1

3 The Term Structure of Interest Rates This chapter presents the preliminaries of the model. A The Economy We consider a frictionless, competitive, and discrete trading economy.

2

By frictionless we mean:no transaction costs, no bid/ask spreads,no restrictions on trade (legal or otherwise) suchas margin requirements or short sale restrictions,and no taxes.

The frictionless markets assumption can bejustified on two grounds.

First, large institutional traders approximatefrictionless markets since their transaction costsare minimal. If these traders determine prices,then this model approximates actual pricing andhedging well.

The second argument is that frictionless markets isa necessary prelude to understandingfriction-filled markets.

3

The markets are assumed to be competitive. Thisimplies that the market for any financial securityis perfectly (infinitely) liquid. This is anidealization more nearly satisfied by large volumetrading on organized exchanges than it is in theover-the-counter markets.

Last, we consider a discrete trading economy withtrading dates {0, 1, 2, ..., }. This assumption is areasonable approximation if is large and thetime interval between trading periods is small.

4

B The Traded Securities

Traded in this economy are zero-coupon bonds ofall maturities {0, ..., } and a money marketaccount.

The price of a zero-coupon bond at time t thatpays a sure dollar at time T t is denoted P(t,T).

All zero-coupon bonds are assumed to be defaultfree and have strictly positive prices.

The money market account represents aninvestment portfolio in the shortest term zero-coupon bond. It is initialized at time 0 with adollar investment, and its time t value is denotedB(t) ( so, B(0) = 1).

5

C Interest Rates

Markets quote bond prices using interest rates.

This section defines the most important of these:yields, forward rates, and spot rates.

As a convention in this book, all rates will bedenoted as one plus a percentage (these aresometimes called dollar returns).

6

T h e y i e l d a t t i m e t o n a T - m a t u r i t y z e r o - c o u p o n b o n d , d e n o t e d y ( t , T ) , i s d e f i n e d b y :

)/(1

),(1),(

tT

TtPTty

w i t h y ( t , T ) > 0 .

( 3 . 1 ) I t i s o f t e n c a l l e d t h e h o l d i n g p e r i o d r e t u r n . A l t e r n a t i v e l y w r i t t e n ,

)(),(1),( tTTty

TtP . ( 3 . 2 )

T h e y i e l d i s t h e i n t e r n a l r a t e o f r e t u r n o n t h e z e r o -c o u p o n b o n d .

7

T h e t i m e t f o r w a r d r a t e f o r t h e p e r i o d [ T , T + 1 ] , d e n o t e d f ( t , T ) , i s d e f i n e d b y

)1,(),(),( TtP

TtPTtf . ( 3 . 3 )

T h e f o r w a r d r a t e c a n b e u n d e r s t o o d f r o m t w o p e r s p e c t i v e s . F i r s t , l o o k i n g a t e x p r e s s i o n ( 3 . 3 ) , t h e o n l y d i f f e r e n c e b e t w e e n P ( t , T ) a n d P ( t , T + 1 ) i s t h a t P ( t , T + 1 ) e a r n s i n t e r e s t f o r o n e m o r e t i m e p e r i o d , t h e p e r i o d [ T , T + 1 ] . T a k i n g t h e r a t i o a s i n e x p r e s s i o n ( 3 . 3 ) i s o l a t e s t h e i m p l i c i t r a t e e a r n e d o n t h e l o n g e r m a t u r i t y b o n d o v e r t h i s l a s t t i m e p e r i o d .

8

The second interpretation of the forward rate isthat it corresponds to the rate that one cancontract at time t for a riskless loan over the timeperiod [T,T+1].

9

)1,(),(

10

)1,(),(

)1,( )1,(

),(

1

)1,(),(

1),(

1

TtPTtP

FLOWCASHTOTAL

TtPTtP

TtPTtPTtP

Tmaturitywithbonds

TtPTtPsell

TtPTmaturitywith

bondbuy

TTtTIME

Table 3.2: A Portfolio Generating a Cash Flow

Equal to Borrowing at the Time t Forward Rate for Date T, f(t,T).

