1

12 Forwards and Futures Under stochastic interest rates, we will show that forward and futures contracts are distinct securities and that forward and futures prices are (usually) different quantities. This distinction is not uniformly understood. There are still textbooks available which confuse these two contacts. One contract cannot be used (in a simple fashion) to arbitrage the other, contrary to occasional practice. The futures contracts analyzed here are simplified versions of the exchange-traded futures contracts (see Chapter 1).

2

A F o r w a r d s

C o n s i d e r a f o r w a r d c o n t r a c t i s s u e d o n aT 2 - m a t u r i t y z e r o - c o u p o n b o n d w i t h d e l i v e r y d a t eT 1 w h e r e 21 TT .

T h e t i m e t f o r w a r d p r i c e o n t h i s c o n t r a c t i sd e n o t e d F ( t , T 1 : T 2 ) .

T h e p u r p o s e o f t h i s s e c t i o n i s t o u s e t h e c o n t i n g e n tc l a i m v a l u a t i o n m e t h o d o l o g y t o g i v e a n a l t e r n a t i v ec h a r a c t e r i z a t i o n o f t h e f o r w a r d p r i c e .

G i v e n t h e r e a r e n o a r b i t r a g e o p p o r t u n i t i e s a n dm a r k e t s a r e c o m p l e t e , t h e s u b s e q u e n t a n a l y s i sp r o c e e d s i n d e p e n d e n t l y o f t h e p a r t i c u l a r e c o n o m y( w i t h o n e , t w o , o r N 3 f a c t o r s ) s t u d i e d .

L e t )( tv d e n o t e t h e v a l u e o f t h i s f o r w a r d c o n t r a c ta t t i m e t , w h e n i t i s i n i t i a t e d .

3

W e v a l u e t h i s f o r w a r d c o n t r a c t u s i n g t h e r i s k -n e u t r a l v a l u a t i o n p r o c e d u r e . T h e t i m e

1T p a y o f f i s

)2:1,()2,1( TTtFTTP .

A p p l y i n g t h e r i s k - n e u t r a l v a l u a t i o n f o r m u l a g i v e s

)()1(

)2:1,()2,1(~)( tB

TB

TTtFTTPtEtv

. ( 1 2 . 1 )

4

B y m a r k e t c o n v e n t i o n , t h e f o r w a r d p r i c e )2:1,( TTtF i s d e t e r m i n e d s u c h t h a t a t i n i t i a t i o n , t h e

f o r w a r d c o n t r a c t h a s z e r o v a l u e .

T h e f o r w a r d p r i c e )2:1,( TTtF i s d e t e r m i n e d s u c h

t h a t .0)()1(

)2:1,()2,1(~

tB

TB

TTtFTTPtE

W e n o w s o l v e t h i s e q u a t i o n .

.0)()1(

)2:1,(~)(

)1(

)2,1(~

tB

TB

TTtFtEtB

TB

TTPtE

( 1 2 . 2 )

5

B u t ,

)()1(

)2,1(~)2,( tB

TB

TTPtETtP

.

S i m i l a r l y ,

)()1(

1~)1,( tB

TBtETtP

.

W e a l s o k n o w t h a t )2:1,( TTtF i s a c o n s t a n t a t t i m e

t .

6

C o m b i n e d , t h e s e f a c t s i m p l y t h a t e x p r e s s i o n ( 1 2 . 2 ) c a n b e r e w r i t t e n a s :

.0)1,()2:1,()2,( TtPTTtFTtP ( 1 2 . 3 )

R e a r r a n g i n g t e r m s g i v e s t h e d e s i r e d r e s u l t :

)1,(

)2,()2:1,(

TtP

TtPTTtF . ( 1 2 . 4 )

T h i s e x p r e s s i o n i s e a s y t o u n d e r s t a n d i f o n e r e c o g n i z e s t h e r a t i o 1 / P ( t , T 1 ) a s t h e f u t u r e v a l u e o f a d o l l a r r e c e i v e d a t t i m e T 1 . T h e n e x p r e s s i o n ( 1 2 . 4 ) c a n b e i n t e r p r e t e d a s t h e f u t u r e v a l u e a t t i m e T 1 o f t h e T 2 - m a t u r i t y z e r o - c o u p o n b o n d – h e n c e , a f o r w a r d p r i c e .

7

The self-financing trading strategy thatsynthetically replicates a forward contract iseasily determined.

