1

10 Coupon Bonds The chapter studies coupon bonds from the perspective of the arbitrage-free pricing methodology. This is in contrast to the classical approach to fixed income analysis or coupon bond pricing that was presented in Chapter 2. The differences between the two approaches are numerous.

2

First, the arbitrage-free pricing methodology canbe used to risk manage a portfolio of bonds givenan arbitrary evolution for the term structure ofinterest rates. The classical approach can onlyhedge parallel shifts in the term structure ofinterest rates.

Second, the arbitrage-free pricing approach can beused to price interest rate derivatives in a mannerconsistent with that used to price coupon-bonds.The classical approach cannot.

Third, the arbitrage-free pricing approach can beextended to handle foreign currency risk andcredit risk. The classical approach cannot.

3

A A Coupon Bond as a Portfolio of Zero-Coupon Bonds This section studies the arbitrage-free pricing of noncallable coupon bonds. The valuation method of this section is independent of the particular evolution of the term structure of interest rates selected; in particular, it does not depend on the number or specification of the factors in the economy, either one, two, or three factors. We define a coupon bond with principle L, coupons C, and maturity T to be a financial security that is entitled to receive coupon payments of C dollars at times 1, …, T with a principal repayment of L at time T. The coupon rate on the bond is c = 1+C/L.

4

Table 10.1: The Cash Flows to a Typical Coupon Bond with Price B(0), Principal L,

Coupon C and Maturity T

Time0 1 2 … T| | | |B(0) C C … C Coupons

L Principal

coupon rate c = 1+C/L

5

T h e c o u p o n b o n d ’ s c a s h f l o w s c a n b e o b t a i n e df r o m a p o r t f o l i o o f z e r o - c o u p o n b o n d s .

T h e d u p l i c a t i n g p o r t f o l i o c o n s i s t s o f C

z e r o - c o u p o n b o n d s m a t u r i n g a t t i m e s = 1 , . . . , T - 1a n d C + L z e r o - c o u p o n b o n d s m a t u r i t y a t t i m e T .

L e t t h e m a r k e t p r i c e o f t h e c o u p o n b o n d b ed e n o t e d B ( t ) .

T h e c o s t o f c o n s t r u c t i n g t h e d u p l i c a t i n g p o r t f o l i oo f z e r o - c o u p o n b o n d s i s :

)T,t(LPT

1ti)i,t(CP

.

I n c o n s t r u c t i n g t h i s p o r t f o l i o , i t i s a s s u m e d t h a tt h e c o n s t r u c t i o n o c c u r s a f t e r t h e c o u p o n p a y m e n th a s b e e n p a i d a t t i m e t ( i . e . , i t r e p r e s e n t s t h ee x - c o u p o n v a l u e a t t i m e t ) .

6

Thus, the arbitrage-free price of the coupon bond is:

)T,t(LPT

1ti)i,t(CP)t(

B . (10.1)

Note that the arbitrage-free price for the coupon bond can be computed without any knowledge of the evolution of the term structure of interest rates. It depends solely on the initial zero-coupon bond price curve. We now illustrate this computation with an example.

7

Table 10.2: An Example of a Time 0 Zero-Coupon Bond Price Curve

P(0,4) = .923845P(0,3) = .942322P(0,2) = .961169P(0,1) = .980392

8

Table 10.3: An Example of the Cash Flows to a Coupon Bond

0 1 2 3 4

$5 $5 $100

time

coupon principal

9

EXAMPLE: COUPON BOND CALCULATION. Consider a coupon bond whose cash flows are as given in Table 10.3. Using expression (10.1), the value of this coupon bond at time 0, ex-coupon, is: B(0) = 5P(0,2) + 5P(0,4)+100P(0,4) = 5[0.961169] + 105[0.923845] = 101.8096. If the market price for the coupon bond differed, an arbitrage opportunity would exist.

10

For example, if the market price of this couponbond were 102.000, then an arbitrage opportunityis represented by: (i) shorting and holding untilmaturity the coupon bond, (ii) buying and holdinguntil maturity five units of the two-periodzero-coupon bond, and (iii) buying and holdinguntil maturity 105 units of the four-periodzero-coupon bond. The initial position brings in102-101.8096 dollars. Subsequently, the cash flowsto the short coupon bond are satisfied by the cashflows from the zero-coupon bond portfolio, leavingno further obligation.

11

The above arbitrage-free pricing technique doesnot depend on a particular evolution for the termstructure of interest rates. This makes theapproach quite useful for pricing

It is less worthwhile, however, for hedging.

In this approach, a synthetic coupon bond isconstructed via a buy and hold strategy involvinga portfolio of zero-coupon bonds.