10

We can derive an expression for the bond's price in terms of forward rates:

1),(

1),( T

tjjtf

TtP (3.4)

11

Table 3.1: Hypothetical Zero-Coupon Bond Prices, Forward Rates and Yields Time to

Maturity (T) Zero-Coupon Bond Prices P(O,T)

Forward Rates f(O,T)

Yields y(O,T)

PANEL A: FLAT

TERM-STRUCTURE

0 1 2 3 4 5 6 7 8 9

1 .980392 .961168 .942322 .923845 .905730 .887971 .870560 .853490 .836755

1.02 1.02 1.02 1.02 1.02 1.02 1.02 1.02 1.02

1.02 1.02 1.02 1.02 1.02 1.02 1.02 1.02 1.02

PANEL B: DOWNWARD

SLOPING TERM-

STRUCTURE

0 1 2 3 4 5 6 7 8 9

1 .976151 .953885 .932711 .912347 .892686 .873645 .855150 .837115 .820099

1.024431 1.023342 1.022701 1.022319 1.022025 1.021794 1.021627 1.021544 1.020748

1.024431 1.023886 1.023491 1.023198 1.022963 1.022768 1.022605 1.022472 1.022281

PANEL C: UPWARD SLOPING

TERM-STRUCTURE

0 1 2 3 4 5 6 7 8 9

1 .984225 .967831 .951187 .934518 .917901 .901395 .885052 .868939 .852514

1.016027 1.016939 1.017498 1.017836 1.018102 1.018312 1.018465 1.018542 1.019267

1.016027 1.016483 1.016821 1.017075 1.017280 1.017452 1.017597 1.017715 1.017887

12

EXAMPLE: COMPUTING FORWARD RATES AND BOND PRICES USING TABLE 3.1.

f(0,3) = P(0,3)/P(0,4),

1.02 = 0.942322/0.923845 Conversely, given the forward rates, we can determine the bond prices.

P(0,3)=1/f(0,0)f(0,1)f(0,2)

0.942322=1/(1.02)(1.02)(1.02).

13

T h e s p o t r a t e , d e n o t e d r ( t ) , i s t h e r a t e c o n t r a c t e d a t t i m e t o n a o n e - p e r i o d r i s k l e s s l o a n s t a r t i n g i m m e d i a t e l y .

),()( ttftr . ( 3 . 5 ) W e c a n n o w r e t u r n t o t h e m o n e y m a r k e t a c c o u n t .

1

0).()1()1()(

t

jjrtrtBtB ( 3 . 7 )

14

D Forward Prices

A forward contract is a financial securityobligating the purchaser to buy a commodity at aprespecified price (determined at the time thecontract is written) and at a prespecified date. Atthe time the contract is initiated, by convention, nocash changes hands. The contract has zero value. The prespecified purchase price is called theforward price. The prespecified date is called thedelivery or expiration date. We are interested inforward contracts on zero-coupon bonds.

There are three dates of importance for forwardcontracts on zero-coupon bonds:the date the contract is written (t),the date the zero-coupon bond is purchased ordelivered (T1), andthe maturity date of the zero-coupon bond (T2).

The dates must necessarily line up as t T1 T2.

15

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

).2,1()2:1,1( TTPTTTF ( 3 . 8 ) T h e b o u n d a r y c o n d i t i o n o r p a y o f f t o t h e f o r w a r d c o n t r a c t o n t h e d e l i v e r y d a t e i s

).2:1,()2,1( TTtFTTP ( 3 . 9 ) T h e f o r w a r d c o n t r a c t h a s u n l i m i t e d g a i n s a n d u n l i m i t e d l o s s p o t e n t i a l .