The synthetic forward contract is constructedwith what is called a "cash and carry" strategy.

8

Table 12.1: Cash Flows to a Cash and Carry Trading Strategy

9

A f t e r t h e i n i t i a t i o n d a t e , t h e f o r w a r d c o n t r a c t c a n h a v e n o n z e r o v a l u e . I n d e e d , a t a s u b s e q u e n t d a t e t < m T 1 , t h e f o r w a r d c o n t r a c t ' s v a l u e i s

)1,()2:1,()2,(

)()1(

)2:1,()2,1(~)(

TmPTTtFTmP

mBTB

TTtFTTPmEmv

( 1 2 . 5 )

w h i c h c a n d i f f e r f r o m z e r o .

10Figure 12.1: An Example of a One-Factor Bond Price Curve Evolution. The Money Market Account Values and Spot Rates are Included on the Tree. Pseudo-Probabilities Are Along Each Branch of the Tree.

.923845

.942322

.961169

.980392 1

.947497

.965127

.982699 1

.937148

.957211

.978085 1

1/2

1/2

1/2

1/2

1/2

1/2

.967826

.984222 1

.960529

.980015 1

.962414

.981169 1

.953877

.976147 1

.985301 1

.981381 1

.982456 1

.977778 1

.983134 1

.978637 1

.979870 1

.974502 1

1

1

1

1

1

1

1

1

1/2

1/2

1/2

1/2

1/2

1/2

1/2

1/2

P(0,4) P(0,3) P(0,2) P(0,1) P(0,0)

=

B(0) 1

1.02

1.02

1.037958

1.037958

1.042854

1.042854

r(0) = 1.02

1.017606

1.022406

1.016031

1.020393

1.019193

1.024436

1.054597

1.054597

1.059125

1.059125

1.062869

1.062869

1.068337

1.068337

time 0 1 2 3 4

11

EXAMPLE: FORWARD PRICE AND FORWARD CONTRACT VALUES. From chapter 9 we know that the evolution is arbitrage-free. Consider a forward contract with expiration date 2 on the three-period zero-coupon bond. Using the formula for the forward price given in expression (12.4), we compute the time 0 forward price to be:

F(0,2:3) = P(0,3)/P(0,2) = .942322/.961169 = .980392.

12

The same formula generates forward prices attimes 1 and 2 to be:

F(1,2:3;u) = P(1,3;u)/P(1,2;u) = .965127/.982699 =.982119

F(1,2:3;d) = P(1,3;d)/P(1,2;d) = .957211/.978085 =.978658

F(2,2:3;uu) = P(2,3;uu) = .984222F(2,2:3;ud) = P(2,3;ud) = .980015F(2,2:3;du) = P(2,3;du) = .981169F(2,2:3;dd) = P(2,3;dd) = .976147.

13

Figure 12.2: An Example of a Forward Contract Initiated at Time 0 on a 3-Period Zero-Coupon Bond. The forward contract expires at time 2, with value (v(t;s t)) and

forward price (F(t,2:3;st)).

F(0,2:3) = .980392 v(0) = 0

F(1,2:3;u) = .982119 v(1;u) = .001696

F(1,2:3;d) = .978658 v(1;d) = –.001696

F(2,2:3;uu) = .984222 v(2;uu) = .003830

F(2,2:3,ud) = .980015 v(2;ud) = –.000377

F(2,2:3;du) = .981169 v(2;du) = .000777

F(2,2:3;dd) = .976147 v(2;dd) = –.004245

1/2

1/2

r(0) = 1.02

1/2

1/2

r(1;u) = 1.017606

1/2

1/2

r(1;d) = 1.022406

time 0 1 2

14

The value of the forward contract can now be determined. There are two approaches. One, we could use the second formula in expression (12.5). This just involves knowledge of the zero-coupon bond price evolution in Figure 12.1. Two, we could use the risk-neutral valuation approach. We choose the later.