Zero-coupon bonds are needed for each date onwhich a cash payment to the coupon bond is made. For example, given a 20 year bond with semi-annual coupon payments, 40 zero-coupon bondsare required.

12

This requirement has two practical problems.

One, the particular zero-coupon bonds that matchthe coupon dates most likely do not trade, makingthe replication impossible.

Two, if they all trade, the initial transaction costswill be quite large.

13

B A Coupon Bond as a Dynamic Trading Strategy

This section shows how to use the HJM model tosynthetically construct a coupon bond usingfewer zero-coupon bonds then the number ofpayment dates.

This approach is dependent, however, on aparticular evolution for the term structure ofinterest rates.

EXAMPLE: SYNTHETIC COUPON BONDCONSTRUCTION IN A ONE-FACTORMODEL.

14Figure 10.1: An Example of a One-Factor Bond Price Curve Evolution. 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

15

The first step in applying this technology is to check to see if the given evolution in arbitrage-free. Fortunately, this determination was already performed in Chapter 9. The pseudo probabilities calculated for each maturity zero-coupon bond/money market account pair at each node in the tree are strictly between zero and one, and equal to each other (i.e., 1/2). We use method 2, risk-neutral valuation, which was illustrated in Chapter 7.

16

S t e p 1 : R i s k - N e u t r a l V a l u a t i o n

T h i s a p p r o a c h p r o c e e d s b y b a c k w a r d i n d u c t i o n .

A t t i m e 4 , w e k n o w t h e c a s h f l o w s t o t h e c o u p o nb o n d . T h e y a r e 1 0 5 d o l l a r s , a c r o s s a l l s t a t e s .

M o v i n g b a c k t o t i m e 3 , a t s t a t e u u u , w e c o m p u t et h e p r i c e o f t h e c o u p o n b o n d a s :

4566.1030149182.1

105)2/1(105)2/1();3(

);4()2/1();4()2/1();3(

uuur

uuuduuuuuuu BBB.

17

Figure 10.2: The Evolution of the Coupon Bond's Price for the Example in Table 10.3.The coupon payment at each date is indicated by the nodes. The Synthetic Coupon-Bond Portfolio (n 0(t;st), n4(t;st)) in the money

market account and four-period zero-coupon bond are given under each node. Pseudo-probabilities along the branches of the Tree.

101.8096

(2.7567,107.218)

104.4006 (4.81516,105.002)

103.2910 (4.79273,105.002)

101.6218 Coupon = 5 (.000067,105)

100.8556 Coupon = 5 (-.001925,105.002)

101.0535 Coupon = 5

(-.023571,105.026)

100.1571 Coupon = 5

(-.010209,105.0112)

103.4566 (98.1006,0)

103.0450 (97.7103,0)

103.1579 (97.3992,0)

102.6667 (96.9354,0)

103.2291 (97.1231,0)

102.7568 (96.6787,0)

102.8864 (96.3052,0)

102.3227 (95.7775,0)

100 Coupon = 5

100 Coupon = 5

100 Coupon = 5

100 Coupon = 5

100 Coupon = 5

100 Coupon = 5

100 Coupon = 5

100 Coupon = 5

1/2

1/2

1/2

1/2 1/2

1/2

1/2

1.02

1.017606

1.016031

1.020393

1.019193

1.022406

1/2

1/2

1/2

1/2 1/2

1/2

1.024436

1/2

time 0 1 2 3 4

18

N e x t , a t t i m e 2 , w e r e p e a t t h e p r e v i o u s p r o c e d u r e .

F o r e x a m p l e , a t t i m e 2 s t a t e u u , t h e c o m p u t a t i o ni s :

6218.101016031.1

0450.103)2/1(4566.103)2/1();2(

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

uur

uuduuuuu BBB.

19

S t e p p i n g b a c k t o t i m e 1 , s t a t e u , w e r e p e a t t h ep r e v i o u s p r o c e d u r e , b u t t h i s t i m e w i t h a t w i s t .

W h e n c o m p u t i n g t h e d i s c o u n t e d e x p e c t e d v a l u e ,w e n e e d t o i n c l u d e b o t h t h e f u t u r e v a l u e a n df u t u r e c a s h f l o w i n t h e n u m e r a t o r .

T h e c a l c u l a t i o n i s :

.4006.104017606.1

]58556.100)[2/1(]56218.101)[2/1();1(

]);2()[2/1(]);2()[2/1();1(

ur

CudCuuu BBB

20

Finally, at time 0, the procedure yields:

8096.10102.1

2910.103)2/1(4006.104)2/1()0(

);1()2/1();1()2/1()0(

r

du BBB.

