1. 2 Figure 7.1: A Graphical Representation of “Arbitraging the Zero-coupon Bond Price Curve”...

1

7 Bond Trading Strategies – An Example A Motivation This chapter shows how to use the concepts of trading strategies, arbitrage opportunities and complete markets to investigate mispricings within the yield curve. Taken as given are the stochastic processes for a few zero-coupon bonds, the stochastic process for the spot rate, and the assumption that there are no arbitrage opportunities. The purpose is to find the arbitrage-free prices of all the remaining zero-coupon bonds.

2Figure 7.1: A Graphical Representation of “Arbitraging the Zero-coupon Bond Price Curve”

Time (T)

Prices P(0,T)

| | | |0 1 2 3 4

1X

FM F=M

X

X given priceF fair priceM market price

3

B Method 1: Synthetic Construction

The first method concentrates on a technique forconstructing the zero-coupon bond synthetically,using the money market account and the 4-periodzero.

4

Figure 7.2: An Example of a Four-Period Zero-Coupon Bond Price Process. The Value of the Money Market Account and the Spot Rates are Included on

the Tree. Actual Probabilities Along Each Branch of the Tree.

1

1

1

1

1

1

1

1

4

1.054597 .985301

1.054597 .981381

1.059125 .982456

1.059125 .977778

1.062869 .983134

1.062869 .978637

1.068337 .979870

1.068337 .974502 3

1.037958 .967826

1.037958 .960529

1.042854 .962414 1.042854 .953877

1.02 .947497

1.02 .937148

2 1 0 time

3/4

1/4

3/4

1/4

3/4

1/4

3/4

1/4

3/4

1/4

3/4

1/4

3/4

1/4

B(0) = 1 P(0,4) = .923845 r(0) = 1.02

1.022406

1.016031

1.020393

1.019193

1.024436

1.017606

5

1 An Arbitrage-free Evolution The first step in the procedure is to investigate whether the exogenous evolution as given in Figure 7.2 is arbitrage-free. We are taking the evolution in Figure 7.2 as given, and then from it, we will derive arbitrage-free prices for the 2- and 3-period zero-coupon bonds. The procedure will be flawed and nonsensical if the basis for the technique itself contains imbedded arbitrage opportunities. The check to determine if the given evolution is arbitrage-free proceeds node by node.

6

First, consider ourselves standing at time 0.

There are two securities available for trade at thisdate, the money market account and the 4-periodzero-coupon bond.

The money market account is riskless over thenext time period earning 1.02.

The 4-period zero-coupon bond is risky.

The return on the 4-period zero-coupon bond inthe up state is (u(0,4) = .947497/.923845 =1.025602) and in the down state it is (d(0,4) =.937148/.923845 = 1.014400).

7

In the up state it earns more than the moneymarket account and in the down state it earns less.

Hence, neither security dominates the other (interms of returns).

There would be no way to form a portfolio ofthese two securities with zero initial investmentthat didn’t have potential losses at time 1.

Furthermore, any portfolio with an initial positivecash flow would have a negative cash flow at time1 in at least one state.

Thus, the fact that neither security dominates theother implies there are no arbitrage opportunitiesbetween time 0 and time 1.

The converse of this statement is also true.

This condition applies to the first time step.

8

B u t , i f a t e a c h a n d e v e r y n o d e i n t h e t r e e , t h i sc o n d i t i o n a p p l i e s , t h e n t h e e n t i r e t r e e – t h e e n t i r ee v o l u t i o n – i s a r b i t r a g e - f r e e .

I n d e e d , t h e n n o m a t t e r w h i c h p a t h i n t h e t r e eo c c u r r e d , a n d a t e a c h n o d e , t h e r e w o u l d b e n ot r a d i n g s t r a t e g y t h a t c o u l d c r e a t e a r b i t r a g ep r o f i t s f r o m t h e r e o n w a r d .

W e h a v e a r g u e d t h a t t h e t r e e i s a r b i t r a g e - f r e e i fa n d o n l y i f

tsandtallfortstdtstrtstu ):4,(),():4,( .

9

Figure 7.3: The Rate of Return Processes for the Four-Period Zero-Coupon Bond and the Money Market Account in Figure 8.2. Actual Probabilities Are

Along Each Branch of the Tree.