15

The value of the forward contract at time 2 is thespot price of the three-period zero-coupon bond attime 2 less the agreed-upon purchase price.

v(2;uu) = P(2,3;uu) F(0,2:3) = .984222 .980392= .003830

v(2;ud) = P(2,3;ud) F(0,2:3) = .980015 .980392= -.000377

v(2;du) = P(2,3;du) F(0,2:3) = .981169 .980392= .000777

v(2;dd) = P(2,3;dd) F(0,2:3) = .976147 .980392= -.004245

16

T h e v a l u e o f t h e f o r w a r d c o n t r a c t a t t i m e 1 i so b t a i n e d v i a t h e r i s k - n e u t r a l v a l u a t i o n p r o c e d u r e :

001696.022406.1/004245.21000777.2

1

;1/2;21~

);1(

001696.017606.1/000377.21003830.2

1

;1/2;21~

);1(

drsvEdv

ursvEuv

F i n a l l y , a t t i m e 0 :

.002.1/001696.21001696.2

1)0(/1;10)0(

rsvEv

T h e f o r w a r d c o n t r a c t h a s z e r o v a l u e a t i n i t i a t i o na s i t s h o u l d .

17Figure 12.3: An Example of a Forward Contract Initiated Time 0 on a Four-Period Zero-Coupon Bond. The forward contract expires at time 3, with value (v(t;st)) and forward price (F(t,3:4;st)).

F(0,3:4) = .980392 v(0) = 0

F(1,3:4;u) = .981733 v(1;u) = .001294

F(1,3:4;d) = .979041 v(1;d) = –.001294

F(2,3:4;uu) = .983341 v(2;uu) = .002903

F(2,3:4;ud) = .980117 v(2;ud) = –.000270

F(2,3:4;du) = .980886 v(2;du) = .000484

F(2,3:4;dd) = .977186 v(2;dd) = –.003130

F(3,3:4;uuu) = .985301 v(3;uuu) = .004909

F(3,3:4;uud) = .981381 v(3;uud) = .000989

F(3,3:4);udu) = .982456 v(3;udu) = .002064

F(3,3:4;udd) = .977778 v(3;udd) = –.002614

F(3,3:4;duu) = .983134 v(3;duu) = .002742

F(3,3:4;dud) = .978637 v(3;dud) = –.00176

F(3,3:4;ddu) = .979870 v(3;ddu) = –.000522

F(3,3:4;ddd) = .974502 v(3;ddd) = –.005890

r(0) = 1.02

r(1;u) = 1.017606

r(1;d) = 1.022406

r(2;uu) = 1.016031

r(2;ud) = 1.020393

r(2;du) = 1.019193

r(2;dd) = 1.024436

1/2

1/2

1/2

1/2

1/2

1/2

1/2

1/2 1/2

1/2

1/2

1/2 1/2

1/2

18

B Futures

Consider the simplified futures contracts issued ona T2-maturity zero-coupon bond with expirationdate T1. The futures price is denoted by F (t, T1 ;T2).

The purpose of this section is to use the contingentclaim valuation methodology to give an alternativecharacterization for the futures price.

Given no arbitrage opportunities and completemarkets, the subsequent analysis proceedsindependently of the particular economy (withone, two, or N 3 factors) studied.

19

Let the futures contract be initiated at time t. Bymarket convention, the futures price for thiscontract is determined such that the futurescontract has zero value.

Furthermore, marking-to-market guarantees thatthe futures contract also has zero value at eachintermediate date over the contract’s life.

20

T o d e t e r m i n e t h e f u t u r e s p r i c e , w e p r o c e e d i n a b a c k w a r d - i n d u c t i v e f a s h i o n . F o r e a s y c o m p a r i s o n w i t h t h e d e r i v a t i o n o f t h e f o r w a r d p r i c e , l e t u s d e n o t e t h e v a l u e o f t h e f u t u r e s c o n t r a c t a t t i m e t t o b e V ( t ) . W e k n o w b y m a r k e t c o n v e n t i o n , h o w e v e r , t h a t tallfor0V(t) . A t t i m e T 1 - 1 t h e v a l u e o f t h e f u t u r e s c o n t r a c t i s i t s d i s c o u n t e d t i m e T 1 p a y o u t . I t i s g i v e n b y :

)11()1(

)2:1,11()2,1(1

1

~)11(

TB

TB

TTTTTPTETV

F.

( 1 2 . 6 )

21

T h e m a r k e t c o n v e n t i o n i s t h a t

t h e f u t u r e s p r i c e )2:1,11( TTT F i s d e t e r m i n e d s u c h

t h a t .0)11()1(

)2:1,11()2,1(11

~

TB

TB

TTTTTPTE

F

W e n o w s o l v e f o r t h e f u t u r e s p r i c e .