21

Step 2: Delta Hedging The next step is to construct the synthetic coupon bond using a dynamic self-financing trading in the four-period zero-coupon bond (n4(t;st)) and the money market account (n0(t;st)). We use the delta approach that was described in Chapter 7. This approach proceeds by backward induction, and it uses the prices of the coupon bond generated in step 1 above.

22

F i r s t , a t t i m e 3 t h e r e i s o n l y o n e z e r o - c o u p o n b o n dt r a d i n g , t h e f o u r - p e r i o d z e r o . I t i s u s e d t oc o n s t r u c t t h e m o n e y m a r k e t a c c o u n t . T h u s w ec a n i n v e s t i n e i t h e r t h e f o u r - p e r i o d z e r o o r t h em o n e y m a r k e t a c c o u n t ( s i n c e b o t h a r e i d e n t i c a l ) . W e c h o o s e , a r b i t r a r i l y , t h e m o n e y m a r k e ta c c o u n t :

T i m e 3 , s t a t e u u u :

1006.98054597.1

4566.103);3(

);4,3();3(4);3();3(0

0);3(4

uuB

uuuPuuunuuuuuun

uuun

B

T i m e 3 , s t a t e u u d :

7103.97054697.1

0450.103);3(

);4,3();3(4);3();3(0

0);3(4

uuB

uudPuudnuuduudn

uudn

B

23

M o v i n g b a c k t o t i m e 2 , w e a p p l y t h e d e l t ac o n s t r u c t i o n a g a i n .

T h e c o m p u t a t i o n i s :

T i m e 2 , s t a t e u u

000067.037958.1

967826.0)105(6218.101);2(

);4,2();2(4);2();2(0

105981381.985301.

0450.1034566.103);4,3();4,3(

);3();3();2(4

uB

uuPuunuuuun

uudPuuuPuuduuuuun

B

BB

24

Time 2, state ud:

001925.037958.1

960529.0)002.105(8556.100);2(

);4,2();2(4);2();2(0

002.105977778.982456.

6667.1021579.103);4,3();4,3(

);3();3();2(4

uB

udPudnududn

uddPuduPudduduudn

B

BB

25

M o v in g b a c k t o t im e 1 , w e g e t

T im e 1 , s t a t e u :

81516.402.1

947497.0)002.105(4006.104)1(

);4,1();1(4);1();1(0

002.105960529.967826.

]58556.100[]56218.101[);4,2();4,2(

]);2([]);2([);1(4

B

uPunuun

udPuuPCudCuuun

B

BB

26

Time 1, state d:

.79273.402.1

937148.0)002.105(291.103)1(

);4,1();1(4);1();1(0

002.105953877.962414.

]51571.100[]50535.101[);4,2();4,2(

]);2([]);2([);1(4

B

dPdndun

ddPduPCddCdudn

B

BB

27

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

.75670.21

923845.0)218.107(8096.101)0(

)4,0()0(4)0()0(0

218.107937148.947497.

291.1034006.104);4,1();4,1(

);1();1()0(4

B

Pnn

dPuPdun

B

BB

T h e c o s t o f c o n s t r u c t i n g t h i s s y n t h e t i c c o u p o nb o n d a t t i m e 0 i s

8096.101)923845(.218.1077567.2

)4,0()0(4)0(0

Pnn.

28

This synthetic construction is more complicatedthan the buy and hold strategy discussedpreviously.

This synthetic construction is dynamic and itinvolves rebalancing the portfolio across time.

The rebalancing is self-financing at times 1 and 3,but it generates a positive cash flow of 5 dollars attime 2. Hence, only at time 2 is the syntheticcoupon bond not self-financing.

29

C Comparison of HJM Hedging versus Duration Hedging This section compares HJM hedging versus the classical duration hedging of Chapter 2. EXAMPLE: ERROR IN MODIFIED DURATION HEDGING ZERO-COUPON BONDS

30

Figure 10.1: An Example of a One-Factor Bond Price Curve Evolution. 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

31

C o n s i d e r h o l d i n g a p o s i t i o n i n c o u p o n b o n d “ a ”a n d d e s i r i n g t o h e d g e t h i s p o s i t i o n w i t h a c o u p o nb o n d “ b ” .

L e t t h e t w o b o n d s ( a , b ) c o n s i d e r e d b e z e r o - c o u p o nb o n d s o f v a r i o u s m a t u r i t i e s ; i . e . ,

.923845.)4,0()0(

961169.)2,0()0(

Pb

andPaBB

32Figure 10.3: A Comparison of HJM Hedging versus Duration Hedging. The Bond Trading Strategy (na(0), nb(0)) is Given.

