Dr. Maddah ENMG 625 Financial Eng g II 11/09/06. Chapter 10 Forwards, Futures, and Swaps (2)

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1 Dr. Maddah ENMG 625 Financial Eng g II 11/09/06 Chapter 10 Forwards, Futures, and Swaps (2) Swaps A swap is an agreement to exchange one cash flow stream for another. In a plain vanilla swap, one party swaps a series of fixedlevel payments for a series of variable-level payments. E.g., a plain vanilla interest swap. Party B (Nc 0, Nc 2,, Nc M 1 ) (Nr, Nr,, Nr) Party A (Nc 0, Nc 2,, N c M 1 ) Party C The amount N is the national principal. The floating interest rates c i are swapped for the fixed interest rate r. Usually, swaps are netted. Only the difference of swapped payments is paid. An example of a commodity swap is the following involving an electric power company hedging the risk of oil prices. 1

2 Value of a commodity swap Consider a swap where party A receives spot price for N units over M periods of a commodity while paying a fixed price X per unit. Let S 1, S 2,, S M denote the spot prices, then the cash flow stream of party A is N(S 1 X,, S t X,, S M X). Each payment in this stream can be seen as resulting from a future contract. For example, at time t the swap stipulates that party A buys N units of commodity at a forward price F 0 = X / unit. Then, the value of the swap (at time 0) is M [ 0 (0, )], i= 1 V = N S Xd i where S 0 is the current spot price. 2

3 Value of a vanilla interest rate swap Under this swap a floating interest stream of (c 0 N, c 1 N,, c M 1 N) is swapped for a fixed-interest stream (rn,, rn) over M periods. Assuming that the interest rates follow the expectation hypothesis (see Chapter 4), this is equivalent to swapping (r 0 N, r 1 N,, r M 1 N) is for (rn,, rn) over M periods, where r t are the short rates estimated at time 0. The value at time 0 of the stream r t is r t d(0, t+1). (Think of a forward contract on a commodity which is the interest rate on $1 at time t+1. The forward price is r t, which implies that the price at time 0 is r t d(0, t+1)). It follows that the value of the swap is M 1 V = N ( rt r) d(0, t+ 1). t= 0 where d(0, t+1) = 1/[(1+r 0 ) (1+r 1 ) (1+r t )]. It can be also shown that M 1 V = 1 d(0, M) r d(0, t+ 1) N. t= 0 3

4 Basics of futures contracts Futures contract are similar to forward contracts but they are traded on an exchange. In order to avoid having similar outstanding contracts with different delivery prices, the delivery price in a futures contract is revised daily. This process of changing the delivery price is called marking the market. It works as follows: o The contract holders (both in long and short positions) open margin accounts with a broker. o If at the end of the trading day, the price of the futures contract went up, the long party receives the price difference times the contract quantity. The short party looses the same amount. (The reverse happens if the price decreases.) Marking the market implies that the futures contract price at the delivery time may be quite different from the price originally stipulated. In practice, however, most contracts are closed before the delivery date. 4

5 Margin accounts Margin accounts guarantee that the parties will not default. An initial margin requirement imposes the minimum starting values of the account. Then if the value of the account drops down below a maintenance margin (around 75% of initial margin requirement) a margin call is placed to the contract holder demanding additional margin. Examples (Source: The Wall Street Journal) 5

6 Futures Prices Fact 1 (convergence) The futures price converges to the spot price at the contract delivery time. Fact 2 (futures-forward equivalence) Futures and forwards prices of corresponding contracts are identical. Proof. Consider two strategies. In strategy A, do the following: o At time 0, go long d(1, T) futures. o At time 1, increase long position to d(2, T) o.. o At time k, increase long position to d(k+1, T) o At time T 1, increase long position to 1. In this strategy the profit at period k+1 from the previous period is (F k+1 F k ) d(k+1, T). If this is invested at the term structure rates it gives F k+1 F k at time T. Therefore, the total profit from A is T 1 ( Fk+ 1 Fk) = FT F0 = ST F0 at time T. k = 0 In strategy B, take a long position 1 in a forward contract, this strategy has profit S T G 0, where G 0 is the forward price. The two strategies should yield the same profit to avoid arbitrage. Then, F 0 = G 0. 6

7 Example Relation to expected spot price Is F 0 = E[S T ]? If F 0 < E[S T ], speculators enter the market on long positions. Otherwise, they enter on short positions. Hedgers are unlikely to be influenced by small discrepancies between F 0 and E[S T ]. If there are more hedgers in short positions than in long positions, then the market will be balanced (by speculators) only if F 0 < E[S T ]. This is the normal backwadation situation. The opposite situation is called conatago. 7

8 The perfect hedge A perfect hedge completely eliminates the risk associated with a future commitment. This is done by taking an equal and opposite position in the futures market. Such as a strategy is possible only if there is a future contract that exactly matches the delivery. For example, the wheat contractor in the above example takes a long position to hedge against wheat price fluctuation. This hedge matches the obligation perfectly. A company who is going to receive 500 M LL in two months could take a short position with a futures contract to sell 500 M LL at 1,550 LL/$ thus guaranteeing a sure amount of $322,580 after two months. This protects the company against exchange rate risk. The minimum-variance hedge It is not always possible to have a perfect hedge. This could be due to (i) Unavailability of a futures contract on the asset being hedged. (ii) Differences in delivery date the contract and commitment. (iii) The committed asset amount may not be an integral multiple of the contract size. 8

9 A measure of a hedge imperfection is the basis defined as basis = hedged asset spot price at maturity futures price. The basis is a random variable in the absence of a perfect hedge. Suppose you face an obligation to deliver W units of a commodity at time T. Let S T be the spot price of the commodity at that time. To hedge the risk of the commodity price, suppose you take a long position with h units of another commodity (the hedging commodity), whose futures price at T is F T (also equal the spot price of the hedging commodity). Let F 0 be the delivery price of this long contract. Then, your total cash flow at time T Then, y = WS + ( F F ) h. 1 T 2 2 var[ y] W var[ ST] 2Whcov( ST, FT) h var[ FT] T = + +. This variance is minimized for cov( ST, FT) h= W = βw, var[ F ] where β = cov( S, F ) / var[ F ]. T T T The corresponding minimum variance is 2 2 [cov( ST, F )] T var[ y*] = W var[ ST ] var[ FT ]. 1 This formula works for both long and short positions with the convention that W < 0 and h > 0 when the hedger takes a long position and W > 0 and h < 0 for the short position. T 0 9