Chapter 7 Forwards and Futures

선생님 강의록Chapter 7Forwards and FuturesCopyrightc2008–2011 Hyeong In Choi, All rights reserved.7.1 Basics of forwards and futuresThe financial assets −typically stocks− we have been dealing withso far are the so-called spot assets. By a spot asset, we mean afinancial asset that is sold and bought for immediate delivery (changeof ownership) in exchange for monetary payment. A market in whichspot assets are traded is called a spot market.In contrast to spot assets, forwards and futures are contracts thatstipulate the delivery of a financial asset at a future date. They aresimilar in spirit but differ in details. Typically forward contracts arestruck up between two parties over-the-counter, while the futuresare bought and sold in and managed and by an officially sanctionedexchange.7.1.1 ForwardsForward contract is an agreement to deliver a financial asset at afuture day, say, at time T. Suppose this contract is entered into attime t < T , and let Stdenote the price process of the underlyingspot asset. When this contract is entered into at time t the buyer ofthis forward contract agrees to pay K at time T to the seller of thisforward contract in exchange for the spot asset whose price at time Tis 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 toboth parties. The holder of this contract should have the profit (or7.1. BASICS OF FORWARDS AND FUTURES 208loss) at time T given by ST−K depending on whether STis greateror less K. Its risk neutral value at t has to beVt= BtEQ(ST− K)BT| Ft= St− KBtEQ1BT| Ft. (7.1)If this contract is fair to both parties, Vthas to be zero. Thereforesetting the right hand of (7.1) equal to zero and solving for K, wehaveK =StBtEQh1BT| Fti.This K is called the forward price and we denote it by Gtor G(t, T )in this Chapter.Now, BtEQh1BT| Ftiis the value at t of a contingent claim thatpays 1 at time T. This contingent claim is called a zero-coupon bondand is denoted by p(t, T ), If the interest rate r is constant, p(t, T )simply isp(t, T ) = e−r(T −t).Therefore we have the formula for the forward price Gt:Gt=Stp(t, T )= er(T −t)St. (7.2)Remark 7.1. The forwards and futures are intricately tied with theinterest rate model. But since we have not yet developed an adequatemodel for it, we will later come back to further issues related toforwards and futures when appropriate.Suppose a forward contract is entered into at a time t1. Assumethe interest r is constant. Then the forward price at time t1isGt1= er(T −t1)St1,which is fixed throughout the duration of this contract. At a laterdate, say, at t2> t1, the holder of this forward contract will face profitor loss depending on whether the price St2at time t2of the underlyingspot asset is greater or less than Gt1. First note that the value at t2of7.1. BASICS OF FORWARDS AND FUTURES 209the money is Gt1payable at T is certainly e−r(T −t2)Gt1= er(t2−t1)St1.Thus the profit or loss at t2has to beSt2− er(t2−t1)St1.It can be also seen by using the risk neutral valuation method.Namely, since the profit or loss at time T of the buyer of this forwardcontract is ST− Gt1, its value at t2has to beBt2EQ(ST− Gt1)BT| 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 processGt= er(T −t)St= erTSt∗,it is certainly a Q-martingale as St∗= e−rtStis. However it is aspecial situation that happens to occur in the case of deterministicinterest rate. In general for stochastic interest rate case Gtis not aQ-martingale. As we shall see later, Gtis a martingale with respectto some other measure called the T-forward measure.7.1.2 FuturesThe 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 differences. First, each futures contract is a standard-ized contract that specifies the asset and the delivery date. Second,for such standardized contract, there are buyers and sellers in themarket at any time during the trading day with the usual bid andask prices. When bid and ask prices coincide, a futures contract istraded and a buyer and a seller of the futures contract is established.In this sense, the futures price can be regarded as a price determinedby the market. Such futures price changes constantly during thetrading day depending on the ebbs and flows of the market.There is a special price called the “daily” or “daily closing” pricethat is used to calculate the daily profit and loss settlement. It canbe a closing price of the day. But to guard against manipulations,the exchange sets a more elaborate rule. Its detail does not concernus here. But one must remember that there is a well-defined “daily”price. Third, using this daily price as a reference, the buyer or sellerof futures contract incurs profit or loss everyday. This profit or losshas to be settled daily by crediting or debiting the appropriate bankaccount. This daily settlement feature is what really distinguishesfutures contract from forward contract.7.1. BASICS OF FORWARDS AND FUTURES 210Let us now look into this daily price. To set up the notation, letFt= F (t, T ) be futures price at t of a futures contract with deliverydate T . Let tibe the ith trading day and let Ftibe the “daily” price ofthat day and so on. Thus at the close of (i+1)st day the buyer of thiscontract incurs the daily profit or loss by Fti+1− Fti. If the marketprice should have been determined in such a way that favors neitherthe buyer nor the seller, Ftiand Fti+1must have been determined sothat the risk neutral value at tiof Fti+1−Ftihas to be zero. NamelyBtiEQFti+1− FtiBti+1| Fti= 0.Now the interest rate process is always postulated to be predictable.(If r is constant, it is a moot point, anyway.) Thus Bti+1∈ Fti.Therefore the above equality must imply EQFti+1− Fti| Fti = 0,which again implies thatFti= EQFti+1| Fti .Extending it to any t, we set thatFt= EQ[FT| Ft] .On the other hand, it is obvious that FT= STas T is the deliveryday. Therefore we have to following very importantFt= F (t, T ) = EQ[ST| Ft] . (7.3)Remark 7.3. Unlike the forwards (7.3) implies that the futures priceprocess Ftis always a Q-martingale even with a stochastic interestrate model.However, if the interest rate is deterministic, the forward