16

Figure 3.1: Payoff Diagram for a Forward Contract with Delivery Date T1 on a T2-maturity Zero-coupon Bond

P(T1, T2)

P(T1, T2) - F(t, T1: T2)

0 F(t, T1: T2)

17

E Futures Prices

A futures contract is an agreement to purchase acommodity at a prespecified date, called thedelivery or expiration date, and for a given price,called the futures price. The futures price is paidvia a sequence of random and unequalinstallments over the contract's life. At the timethe contract is initiated, by convention, no cashchanges hands. The contract has zero value atinitiation. A cash payment, however, is made atthe end of each trading interval, and it is equal tothe change in the futures price over that interval. This cash payment resets the value of the futurescontract to zero.

18

D e n o t e t h e t i m e t f u t u r e s p r i c e o n a c o n t r a c t w i t h d e l i v e r y d a t e T 1 o n a T 2 - m a t u r i t y z e r o - c o u p o n b o n d a s )2:1,( TTtF . B y d e f i n i t i o n , t h e f u t u r e s p r i c e f o r a c o n t r a c t w i t h i m m e d i a t e d e l i v e r y 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 i s t h e b o n d ' s p r i c e ; i . e . ,

)2,1()2:1,1( TTPTTT F . ( 3 . 1 0 )

19

T h e c a s h f l o w t o 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 + 1 i s t h e c h a n g e i n 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 o v e r t h e p r e c e d i n g p e r i o d [ t , t + 1 ] , i . e .

).2:1,()2:1,1( TTtTTt FF ( 3 . 1 1 ) T h i s p a y m e n t o c c u r s a t t h e e n d o f e v e r y p e r i o d o v e r t h e f u t u r e s c o n t r a c t ’ s l i f e . T h i s c a s h p a y m e n t t o t h e f u t u r e s c o n t r a c t i s c a l l e d m a r k i n g t o t h e m a r k e t .

20

T i m e

F o r w a r d C o n t r a c t

F u t u r e s C o n t r a c t

t

t + 1

t + 2

T 1 1

T 1

0

0

0

0

P T 1 , T 2 F t , T 1 : T 2

0

F t 1 , T 1 : T 2 – F t , T 1 : T 2

F t 2 , T 1 : T 2 – F t 1 , T 1 : T 2

F T 1 1 , T 1 : T 2 – F T 1 2 , T 1 : T 2

P T 1 , T 2 F T 1 1 , T 1 : T 2

S U M

P T 1 , T 2 F t , T 1 : T 2

P T 1 , T 2 F t , T 1 : T 2

T a b l e 3 . 3 : C a s h F l o w C o m p a r i s o n o f a F o r w a r d a n d F u t u r e s C o n t r a c t

21

T h e r e i s a h e u r i s t i c a r g u m e n t t h a t h e l p s f o r m i n t u i t i o n f o r t h e r e l a t i o n b e t w e e n f o r w a r d a n d f u t u r e s p r i c e 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 w i t h d e l i v e r y d a t e 1T o n t h e 2T - m a t u r i t y b o n d w i t h f o r w a r d p r i c e

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

22

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

L e t u s d e c i d e w h e t h e r a l o n g p o s i t i o n i n a f o r w a r dc o n t r a c t i s p r e f e r r e d t o a l o n g p o s i t i o n i n a f u t u r e sc o n t r a c t w i t h d e l i v e r y d a t e 1T o n t h e s a m e 2T -m a t u r i t y b o n d . I f t h e f o r w a r d c o n t r a c t i sp r e f e r r e d , t h e n t h e f o r w a r d p r i c e s h o u l d b eg r e a t e r t h a n t h e f u t u r e s p r i c e . i . e .

)2:1,()2:1,( TTtFTTt F .

23

We claim that the forward contract on a zero-couponbond is preferred to a futures contract.

Suppose the spot rate increases, then:(i) the zero-coupon bond price falls,(ii) the current futures price falls,(iii) the change in the futures price is negative,

thus(iv) the futures contract has a negative cash

outflow.

But, to get this cash to cover the futures contract’sloss, we need to borrow, and spot rates are high. This is a negative compared to the forward contractthat has no cash flow and an implicit borrowing rateset before rates increased.