R e c o g n i z i n g t h a t t h e m o n e y m a r k e t a c c o u n t ' sv a l u e B ( T 1 ) i s k n o w n a t t i m e T 1 – 1 , i t c a n b eb r o u g h t o u t s i d e t h e e x p e c t a t i o n s o p e r a t o r .

T h e n , m u l t i p l y i n g b o t h t h e l e f t a n d r i g h t s i d e o ft h i s e x p r e s s i o n b y B ( T 1 ) r e m o v e s i t f r o m t h ee x p r e s s i o n .

F i n a l l y , n o t e t h a t t h e f u t u r e s p r i c e a t e x p i r a t i o n i st h e s p o t p r i c e , i . e . , F ( T 1 , T 1 ; T 2 ) = P ( T 1 , T 2 ) .

22

S u b s t i t u t i o n g i v e s :

02:1,112:1,111

~

TTTTTTTE FF .

( 1 2 . 7 ) G i v e n t h a t )2:1,11( TTT F i s k n o w n a t t i m e T 1 – 1 , w e

g e t :

2:1,111

~2:1,11 TTTTETTT FF .

( 1 2 . 8 )

23

N e x t , a t t i m e T 1 - 2 :

2)1B(T1)1B(T

)]2T:1T2,1F(T)2T:1T1,1[F(T1)1V(T2

1TE

2)1V(T

~ ( 1 2 . 9 a )

B u t , b y m a r k e t c o n v e n t i o n , t h e f u t u r e s c o n t r a c t ’ s v a l u e i s a l w a y s z e r o .

2)1B(T1)1B(T

)]2T:1T2,1F(T)2T:1T1,1[F(T2

1TE0

~ ( 1 2 . 9 b )

A l g e b r a y i e l d s

2:1,1121

~2:1,21 TTTTETTT FF . ( 1 2 . 1 0 )

24

C o n t i n u i n g t h e s a m e a r g u m e n t i n d u c t i v e l y b a c k w a r d i n t i m e g e n e r a t e s o u r f i n a l r e s u l t :

2:1,1

~2:1, TTttETTt FF . ( 1 2 . 1 1 )

T h i s s a y s t h a t u n d e r t h e p s e u d o p r o b a b i l i t i e s , f u t u r e s p r i c e s a r e m a r t i n g a l e s . U s i n g t h e l a w o f i t e r a t e d e x p e c t a t i o n s ( b a c k w a r d s u b s t i t u t i o n ) , t h i s m a r t i n g a l e p r o p e r t y i m p l i e s t h a t

2,1

~2:1, TTPtETTtF . ( 1 2 . 1 2 )

25Figure 12.1: An Example of a One-Factor Bond Price Curve Evolution. The Money Market Account Values and Spot Rates are Included on the Tree. Pseudo-Probabilities Are Along Each Branch of the Tree.

.923845

.942322

.961169

.980392 1

.947497

.965127

.982699 1

.937148

.957211

.978085 1

1/2

1/2

1/2

1/2

1/2

1/2

.967826

.984222 1

.960529

.980015 1

.962414

.981169 1

.953877

.976147 1

.985301 1

.981381 1

.982456 1

.977778 1

.983134 1

.978637 1

.979870 1

.974502 1

1

1

1

1

1

1

1

1

1/2

1/2

1/2

1/2

1/2

1/2

1/2

1/2

P(0,4) P(0,3) P(0,2) P(0,1) P(0,0)

=

B(0) 1

1.02

1.02

1.037958

1.037958

1.042854

1.042854

r(0) = 1.02

1.017606

1.022406

1.016031

1.020393

1.019193

1.024436

1.054597

1.054597

1.059125

1.059125

1.062869

1.062869

1.068337

1.068337

time 0 1 2 3 4

26

EXAMPLE: FUTURES PRICE COMPUTATIONS. We know from Chapter 9 that the evolution is arbitrage-free. Consider a futures contract on a three-period zero-coupon bond with delivery date time 2. By definition, at time 2, the futures price is the spot price: i.e.,

F (2,2:3;uu) = P(2,3;uu) = .984222 F (2,2:3;ud) = P(2,3;ud) = .980015 F (2,2:3;du) = P(2,3;du) = .981169 F (2,2:3;dd) = P(2,3;dd) = .976147.

