Chapter 12. Forwards, Futures, and Swaps

Transcription

1 IE Chapter 12. Forwards, Futures, and Swaps

2 IE Pricing Principles Suppose that your uncle promises that he will give you an ounce of gold 1 year from now, which is worth $1,000 today. How should you evaluate this promised gift? Pricing principles: Use the market. The gold market may be fluctuating, but one can purchase the gold and give it to you next year. So using the market, its value today is $1,000. Discounting certain cash at the current rate of interest. Suppose your uncle promises to give you $1,000 next year, and the interest rate is 10%. Then, the value of the cash is $909. If asset A has value V A, and B has value V B, then the value of a units of A and b units of B is av A + bv B.

3 IE Forward Contracts A forward contract on a commodity is a contract to purchase or sell a specific amount of the commodity at a specific price and at a specific time in the future. Long position: buyer. Short position: seller. Spot market. Forward market.

4 IE Forward interest rate Example Suppose that you wish to arrange to loan money for 6 months beginning 3 months from now. Suppose that the forward rate for that period is 10%. A suitable contract that implements this loan would be an agreement for a bank to deliver to you, 3 months from now, a 6-month Treasury bill. The price would be agreed upon today for this delivery, and the Treasury bill would pay its face value of, say $1000, at maturity. The value of the T-bill would be $1000/1.05=$ This is the price you would agree to pay in 3 months when the T-bill is delivered to you. Six months later you receive the $1000 face value.

5 IE Forward prices Forward price F. Current value of a forward contract. Delivery time T. Spot market price S. Forward price formula. Suppose an asset can be stored at no cost and also sold short. The theoretical forward price is F = S/d(0, T ).

6 IE Proof of forward price formula: One of the following two strategies will provide an arbitrage opportunity if the formula did not hold: t = 0 initial cost final receipt borrow $S S S/d(0, T ) buy 1 unit and store S 0 short 1 forward 0 F Total: 0 F S/d(0, T ) t = 0 initial cost final receipt short 1 unit S 0 lend $S S S/d(0, T ) go long 1 forward 0 F Total: 0 S/d(0, T ) F

7 IE Costs of Carry Forward price formula with carrying costs. Suppose that an asset has a holding cost of c(k) per unit in period k, and the asset can be sold short. Suppose the initial spot price is S. Then the theoretical forward price is F = M 1 S d(0, M) + k=0 c(k) d(k, M), where d(k, M) is the discount factor from k to M. Equivalently S = M 1 k=0 d(0, k)c(k) + d(0, M)F.

8 IE t = 0 time 0 cost time k cost receipt at time M short 1 unit 0 0 F borrow $S S 0 S/d(0, M) buy 1 unit spot S 0 0 borrow c(k) s forward c(0) c(k) M 1 k=0 pay storage c(0) c(k) 0 c(k) d(k,m) Total: 0 0 F S d(0,m) M 1 k=0 c(k) d(k,m)

9 IE Example The current price of sugar is 12 cents per pound. We wish to find the forward price of sugar to be delivered in 5 months. The carrying cost of sugar is 0.1 cent per pound per month, to be paid at the beginning of the month, and the interest rate is constant at 9% per annum. The monthly interest rate is 0.09/12 = 0.75%. The reciprocal of the 1-month discount rate is Therefore, F = ( ) = = cents.

10 IE Tight Market According to the above analysis, the forward prices should be increasing with M. However, it may not necessarily be the case in practice. The main reason for this is that it is difficult or even impossible to reverse the positions: short-selling with the spot price especially when the market on the commodity is tight, and charge the storage costs to someone else. Hence, we will only be able to establish F M 1 S d(0, M) + k=0 c(k) d(k, M). The so-called convenience yield y is the slack to make the above an equality: F = M 1 S d(0, M) + k=0 M 1 c(k) d(k, M) k=0 y d(k, M).

11 IE Investment Assets We can roughly distinguish the commodities by their nature: (1) consumption assets (such as food, cotton, oil,...); (2) investment assets (such as gold, silver, or other precious metals). The main difference is that many people are holding investment assets for profit, and so the market is less likely to be tight. The construction of an arbitrage such as the following is more likely: t = 0 initial cost final receipt short 1 unit S 0 lend $S S S/d(0, T ) go long 1 forward 0 F Total: 0 S/d(0, T ) F Hence, the equation F = S/d(0, T ) is more likely; or, the convenience yield for investment assets is small.

12 IE The value of a forward contract The value of a forward contract. Suppose a forward contract for delivery at time T in the future has a delivery price F 0 and a current forward price F t. The value of the contract is f t = (F t F 0 )d(t, T ), where d(t, T ) is the risk-free discount factor from t to T. Proof. Form the following portfolio at time t: one unit long of a new forward contract with delivery price F t maturing at time T, and one unit short of the old contract with delivery price F 0. The initial cash flow of this portfolio is f t. The final cash flow at time T is F 0 F t. The present value of the portfolio is f t + (F 0 F t )d(t, T ), which must be zero.

13 IE Swaps A swap is an agreement to exchange one cash flow stream for another. Consider an electric power company that must purchase oil every month for its power generation facility.

