Bonds, forward contracts and futures

Chapter 27 Bonds, forward contracts and futures Let fW t; F t; 0  t  T g be a Brownian motion (Wiener process) on some ; F ; P . Con- sider an asset, which we call a stock, whose price satisfies dS t= rtStdt + tS t dW t: Here, r and  are adapted processes, and we have already switched to the risk-neutral measure, which we call IP . Assume that every martingale under IP can be represented as an integral with respect to W . Define the accumulation factor  t = exp  Z t 0 ru du  : A zero-coupon bond, maturing at time T , pays 1 at time T and nothing before time T . According to the risk-neutral pricing formula, its value at time t 2 0;T is B t; T =tIE  1 T     Ft  =IE  t T     Ft  =IE " exp  , Z T t ru du      F t  : Given B t; T  dollars at time t , one can construct a portfolio of investment in the stock and money 267 268 market so that the portfolio value at time T is 1 almost surely. Indeed, for some process  , B t; T =tIE  1 T     Ft  | z  martingale =  t  IE  1  T   + Z t 0  u dW u  =  t  B 0;T+ Z t 0 u dW u  ; dB t; T =rtt  B0;T+ Z t 0 u dW u  dt +  t t dW t = rtB t; T  dt +  t t dW t: The value of a portfolio satisfies dX t= tdS t+ rtX t , tS tdt = rtX t dt +ttStdW t: (*) We set t= tt tSt : If, at any time t , X t= Bt; T  and we use the portfolio u;tuT , then we will have X T = BT; T = 1: If rt is nonrandom for all t ,then B t; T  = exp  , Z T t ru du  ; dB t; T =rtBt; T  dt; i.e.,  =0 .Then  given above is zero. If, at time t , you are given B t; T  dollars and you always invest only in the money market, then at time T you will have B t; T  exp  Z T t ru du  =1: If rt is random for all t ,then  is not zero. One generally has three different instruments: the stock, the money market, and the zero coupon bond. Any two of them are sufficient for hedging, and the two which are most convenient can depend on the instrument being hedged. CHAPTER 27. Bonds, forward contracts and futures 269 27.1 Forward contracts We continue with the set-up for zero-coupon bonds. The T -forward price of the stock at time t 2 0;T is the F t -measurable price, agreed upon at time t , for purchase of a share of stock at time T , chosen so the forward contract has value zero at time t .Inotherwords, IE  1  T  S T  , F t     F t  =0; 0tT: We solve for F t : 0=IE  1 T ST,Ft     F t  = IE  S T   T      F t  , F t  t IE   t  T      F t  = S t  t , F t  t B t; T : This implies that F t= St Bt; T  : Remark 27.1 (Value vs. Forward price) The T -forward price F t is not the value at time t of the forward contract. The value of the contract at time t is zero. F t is the price agreed upon at time t which will be paid for the stock at time T . 27.2 Hedging a forward contract Enter a forward contract at time 0, i.e., agree to pay F 0 = S 0 B 0;T  for a share of stock at time T . At time zero, this contract has value 0. At later times, however, it does not. In fact, its value at time t 2 0;T is V t=tIE  1 T ST,F0     F t  =  t IE  S T   T      F t  , F 0 IE   t  T      F t  =  t S t  t , F 0B t; T  = S t , F 0B t; T : This suggests the following hedge of a short position in the forward contract. At time 0, short F 0 T -maturity zero-coupon bonds. This generates income F 0B 0;T= S0 B 0;T B0;T=S0: 270 Buy one share of stock. This portfolio requires no initial investment. Maintain this position until time T , when the portfolio is worth S T  , F 0B T; T = ST,F0: Deliver the share of stock and receive payment F 0 . A short position in the forward could also be hedged using the stock and money market, but the implementation of this hedge would require a term-structure model. 27.3 Future contracts Future contracts are designed to remove the risk of default inherent in forward contracts. Through the device of markingtomarket, the value of the future contract is maintained at zero at all times. Thus, either party can close out his/her position at any time. Let us first consider the situation with discrete trading dates 0=t 0 t 1  :::  t n = T: On each t j ;t j+1  , r is constant, so  t k+1  = exp  Z t k+1 0 ru du  = exp 8  : k X j =0 rt j t j +1 , t j  9 = ; is F t k  -measurable. Enter a future contract at time t k , taking the long position, when the future price is t k  . At time t k+1 , when the future price is t k+1  , you receive a payment t k+1  , t k  . (If the price has fallen, you make the payment ,t k+1  , t k  . ) The mechanism for receiving and making these payments is the margin account held by the broker. By time T = t n , you have received the sequence of payments t k+1  , t k ; t k+2  , t k+1 ; ::: ; t n  , t n,1  at times t k+1 ;t k+2 ;::: ;t n . The value at time t = t 0 of this sequence is  t IE 2 4 n,1 X j =k 1  t j +1  t j +1  , t j      F t 3 5 : Because it costs nothing to enter the future contract at time t , this expression must be zero almost surely. CHAPTER 27. Bonds, forward contracts and futures 271 The continuous-time version of this condition is  t IE " Z T t 1  u du     F t  =0; 0tT: Note that  t j +1  appearing in the discrete-time version is F t j  -measurable, as it should be when approximating a stochastic integral. Definition 27.1 The T -future price of the stock is any F t -adapted stochastic process ft; 0  t  T g ; satisfying T =ST a.s., and (a) IE " Z T t 1  u du     F t  =0; 0tT: (b) Theorem 3.66 The unique process satisfying (a) and (b) is t=IE  ST     Ft  ;0t T: Proof: We first show that (b) holds if and only if  is a martingale. If  is a martingale, then R t 0 1  u du is also a martingale, so IE " Z T t 1  u du     F t  = IE  Z t 0 1  u du     F t  , Z t 0 1  u du =0: On the other hand, if (b) holds, then the martingale M t=IE " Z T 0 1 u du     F t  satisfies M t= Z t 0 1 u du+IE " Z T t 1 u du     F t  = Z t 0 1  u du; 0  t  T: this implies dM t= 1 t dt; dt=tdM t; 272 and so  is a martingale (its differential has no dt term). Now define t=IE  ST     Ft  ; 0tT: Clearly (a) is satisfied. By the tower property,  is a martingale, so (b) is also satisfied. Indeed, this  is the only martingale satisfying (a). 27.4 Cash flow from a future contract With a forward contract, entered at time 0, the buyer agrees to pay F 0 for an asset valued at S T  . The only payment is at time T . With a future contract, entered at time 0, the buyer receives a cash flow (which may at times be negative) between times 0 and T . If he still holds the contract at time T ,thenhepays S T  at time T for an asset valued at S T  . The cash flow received between times 0 and T sums to Z T 0 du=T,0 = S T  , 0: Thus, if the future contract holder takes delivery at time T , he has paid a total of 0 , S T  + S T  = 0 for an asset valued at S T  . 27.5 Forward-future spread Future price: t=IE  ST     Ft  . Forward price: F t= St Bt; T  = S t  tIE  1  T      F t  : Forward-future spread: 0 , F 0 = IE S T  , S 0 IE h 1  T  i = 1 IE  1  T    IE  1  T   IE S T  , IE  S T   T   : If 1  T  and S T  are uncorrelated, 0 = F 0: CHAPTER 27. Bonds, forward contracts and futures 273 If 1  T  and S T  are positively correlated, then 0  F 0: This is the case that a rise in stock price tends to occur with a fall in the interest rate. The owner of the future tends to receive income when the stock price rises, but invests it at a declining interest rate. If the stock price falls, the owner usually must make payments on the future contract. He withdraws from the money market to do this just as the interest rate rises. In short, the long position in the future is hurt by positive correlation between 1  T  and S T  . The buyer of the future is compensated by a reduction of the future price below the forward price. 27.6 Backwardation and contango Suppose dS t= S t dt + St dW t: Define  = ,r  ; f W t=t + W t , Z T  = expf,W T  , 1 2  2 T g f IP A= Z A ZTdIP; 8A 2FT: Then f W is a Brownian motion under f IP ,and dS t=rS t dt + S t d f W t: We have  t=e rt S t=S0 expf , 1 2  2 t + W tg = S0 expfr , 1 2  2 t +  f W tg Because 1  T  = e ,rT is nonrandom, S T  and 1  T  are uncorrelated under f IP . Therefore, t= f IEST     Ft = F t = S t B t; T  = e rT ,t S t: The expected future spot price of the stock under IP is IEST = S0e T IE h exp n , 1 2  2 T + W T  oi = e T S 0: 274 The future price at time 0 is 0 = e rT S 0: If r ,then 0 IEST : This situation is called normal backwardation (see Hull). If r , then 0 IEST  . This is called contango. [...]... = expf,W T  , 1 2 T g 2 Z f IP A = Z T  dIP; 8A 2 F T : A f f Then W is a Brownian motion under I , and P f dS t = rS t dt + S t dW t: We have t = ert 1 S t = S 0 expf , 2 2t + W tg f = S 0 expfr , 1 2 t + W tg 2 Because 1  T f = e,rT is nonrandom, S T  and 1T  are uncorrelated under IP Therefore, f t = IE S T  F t = F t  = BSt;tT  = erT ,tS t: . hedged. CHAPTER 27. Bonds, forward contracts and futures 269 27.1 Forward contracts We continue with the set-up for zero-coupon bonds. The T -forward price of. T   : If 1  T  and S T  are uncorrelated, 0 = F 0: CHAPTER 27. Bonds, forward contracts and futures 273 If 1  T  and S T  are positively