27

Using expression (12.11), we obtain

F (1,2:3;u) = [(1/2).984222 + (1/2).980015] = .982119

and F (1,2:3;d) = [(1/2).981169 + (1/2).976147] =

.978658. The time 0 futures price is

F (0,2:3) = [(1/2).982119 + (1/2).978658] = .980388.

28

The cash flow at time t, state st is the change in thefutures price, i.e.,

F (t,2:3;st) – F (t-1,2:3;st-1).

For example, at time 2, state uu,

cash flow = F (2,2:3;uu) – F (1,2:3;u) =.984222 - .982119 = .002104.

29

Figure 12.4: An Example of a Futures Contract with Expiration Date 2 on the 3-Period Zero-Coupon Bond. Futures prices (F(t,2:3)) are given at each node. Pseudo-probabilities are on each branch of the tree. The synthetic futures contract (n0(t;st), n3(t;st)) in the money market account

and the 3-period bond are also provided.

(0,2:3) = .980388 (–.411963, .437178)

(1,2:3;u) = .982119 cash flow = .00173 (–.946203, 1)

(1,2:3;d) = .978658 cash flow = –.00173 (–.938442, 1)

(2,2:3;uu) = .984222 cash flow = .002104

(2,2:3;ud) = .980015 cash flow = –.002104

(2,2:3;du) = .981169 cash flow = .002511

(2,2:3;dd) = .976147 cash flow = –.002511

1/2

1/2

1/2

1/2

1/2

1/2

F

F

F

F

F

F

F

r(0) = 1.02

1.017606

1.022406

time 0 1 2

30

To create a synthetic futures contract, we need to create the cash flows in Figure 12.4. The only difference between this procedure and what was previously done is that here we are duplicating cash flows instead of values. Otherwise, it is the same. The procedure for computing the synthetic futures contract is to use the delta approach.

31

A t t i m e 1 , s t a t e u w e c h o o s e n 3 ( 1 ; u ) u n i t s o f t h et h r e e - p e r i o d z e r o - c o u p o n b o n d v i a

1980015.984222.

)002104.(002104.);3,2();3,2(

);2( );2( );1(3

udPuuPudflowcashuuflowcashun

a n d t h e p o s i t i o n i n t h e m o n e y m a r k e t a c c o u n t i s :

.946203.02.1/)965127(.1

)1();3,1();1(30);1(0

BuPunun

32

C o n t i n u i n g , a t t i m e 1 , s t a t e d ,

1976147.981169.

)002511.(002511.);3,2();3,2(

);2( );2( );1(3

ddPduPddflowcashduflowcashdn

a n d

.938442.02.1/)957211(.1

)1()];3,1();1(30[);1(0

BdPdndn

33

F i n a l l y , a t t i m e 0 ,

437178.957211.965127.

)00173.(00173.);3,1();3,1(

);1( );1( )0(3

dPuPdflowcashuflowcashn

a n d

.411963.)942322)(.437178(.0

)3,0()0(30)0(0

Pnn

34

Note that the cost of forming this portfolio is:

n0(1;u)B(1) + n3(1;u)P(1,3;u)= -.946203(1.02) + 1(.965127) = 0.

This is the time 0 arbitrage-free value of thefutures contract, as it should be!

35

Figure 12.5: An Example of a Futures Contract with Expiration Date 3 on a 4-Period Zero-Coupon Bond. Futures prices (F(t,3:4)) are given at each node. Pseudo-probabilities are along the

branches of the tree. The synthetic futures contract (n0(t;st), n4(t;st)) in the money market account and 4-period bond is also provided.

time 0 1 2 3

(0,3:4) = .980383 (–.240452, +.260273)

(1,3:4;u) = .981729 cash flow = .001347 (–.410432, +.441839)

(1,3:4;d) = .979036 cash flow = –.001347 (–.398134, +.433332)

(2,3:4;uu) = .983341 cash flow = .001612 (–.932433, 1)

(2,3:4;ud) = .980117 cash flow = –.001612 (–.925403, 1)

(2,3:4;du) = .980886 cash flow = .001850 (–.922866, 1)

(2,3:4;dd) = .977186 cash flow = –.001850 (–.914680, 1)