14 IE Value of a commodity swap. Consider an agreement where party A receives spot price for N units of a commodity each period while paying a fixed amount X per unit for N units. If the agreement is made for M periods, the net cash flow received by A is (S 1 X, S 2 X,, S M X) multiplied by N, where S i is the spot price at time i. The current value of receiving S i at time i is d(0, i)f i. Hence, the total value of the stream is V = M d(0, i)(f i X)N. i=1

15 IE Example Consider an agreement by an electronic firm to receive spot value for gold in return for fixed payments. We assume that gold is in ample supply and can be stored without cost; in that case we know that the forward price is F i = S 0 /d(0, i). Therefore [ ] M V = MS 0 d(0, i)x N. Suppose the price of a bond of maturity M, face value F, coupon C per period is B(M, C). Then, V = {MS 0 XC } [B(M, C) F d(0, M)] N. i=0

16 IE Value of an interest rate swap. Party A agrees to make payments of a fixed rate r of interest on principal N while receiving floating rate payments on the same notional principal for M periods. The cash flow stream received by A is (c 0 r, c 1 r,, c M r) N where c i is the floating rate in period i. The initial value of a floating rate bond is par. The value of the floating rate portion of the swap is par minus the present value of the principal received at M. Hence, the value of the floating rate portion of the swap is N d(0, M)N. The overall value of the swap is [ V = 1 d(0, M) r ] M d(0, i) N. i=1

17 IE Basics of futures contracts Futures market. Marking to market. Margin account. Margin call. Example 12.7 (Margin). Mr. Smith takes a long position of one contract of corn (5,000 bushels) for March delivery at a price of $2.10 per bushel. The broker requires margin of $800 with a maintenance margin of $600. The next day the price drops to $2.07, representing a loss of , 000 = $150. The broker takes this amount from the margin account, leaving a balance of $650. The following day it further drops to $2.05, representing an additional loss of $100. At this point the broker calls Mr. Smith telling him that he must deposit at least $50 in his margin account, or his position will be closed out.

18 IE Futures prices Futures-forward equivalence. Suppose that the interest rates are known to follow expectation dynamics. Then the theoretical futures and forward prices of corresponding contracts are identical.

19 IE Let F 0 be the initial futures price. Let G 0 be the forward price (to be paid at delivery). Consider the following two strategies: Strategy A: Time 0: Go long d(1, T ) futures. Time 1: Increase position to d(2, T ). Time k: Increase position to d(k + 1, T ). Time T 1: Increase position to 1. The profit at time k + 1 from the previous period is (F k+1 F k )d(k + 1, T ). This amounts to the final payment (F k+1 F k )d(k + 1, T ) d(k + 1, T ) = F k+1 F k.

20 IE Therefore, the total final settlement is: T 1 (F k+1 F k ) = F T F 0 = S T F 0. k=0 Strategy B: Take a long position in one forward contract. This requires no initial payment and the final settlement will be S T G 0. Now, let us consider a new strategy: A B. This new strategy requires no cash flow until T, when the value is: G 0 F 0. According to the no-arbitrage principle, we must have G 0 = F 0. We have shown that the initial futures price must be equal to the forward contract value to be delivered in the end.

21 IE The perfect hedge Example A U.S. electronics firm has received an order to sell equipment to a German customer in 90 days. The price of the order is specified as 500,000 euros, which will be paid upon delivery. The U.S. firm faces the exchange risk. The firm can hedge this exchange rate risk with four euros contracts (125,000 per contract) with a 90-day maturity date. Effectively, the firm hedges the risk by taking a short position on four contracts.

22 IE The minimum-variance hedge Sometimes it is not possible to hedge the risk perfectly. The lack of hedging perfection can be measured by the so-called basis: basis = spot price of asset to be hedged - futures price of contract used. Suppose x to be the cash to occur at T, h to be the futures position taken. Then, the cash flow at T is y = x + (F T F 0 )h, with var (y) = var (x) + 2cov (x, F T )h + var (F T )h 2. The minimum-variance hedging formula: h = cov (x, F T ) var (F T ) var (y) = var (x) cov (x, F T ) 2. var (F T )

23 IE Optimal Hedging If a utility function U is available, then it is appropriate to solve max h E[U(x + h(f T F 0 ))]. Suppose the utility function is a quadratic function. Then, the objective becomes E[x + h(f T F 0 )] rvar (x + hf T ). The optimal solution is h = E[F T ] F 0 2rvar (F T ) cov (x, F T ) var (F T ).

24 IE Example In January a large producer of commercial flour and bread wishes to lock in the price for a large order of wheat. The producer would like to buy 500,000 bushels of wheat forward for May delivery. The current futures price for May delivery is $3.30 per bushel. Suppose this producer expects the price of wheat to increase by 5% in 3 months, and the wheat market has approximately 30% volatility per year, so the producer assigns a 15% volatility to the 3-month forecast (15% = 30%/ 4). Using x = 500, 000F T, we have h = 500, E[F T ] F 0 2rvar (F T ) = 500, = 500, E[F T ] F 0 1 2rF 0 var (F T /F 0 ) = 500, r r Suppose r = 1/1, 000, 000, we have h = 164, 000..

25 IE Hedging Nonlinear Risk Example (A corn farmer case). The amount of corn harvested by every farmer depends on the weather, and the price of corn per bushel is determined by the equation P = 10 D/100, 000 where D is the total supply. We assume the amount of corn grown on each farm is C and varies between 0 to 6,000 bushels with E[C] = 3, 000. There are a total of 100 farms, and so D = 100C. The revenue of a farmer will be R = P C = 10C C2 1, 000. Suppose $7 per bushel is the current futures price. Let h be the futures market position. The farmer s revenue will then be P C + h(p P 0 ) = 10C C2 1, E[C] C 1, 000 h. One may wish to find h by maximizing E[U(10C C2 1,000 + E[C] C 1,000 h)].