Dr. Maddah ENMG 625 Financial Eng g II Chapter 12 Forwards, Futures, and Swaps Forward Contracts A forward contract on a commodity is a contract to purchase or sell a specific amount of an underlying commodity at a specific price and at a specific time in the future. E.g., purchase $100 K at1,555 LL/$ on 06/07/2017. The contract is between a buyer and a seller. o The buyer is said to be in a long position (e.g., long $100K.) o The seller is said to be in a short position. A forward contract is a legal document which is binding. Almost always, the initial payment is zero. The forward price is the price that applies at delivery (e.g., 1,555 LL/$ at 06/07/2017). The open market for the underlying asset is called the spot market. The price of the underlying asset in this market is called the spot price. The forward contract itself is traded in the forward market. Forward contracts are priced based on the forward rates which can be determined from the prevailing term structure (see Chapter 4, in Luenberger, ENMG 602). 1
Forward contract payoff Consider a forward contract that matures at time T. Denote by S T the price of the underlying asset at maturity of the contract, and let F be the forward price. Then the payoff of the contract is S T F for the long position (the buyer) and F S T for the short position (the seller). For our $/LL exchange forward contract these payoffs are as follows. 300 200 Long_PayoffS T Short_Payoff S T 0 S T 100 0 100 200 300 1300 1400 1500 1600 1700 1800 S T Forward Prices The forward price, F, of a forward contract, is determined such that the initial contract value (at the time the contract is signed), f, is zero. 2
After the initial time, the contract value, f, may vary depending on the fluctuations of the spot price. Assume a perfect market, i.e., zero transaction costs, divisible assets, zero holding costs, and shorting allowed. The forward price, F, is determined based on the current spot price, S, as F S / d(0, T), where d(0, T) is the discount factor 1 between the initial time (t = 0), and the time the contract is executed (t = T). This forward price formula can be formally proved utilizing an arbitrage argument. For example, suppose F < S/d(0, T), then one can do the following a time t = 0, o Sell short 1 unit of the commodity at a price S. o Lend the sales revenue S for a time T. o Take a one-unit long position in the forward market. 2 Then, at time t = T, (i) one collects the loan earning S/d(0, T), (ii) execute the contract at a cost of F, (iii) then return the shorted unit. Then, the arbitrage gain is S/d(0, T) F. The relationship between the forward price, F, and the spot price, S, is illustrated in the following figure, where S t and F t denoted prices at time t. 1 If f 0,T is the forward rate between time 0 and time T, then d(0,t) = 1/(1+ f 0,T ). 2 I.e., enter a contract to receive one unit of the commodity at a price F at time T. 3
F 0 = S 0 /d(0,t) S T S 0 0 T Carrying costs In certain situations storage costs apply when holding a commodity. In a discrete time framework, suppose that it costs c(k) to carry the commodity from period k to period k + 1. Suppose also that the contract is to be executed in T periods. Then, the forward price is given by T 1 S c( k) F d (0, T ) d ( k, T ), where d(k, T) is discount factors between periods k and T. This formula can be proven with an arbitrage argument as it was done in the case where no storage costs apply. Value of a forward contract The value of a long forward contract with delivery time T and delivery price F 0 at time t is k 0 f ( F F ) d( t, T) S F d( t, T), 3 t t 0 t 0 where F t is the forward price at t. This is shown as follows. 3 The value of a short forward contract at time t is the opposite at (F t F 0 )d(t, T). 4
Suppose f t < (F t F 0 )d(t, T), then do the following at time t. o Borrow f t dollars from the bank o Buy 1 long contract with forward price F 0 (for f t ) o Take a short position on a contract with forward price F t Then, at time T, o Purchase the commodity for F 0 according to the long contract bought at time t. o Sell the commodity for F t according to the short contract signed at t. o Return f t / d(t, T) to the bank. The profit from this strategy is (F t F 0 ) f t / d(t, T) > 0. This represents an arbitrage. Suppose f t > (F t F 0 )d(t, T). Then, do the following: (i) Enter both short and long contracts at time 0; (ii) Sell the long contract at time t and deposit f t in a bank; (iii) Enter a new long contract time at t with forward price F t. (iv) Execute contracts at time T. This makes you an arbitrage profit of f t /d(t, T) F t + F 0 > 0. 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. 5
An example of a commodity swap is the following involving an electric power company hedging the risk of oil prices. 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. 6
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. 7
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) 8
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. Example 9
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. 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. 10
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. 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. 11
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. 4 T 2 2 var[ y] W var[ ST ] 2Wh cov( ST, FT ) h var[ FT ] T. This variance is minimized for cov( ST, FT) h W W, var[ F ] where 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 ]. T 0 4 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. 12