrow \$950 to buy gold spot for \$950/1oz, sell it forward for 960 and make \$625 per ounce ( = \$960 - \$950 (1+36%+12%) 1/12 ) profits even taking into account the cost of interest on borrowed funds and the cost to store the gold for a month It would cost you \$30 to trade the gold, however (\$15 to buy spot and \$15 to resell it forward), which would wipe out arb profits Question 2 (15 points) Suppose that the nearby WTI sweet crude oil futures contract (ie, the benchmark contract with the closest expiration date) matures in a month, and that the nearby futures price is \$98 per barrel (i) Based on this information, what should be the OTC (over-the-counter) price of a barrel of WTI for delivery in a month ie, what should the 1-month forward price be? : The price should be \$98 As mentioned in class (and as is explained in more details in LN 4&5 and in the last page of the class handout on futures marking to market), the prices of 4

9 a Can you make money out of those quotes? Explain thoroughly There are two ways to carry out arbitrage ( arb ) strategies in this case: 1 either borrow \$, convert them spot into Yen, deposit the Yen, and sell the Yen forward for \$ in order to repay the dollar loan (in an attempt to make a small profit) 2 or borrow, convert them spot into dollars, deposit the dollars, and sell the dollars forward for in order to repay the Yen loan (in an attempt to make a small profit) The brute-force method to solve this problem is to try both ways, and see if either strategy generates a profit: that is how a computer-based arbitraging program would proceed When proceeding manually, however, note that at most one (if any) strategy can yield a profit, the faster way is to try to assess whether strategy 1 or strategy 2 should be the profitable one It appears that you should be able to make small arbitrage ( arb ) gains, because covered IRP does not seem to hold in this case To see this quickly, let us focus on the round numbers: (i) the \$ is selling at about a 3% 6-month forward discount to the Yen (the 3% figure is obtained by expressing the swap rate of 3 /1\$ (=97-100) as a fraction of the bid spot rate (10000 /1\$); (ii) the interest rate differential, however, is smaller: again concentrating on round figures, the IR diff is about 5% per year annualized (= 55%-05%), or 25% per six months Put differently, it looks as though the dollar is trading at too steep a forward discount to the Yen given the observed interest rate differential This suggests the direction of the possible arbitrage: you need to buy low (buy dollars forward) and sell high (ie, sell Yen forward) In other words, strategy 1 seems like the way to go Assuming that this is the right way to go, you know what else you need to do: in order to get the Yen that you ll be delivering forward, you need to invest Yen for 6 months today; you get those Yen spot, by purchasing them with dollars You don t have dollars, so you borrow them In sum, the arb loop is to borrow dollars at 55%, convert them spot for Yen at 100 (or 001 \$/1 ), deposit the Yen at 05%, and sell the Yen forward at 9750 (or \$/1 ) Formally, the forward rate implied by the interest rate differential and the spot rate is: Now compare this with the 6-month forward rate quoted directly by Daiwa: \$ / 1 Clearly, you should sell Yen forward at \$/1 and buy the Yen forward synthetically at \$/1 by borrowing \$ at 55%, buying spot with the borrowed dollars at 100 /1\$, and investing the at the rate of 05% 9

10 To conclude, let us make sure that the cash-flows all work out: cash-flows today cash-flows in 6 months a + 1\$ (borrow 1\$) \$ (loan repayment incl 6-mo interest of 55%) b - 1\$ (exchange \$ spot for ) none none c (invest at 05% for 6 mo) d none (sell forward for \$) none \$ total \$ The cost is nothing (0 net cash-flow today) For every dollar borrowed, however, the sure gains in six months are 00007\$, ie, a 007% profit margin Note: As an added exercise, you should prove that the reverse strategy (borrowing at 0625%, converting the spot for \$ at /1\$, investing the \$ at 5375% and selling forward the anticipated \$ proceeds for at 9700 /1\$) would lead to a loss b Suppose you must borrow \$1m on behalf of your employer, JP Morgan What would be your total borrowing cost (ie, what is the total number of \$ you would pay on your \$1m loan)? (Hint: what are your borrowing choices?) Your borrowing choices are the following: (1) either borrow \$ at LIBOR (55%): the total \$ cost in 6 months would be \$27,500 (2) or create a similar pattern of cash-flows, borrowing in, converting the into \$, and locking in the \$ cost of the loan through a forward contract Here, the cost would be as follows: - you need \$1m today, hence you borrow 100,500,000 and sell them spot for \$1m (ie, you buy \$1m at the asked price of 10050/1\$) - in 6 months, you will need to pay back 100,814,063; you can lock in today the \$ cost of this repayment by buying the Yen with (ie, by selling) \$1,039,320 six-month forward The total \$ cost would be: \$39,320 The operation I have just described is called a swap Since borrowing directly in \$ is cheaper (\$27,000 vs \$39,320), you should borrow \$ 10

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