24

Next, consider the case where spot rates fall, then:(i) the zero-coupon bond price rises,(ii) the current futures price rises,(iii) the change in the futures price is positive, thus(iv) the futures contract has a positive cash inflow.

But, after getting this cash profit, we need to invest it andspot rates are low. This is a negative compared to theforward contract that has no cash flow and an implicitinvestment rate set before rates decreased.

In both cases the forward contract is preferred to thefutures contract.

The formal proof of these relations requires the fullpower of the theory presented in this book, and thusawaits a subsequent chapter.

25

F Option Contracts

1 Definitions

A call option of the European type is a financialsecurity that gives its owner the right (but not theobligation) to purchase (call) a commodity at aprespecified price (determined at the time thecontract is written) and at a predetermined date. The prespecified price is called the strike price orexercise price. The predetermined date is calledthe maturity date or expiration date.

A call option of the American type is identical tothe European call except that it allows thepurchase decision to be made at any time fromthe date the contract is written until the maturitydate.

26

A put option of the European type is identical tothe European call except that it gives the right tosell (put) the commodity.

A put option of the American type is identical tothe European put except that it allows the selldecision to be made at any time from the date thecontract is written until the maturity date.

Option contracts are written on manycommodities. Examples include zero-couponbonds, coupon bonds, futures contracts, interestrates, and swaps.

27

T o u n d e r s t a n d t h e s e c o n t r a c t s i n m o r e d e t a i l , w ec o n c e n t r a t e o n l y o n o p t i o n s w r i t t e n o n a z e r o -c o u p o n b o n d .

1 P a y o f f D i a g r a m s

L e t t h e u n d e r l y i n g z e r o - c o u p o n b o n d h a v em a t u r i t y d a t e 2T . I t s t i m e t p r i c e i s d e n o t e d

).2,( TtP

C o n s i d e r a E u r o p e a n c a l l o p t i o n w i t h s t r i k e p r i c eK a n d m a t u r i t y d a t e 21 TT w r i t t e n o n t h i s z e r o -c o u p o n b o n d . I t s t i m e t p r i c e i s d e n o t e d

)2:,1,( TKTtC .

28

A t m a t u r i t y i t s p a y o f f o r b o u n d a r y c o n d i t i o n i s :

]0,)2,1(max[)2:,1,1( KTTPTKTTC . ( 3 . 1 2 ) .

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

29

Figure 3.2: Payoff Diagram for a European Call Option on the T2-maturity Zero-coupon Bond with Strike K and Expiration Date T1

KP(T1, T2)

C(T1, T1, K: T2) = max [P(T1, T2) - K, 0]

30

C o n s i d e r a E u r o p e a n p u t o p t i o n w i t h s t r i k e K a n d m a t u r i t y d a t e 21 TT w r i t t e n o n t h i s s a m e z e r o -c o u p o n b o n d . L e t i t s t i m e t p r i c e b e d e n o t e d

)2:,1,( TKTtP . A t m a t u r i t y i t s p a y o f f o r b o u n d a r y c o n d i t i o n i s :

]0),2,1(max[)2:,1,1( TTPKTKTT P ( 3 . 1 3 ) . T h e p a y o f f t o t h e p u t i s n o n - l i n e a r a n d t h e l o s s t o t h e p u t i s b o u n d e d b e l o w b y z e r o . I n c o n t r a s t t o t h e c a l l , t h e p u t ’ s g a i n s a r e b o u n d e d a b o v e b y t h e s t r i k e p r i c e K . T h i s s i m p l e o b s e r v a t i o n s h o w s t h a t a c a l l o p t i o n i s n o t e q u i v a l e n t t o s h o r t i n g a p u t .

31

Figure 3.3: Payoff Diagram for a European Put Option on the T2-maturity Zero-coupon Bond with Strike K and Expiration Date T1

K P(T1, T2)

P(T1, T2, K: T2) = max [K - P(T1, T2), 0]

K