1/2

1/2

1/2

1/2

1/2

1/2

F

F

F

F

F

F

F

(3,3:4;uuu) = .985301 cash flow = .001960 F

(3,3:4;uud) = .981381 cash flow = –.001960 F

(3,3:4;udu) = .982456 cash flow = .002339 F

(3,3:4;udd) = .977778 cash flow = –.002339 F

(3,3:4;duu) = .983134 cash flow = .002249 F

(3,3:4;dud) = .978637 cash flow = –.002249 F

(3,3:4;ddu) = .979870 cash flow = .002684 F

(3,3:4;ddd) = .974502 cash flow = –.002684 F

1/2

1/2

1/2

1/2

1/2

1/2

1/2

1/2

r(0) = 1.02

1.017606

1.016031

1.020393

1.019193

1.022406

1.024436

36

C The Relationship Between Forward and Futures Prices This section formalizes the intuition given in Chapter 3 relating forward and futures prices. EXAMPLE: RELATIONSHIP BETWEEN FORWARD AND FUTURES PRICES. This example uses the forward contracts and futures contracts contained in Figures 12.2- 12.5

37

Table 12.2: A Comparison of Forward and Futures Prices for a Two-Period Contract on the Three-Period

Zero-Coupon Bond. Spot Prices are also Included.

(time, state) Forward Price Futures Price Spot Price

0

(1,u)

(1,d)

(2,uu)

(2,ud)

(2,du)

(2,dd)

.980392

.982119

.978658

.984222

.980015

.981169

.976147

.980388

.982119

.978658

.984222

.980015

.981169

.976147

.942322

.965127

.957211

.984222

.980015

.981169

.976147

38

Table 12.3: A Comparison of Forward and Futures Prices for a Three-Period Contract on the Four-Period

Zero-Coupon Bond. Spot Prices are also Included.

39

W e c a n n o w u s e t h e t h e o r y o f t h e p r e c e d i n g s e c t i o n s t o r e l a t e f o r w a r d a n d f u t u r e s p r i c e s . F r o m e x p r e s s i o n ( 1 2 . 4 ) , u s i n g t h e f a c t t h a t

)()1(

)2,1(~)2,( tB

TB

TTPtETtP

,

w e o b t a i n

)1,()(

)1(2,1~

2:1,TtPtB

TB

TTPtETTtF

.

T h i s i m p l i e s :

.

1,1

1,2,1v~co

1,

)(

1

1~2,1

~

2:1,

TtPtB

TBTTP

TtP

tB

TBtETTPtE

TTtF

( 1 2 . 1 4 )

40

F i n a l l y , u s i n g e x p r e s s i o n ( 1 2 . 1 2 ) g i v e s

1Tt,P1

1)1r(Tr(t)1,2T,1TPvco2T:1Tt,F2T:1Tt,F

~ ( 1 2 . 1 5 )

W h e n t h e c o v a r i a n c e i s p o s i t i v e , f o r w a r d p r i c e s e x c e e d f u t u r e s p r i c e s . T h e c o v a r i a n c e i s p o s i t i v e , i n g e n e r a l , w h e n i n t e r e s t r a t e s a r e n e g a t i v e l y c o r r e l a t e d w i t h z e r o - c o u p o n b o n d p r i c e s .

41

D Options on Futures

Consider a futures contract with delivery date T1

on a T2-maturity zero-coupon bond with T1 T2. Consider a European call option on this futures.

Let the European call option have a maturitydate of * < T1 and an exercise price of K > 0.

By definition, the payoff to the option contract atmaturity is

max(F (*,T1:T2 ) - K,0).

42

U s i n g t h e r i s k - n e u t r a l v a l u a t i o n p r o c e d u r e w e o b t a i n

.tB*B/0,K2T:1T*,maxtE~

tC

F ( 1 2 . 1 7 )

T h i s c a l l c a n b e c r e a t e d s y n t h e t i c a l l y u s i n g a n y T - m a t u r i t y b o n d w i t h T > * a n d t h e m o n e y m a r k e t a c c o u n t . I t c a n a l s o b e c r e a t e d s y n t h e t i c a l l y u s i n g t h e u n d e r l y i n g f u t u r e s c o n t r a c t a n d t h e m o n e y m a r k e t a c c o u n t .

43Figure 12.1: An Example of a One-Factor Bond Price Curve Evolution. The Money Market Account Values

and Spot Rates are Included on the Tree. Pseudo-Probabilities Are Along Each Branch of the Tree.