210time

3/4

r(0) = 1.02

1/4

u(0,4) = 1.025602

d(0,4) = 1.014400

u(1,4;u) = 1.021455

d(1,4;u) = 1.013754

r(1;u) = 1.017606

3/4

1/4

3/4

u(1,4;d) = 1.026961

d(1,4;d) = 1.017851

r(1;d) = 1.022406

1/4

3/4

1/4

3/4

1/4

3/4

1/4

3/4

1/4

u(2,4;uu) = 1.018056

d(2,4;ud) = 1.014006

r(2;uu) = 1.016031

r(2;ud) = 1.020393

u(2,4;ud) = 1.022828

d(2,4;ud) = 1.017958

u(2,4;du) = 1.021529

d(2,4;du) = 1.016857

r(2;du) = 1.019193

u(2,4;dd) = 1.027250

d(2,4;dd) = 1.021622

r(2;dd) = 1.024436

3

10

2 Complete Markets Given the tree is arbitrage-free, we next study the pricing of the 2- and 3-period zero-coupon bonds. Figure 7.3 has that in all states and times, u(t,4) > r(t) > d(t,4). Besides implying that the tree is arbitrage-free, this will be seen to be a sufficient condition for market completeness.

11

First, let us construct a synthetic two-periodzero-coupon bond.

The payoff to this contingent claim is one dollar attime 2 across all states (uu, ud, du, dd).

To construct this contingent claim, we workbackwards through the tree, starting at time 1.

Consider first the node at time 1 state u.

12

T h e d e s i r e i s t o d e t e r m i n e n 0 ( 1 ; u ) a n d n 4 ( 1 ; u ) , t h e n u m b e r o f u n i t s o f t h e m o n e y m a r k e t a c c o u n t a n d t h e f o u r - p e r i o d z e r o -c o u p o n b o n d , s u c h t h a t t h e p o r t f o l i o ' s r e s u l t i n g t i m e 2 v a l u e e q u a l s o n e d o l l a r i n b o t h s t a t e s ; i . e . ,

1967826.0);1(4037958.1);1(0);4,2();1(4);2();1(0 ununuuPunuBun

( 7 . 1 a ) a n d

1960529.0);1(4037958.1);1(0);4,2();1(4);2();1(0 ununudPunuBun

( 7 . 1 b )

T h e s o l u t i o n t o t h i s s y s t e m o f e q u a t i o n s i s :

963430.037958.11);1(0 un ( 7 . 2 a )

0);1(4 un . ( 7 . 2 b )

13

T h e t i m e 1 c o s t a t s t a t e u o f 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 s

982699.0

947947.0);1(402.1);1(0);4,1();1(4)1();1(0

ununuPunBun

( 7 . 3 ) T h i s c o s t o f c o n s t r u c t i o n m u s t b e t h e a r b i t r a g e - f r e e v a l u e o f a t w o - p e r i o d z e r o - c o u p o n b o n d ; i . e . ,

982699.0);2,1( uP . ( 7 . 4 )

14

A s i m i l a r c o n s t r u c t i o n a t t i m e 1 u n d e r s t a t e d g i v e s :

1962414.0);1(4042854.1);1(0

);4,2();1(4);2();1(0

dndn

duPdndBdn ( 7 . 5 a )

a n d

1953877.0);1(4042854.1);1(0

);4,2();1(4);2();1(0

dnun

ddPdndBdn. ( 7 . 5 b )

15

T h e s o l u t i o n t o t h i s s y s t e m i s :

958907.042854.11);1(0 dn ( 7 . 6 a )

0);1(4 dn . ( 7 . 6 b )

I t s t i m e 1 , s t a t e d , a r b i t r a g e - f r e e v a l u e m u s t b e

.978085.

937148).;1(402.1);1(0

);4,1();1(4)1();1(0 );2,1(

dndn

dPdnBdndP

( 7 . 7 )

16

N o w , m o v i n g b a c k w a r d i n t h e t r e e t o t i m e 0 . L e t n 0 ( 0 ) a n d n 4 ( 0 ) b e t h e n u m b e r o f u n i t s o f t h e m o n e y m a r k e t a c c o u n t a n d o f t h e f o u r - p e r i o d z e r o - c o u p o n b o n d , r e s p e c t i v e l y , n e e d e d a t t i m e 0 s u c h t h a t

);2,1( 982699.