HJM .549287(1, -.445825)

Duration .480585(1, -.52020)

Actual PayoffHJM .56027Duration .489811

HJM .56027Duration .490581

Investment

1/2

1/2

r(0) = 1.02

time 0 1

Duration hedge (if corrcct)1.02(.480585)=.490197

33

1 T h e H J M H e d g e

F r o m C h a p t e r 8 , w e k n o w t h a t a h e d g e d p o r t f o l i oi n v o l v i n g t h e s e t w o z e r o s c a n o n l y b e o b t a i n e d b yc r e a t i n g t h e “ a ” b o n d s y n t h e t i c a l l y u s i n g t h e “ b ”b o n d .

T h e d e l t a g i v e s t h e a p p r o p r i a t e p o s i t i o n i n t h e “ b ”b o n d .

T h e e n t i r e p o r t f o l i o i s t h e n l o n g 1 u n i t o f b o n d “ a ”a n d s h o r t bn u n i t s o f b o n d “ b ” , i . e

.445835.);4,1();4,1();2,1();2,1(

1

dPuPdPuP

bn

andan

T h e i n i t i a l i n v e s t m e n t i n t h i s h e d g e d p o r t f o l i o i s :

.549287.)923845(.445835.)961169(.1

)4,0()2,0(

PbnPan

34

I f t h e p o r t f o l i o i s r i s k l e s s , t h e n t o a v o i d a r b i t r a g ea t t i m e 1 , i t s v a l u e s h o u l d b e t h e i n i t i a l i n v e s t m e n tt i m e s t h e s p o t r a t e o f i n t e r e s t o v e r [ 0 , 1 ] , i . e .

. 5 4 9 2 8 7 ( 1 . 0 2 ) = . 5 6 0 2 7 .

T h e v a l u e o f t h e H J M h e d g e d p o r t f o l i o a t t i m e 1c a n b e c o m p u t e d a s f o l l o w s :

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

.56027.)947497(.445835.)982699(.1

);4,1();2,1(

uPbnuPan

T i m e 1 , s t a t e d

.56027.)937148(.445835.)978085(.1

);4,1();2,1(

dPbndPan

T h e v a l u e s a r e e x a c t l y a s n e e d e d t o g e n e r a t e ar i s k l e s s p o r t f o l i o . S o t h e H J M h e d g e w o r k s !

35

2 T h e D u r a t i o n H e d g e

W e n o w c a l c u l a t e t h e h e d g e b a s e d o n m o d i f i e dd u r a t i o n .

F o r t h i s e x a m p l e i t i s e a s y t o s h o w t h a t t h a td u r a t i o n o f b o n d s “ a ” a n d “ b ” a r e :

.4)0( 2)0( bDa ndaD

T h i s f o l l o w s b e c a u s e a z e r o - c o u p o n b o n d ' sd u r a t i o n i s a l w a y s e q u a l t o i t s t i m e t o m a t u r i t y .

G i v e n t h a t t h e f o r w a r d r a t e c u r v e i s f l a t a t 1 . 0 2 ,w e h a v e t h a t t h e y i e l d o n b o t h b o n d s “ a ” a n d “ b ”a r e i d e n t i c a l a n d e q u a l t o Y a ( 0 ) = Y b ( 0 ) = 1 . 0 2 .

T h e b o n d s ’ m o d i f i e d d u r a t i o n s a r e :

.0 2.1/4)0(,0 2.1/2)0(, bMDa ndaMD

36

T h e m o d i f i e d - d u r a t i o n h e d g e i s d e t e r m i n e d f r o m e x p r e s s i o n ( 2 . 1 3 ) i n C h a p t e r 2 . I t i s g i v e n b y

52020.02.1/)923845(.402.1/)961169(.2

)0()0(,

)0()0(,

1

bbMDaaMD

bn

an

B

B

T h e i n i t i a l i n v e s t m e n t i n t h i s p o r t f o l i o i s :

.480585.)923845(.52020.)961169(.1

)4,0()2,0(

PbnPan

37

Again, if the portfolio is riskless, then to avoidarbitrage at time 1, its value should be the initialinvestment times the spot rate of interest over[0,1], i.e.

(.480585)1.02 = .490197.

38

T h e v a l u e o f t h e d u r a t i o n h e d g e d p o r t f o l i o a t t i m e1 i s :

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

.489811.)947497(.52020.)982699(.1

);4,1();2,1(

uPbnuPan

T i m e 1 , s t a t e d

.490581.)937148(.52020.)978085(.1

);4,1();2,1(

dPbndPan

T h i s p o r t f o l i o i s n o t r i s k l e s s ! I t e a r n s m o r e i n t h ed o w n s t a t e a n d l e s s i n t h e u p s t a t e t h e n t h e s p o tr a t e o f i n t e r e s t .

T h i s i l l u s t r a t e s t h a t t h e d u r a t i o n h e d g e d o e s n o tw o r k .