.923845

.942322

.961169

.980392 1

.947497

.965127

.982699 1

.937148

.957211

.978085 1

1/2

1/2

1/2

1/2

1/2

1/2

.967826

.984222 1

.960529

.980015 1

.962414

.981169 1

.953877

.976147 1

.985301 1

.981381 1

.982456 1

.977778 1

.983134 1

.978637 1

.979870 1

.974502 1

1

1

1

1

1

1

1

1

1/2

1/2

1/2

1/2

1/2

1/2

1/2

1/2

P(0,4) P(0,3) P(0,2) P(0,1) P(0,0)

=

B(0) 1

1.02

1.02

1.037958

1.037958

1.042854

1.042854

r(0) = 1.02

1.017606

1.022406

1.016031

1.020393

1.019193

1.024436

1.054597

1.054597

1.059125

1.059125

1.062869

1.019193

1.068337

1.068337

time 0 1 2 3 4

44

EXAMPLE: OPTIONS ON FUTURES. This is the same evolution as in Chapter 9, hence, the evolution is arbitrage-free. Consider the futures contract of Fig. 12.5. This is a three-period futures contract on a four-period zero-coupon bond. Let us value a European call option on this futures price with maturity date 2 and strike price K = 0.981000.

45

The payoffs to the option at date 2 are

C(2;uu) = max(.983341-.981000,0) = .002341

C(2;ud) = max(.980117-.981000,0) = 0

C(2;du) = max(.980886-.981000,0) = 0

C(2;dd) = max(.977186-.981000,0) = 0.

46

The values at time 1 are

C(1;u) = [(1/2).002341 + (1/2)0]/1.017606 =.001150

C(1;d) = [(1/2)0 + (1/2)0]/1.022406 = 0.

Finally, its value at time 0 is

C(0) = [(1/2).001150 + (1/2)0]/1.02 = .000564.

47

T h e s y n t h e t i c f u t u r e s o p t i o n s c a n b e g e n e r a t e d b yi n v e s t i n g i n t h e t h r e e - p e r i o d z e r o - c o u p o n b o n d( n 3 ( t ; s t ) ) a n d t h e m o n e y m a r k e t a c c o u n t ( n 0 ( t ; s t ) ) a sf o l l o w s :

556460.980015.984222.

00023410.);3,2();3,2(

);2();2();1(3

udPuuP

udCuuCun

525400.

02.1/)965127(.556460.001150.

)1();3,1();1(3);1();1(0

BuPunuCun

A t t i m e 1 , s t a t e d , h o l d z e r o u n i t s o f b o t h s i n c e t h ec a l l h a s z e r o v a l u e .

48

A t t i m e 0 ,0145324.

957211.965127.0001150.

);3,1();3,1();1();1()0(3

dPuPdCuCn

.136378.)942322(.145324.000564.

)3,0()0(3)0()0(0

PnCn

49

I n s t e a d , w e c o u l d h a v e h e d g e d t h i s o p t i o n u s i n gt h e u n d e r l y i n g f u t u r e s c o n t r a c t .

T h e s y n t h e t i c p o s i t i o n i n t h e f u t u r e s c o n t r a c t));(( tstFn a n d t h e m o n e y m a r k e t a c c o u n t ));(0( tstn

c a n b e o b t a i n e d a s f o l l o w s :

A t t i m e 1 , s t a t e u ,

726117.)001612.(001612.

0002341.);2( );2(

);2();2();1(

udflowcashuuflowcashudCuuCuFn

a n d

.001127.02.1/]001150[.

)1(0•);1();1(();1(0

BuFnuCun

A t t i m e 1 , s t a t e d , h o l d z e r o u n i t s o f b o t h a s s e t s a st h e o p t i o n h a s z e r o v a l u e .

50

A t t i m e 0 ,

426875.)001347.(001347.

0001150.);1( );1(

);1();1()0(

dflowcashuflowcashdCuC

Fn

a n d

.000564.

0•)0()0()0(0

FnCn

51

E Exchange-Traded Treasury Futures Contracts The futures contracts analyzed in this chapter are simplified versions of the Treasury futures contracts actually traded on organized exchanges. Exchange-traded Treasury futures contracts have various imbedded options within them related to the delivery procedure. These imbedded options are known as (i) the delivery option, (ii) the wildcard option, and (iii) the quality option. These can significantly influence the futures price. To value and hedge these futures contracts with their imbedded options, one can utilize the theory of Chapter 9 in conjunction with an explicit description of the exchange-traded Treasury futures contracts.