Chapter 7 Forwards and Futures Copyright c cid 13 2008 2011 Hyeong In Choi All rights reserved 7 1 Basics of forwards and futures The nancial assets typically stocks we have been dealing with so far are the so called spot assets By a spot asset we mean a nancial asset that is sold and bought for immediate delivery change of ownership in exchange for monetary payment A market in which spot assets are traded is called a spot market In contrast to spot assets forwards and futures are contracts that stipulate the delivery of a nancial asset at a future date They are similar in spirit but di er in details Typically forward contracts are struck up between two parties over the counter while the futures are bought and sold in and managed and by an o cially sanctioned exchange 7 1 1 Forwards Forward contract is an agreement to deliver a nancial asset at a future day say at time T Suppose this contract is entered into at time t T and let St denote the price process of the underlying spot asset When this contract is entered into at time t the buyer of this forward contract agrees to pay K at time T to the seller of this forward contract in exchange for the spot asset whose price at time T is obviously ST Furthermore this price K called the forward price is determined at the time when this contract is made i e at time t The question is what this K has to be in order for it to be fair to both parties The holder of this contract should have the pro t or 7 1 BASICS OF FORWARDS AND FUTURES 208 loss at time T given by ST K depending on whether ST is greater or less K Its risk neutral value at t has to be Vt BtEQ cid 20 ST K BT cid 21 Ft cid 20 1 BT cid 21 Ft St KBtEQ 7 1 If this contract is fair to both parties Vt has to be zero Therefore setting the right hand of 7 1 equal to zero and solving for K we have K St cid 104 1 BT cid 105 Ft BtEQ This K is called the forward price and we denote it by Gt or G t T in this Chapter Now BtEQ is the value at t of a contingent claim that pays 1 at time T This contingent claim is called a zero coupon bond and is denoted by p t T If the interest rate r is constant p t T simply is cid 104 1 BT Ft cid 105 Therefore we have the formula for the forward price Gt p t T e r T t Gt St p t T er T t St 7 2 Remark 7 1 The forwards and futures are intricately tied with the interest rate model But since we have not yet developed an adequate model for it we will later come back to further issues related to forwards and futures when appropriate Suppose a forward contract is entered into at a time t1 Assume the interest r is constant Then the forward price at time t1 is Gt1 er T t1 St1 which is xed throughout the duration of this contract At a later date say at t2 t1 the holder of this forward contract will face pro t or loss depending on whether the price St2 at time t2 of the underlying spot asset is greater or less than Gt1 First note that the value at t2 of 7 1 BASICS OF FORWARDS AND FUTURES 209 the money is Gt1 payable at T is certainly e r T t2 Gt1 er t2 t1 St1 Thus the pro t or loss at t2 has to be St2 er t2 t1 St1 It can be also seen by using the risk neutral valuation method Namely since the pro t or loss at time T of the buyer of this forward contract is ST Gt1 its value at t2 has to be Bt2EQ cid 20 ST Gt1 BT cid 21 Ft St2 e r T t2 er T t1 St1 St2 er t2 t1 St1 Remark 7 2 If we look at the forward price process Gt er T t St erT St e rtSt is However it is a it is certainly a Q martingale as St special situation that happens to occur in the case of deterministic interest rate In general for stochastic interest rate case Gt is not a Q martingale As we shall see later Gt is a martingale with respect to some other measure called the T forward measure 7 1 2 Futures The futures contract is an agreement to deliver a spot asset at a fu ture date In this respect it is similar to forward contract However there are many di erences First each futures contract is a standard ized contract that speci es the asset and the delivery date Second for such standardized contract there are buyers and sellers in the market at any time during the trading day with the usual bid and ask prices When bid and ask prices coincide a futures contract is traded and a buyer and a seller of the futures contract is established In this sense the futures price can be regarded as a price determined by the market Such futures price changes constantly during the trading day depending on the ebbs and ows of the market There is a special price called the daily or daily closing price that is used to calculate the daily pro t and loss settlement It can be a closing price of the day But to guard against manipulations the exchange sets a more elaborate rule Its detail does not concern us here But one must remember that there is a well de ned daily price Third using this daily price as a reference the buyer or seller of futures contract incurs pro t or loss everyday This pro t or loss has to be settled daily by crediting or debiting the appropriate bank account This daily settlement feature is what really distinguishes futures contract from forward contract 7 1 BASICS OF FORWARDS AND FUTURES 210 Let us now look into this daily price To set up the notation let Ft F t T be futures price at t of a futures contract with delivery date T Let ti be the ith trading day and let Ftibe the daily price of that day and so on Thus at the close of i 1 st day the buyer of this contract incurs the daily pro t or loss by Fti 1 Fti If the market price should have been determined in such a way that favors neither the buyer nor the seller Fti and Fti 1 must have been determined so that the risk neutral value at ti of Fti 1 Fti has to be zero Namely BtiEQ cid 20 Fti 1 Fti Bti 1 cid 21 Fti 0 Now the interest rate process is always postulated to be predictable If r is constant it is a …