947497).0(402.1)0(0);4,1()0(4)1()0(0uP

nnuPnBn

( 7 . 8 a )

a n d

);2,1( 978085.

937148).0(402.1)0(0);4,1()0(4)1()0(0dP

nndPnBn

( 7 . 8 b )

17

A s o l u t i o n t o t h i s s y s t e m e x i s t s b e c a u s e u ( 0 , 4 ) = . 9 4 7 4 9 7 / . 9 2 3 8 4 5 = 1 . 0 2 5 6 0 2 > d ( 0 , 4 ) = . 9 3 7 1 4 8 / . 9 2 3 8 4 5 = 1 . 0 1 4 3 9 9 . T h i s i s t h e s u f f i c i e n c y c o n d i t i o n m e n t i o n e d e a r l i e r f o r m a r k e t c o m p l e t e n e s s . T h e s o l u t i o n i s :

549286.0)0(0 n ( 7 . 9 a )

445835.0)0(4 n . ( 7 . 9 b )

I t s t i m e 0 a r b i t r a g e - f r e e v a l u e i s 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 p o r t f o l i o , i . e . ,

961169.923845).0(41)0(0

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

nn

PnBnP. ( 7 . 1 0 )

18

Figure 7.4: Arbitrage-Free Values (P(t,2;st)) and Synthetic Portfolio Positions (n0(t;st), n3(t;st)) for the Two-Period Zero-Coupon Bond. The actual probabilities

are along each branch of the tree with the pseudo-probabilities in parentheses.

.982699 (.963430, 0)

.978085 (.958907, 0)

.961169 (.549286, .445835)

2 1 0 time

3/4 (1/2)

1/4 (1/2)

3/4 (1/2)

1/4 (1/2)

3/4 (1/2)

1/4 (1/2)

1 1

1 1

1.02

1.017606

1.022406

19

It is important to note that the value for the two-period zero-coupon bond does not depend on theactual probabilities for the evolution of the four-period zero-coupon bond and the money marketaccount.

Regardless of the state which occurs (or itsprobability of occurrence), the 2-period zero-coupon bond’s values are duplicated.

Hence, the cost of construction, the arbitrage-freeprice, is determined independent of the actualprobabilities of the future states!

This completes the presentation of method 1,valuation by the cost of synthetic construction.

This method always works and in many ways, it isthe most intuitive of the two approaches

20Figure 7.5: Arbitrage-Free Values (P(t,3;st)) and Synthetic Portfolio Positions (n0(t;st), n4(t;st)) for the Three-

Period Zero-Coupon Bond. The actual probabilities are along each branch of the tree with the pseudo-probabilities in parentheses.

1 1 1 1 1 1 1 1 3

.984222 (.948223, 0)

.980015 (.944176, 0)

.981169 (.940850, 0)

.976147 (.936034, 0)

.965127 (.410592, .576597)

.957211 (.397975, .588249)

.942322 (.235620; .764957)

2 1 0 time

3/4 (1/2)

1/4 (1/2)

3/4 (1/2)

3/4 (1/2)

1/4 (1/2)

1/4 (1/2)

3/4 (1/2)

1/4 (1/2)

3/4 (1/2)

1/4 (1/2)

3/4 (1/2)

1/4 (1/2)

1.02

1.017606

1.016031

1.020393

1.022406

1.019193

1.024436

1/4 (1/2)

3/4 (1/2)

21

C Method 2: Risk Neutral Valuation This method is the most useful for computation. It values the 2-and 3-period zero-coupon bonds by deriving a present value operator. This present value operator is related to the present value form of the expectations hypothesis discussed in chapter 5.

22

1 R i s k N e u t r a l P r o b a b i l i t i e s T h e f i r s t s t e p i n t h i s m e t h o d , a s w i t h m e t h o d 1 , i s t o g u a r a n t e e t h a t t h e e v o l u t i o n o f t h e m o n e y m a r k e t a c c o u n t a n d 4 - p e r i o d z e r o - c o u p o n b o n d p r i c e p r o c e s s i s a r b i t r a g e - f r e e . W e s a w b e f o r e t h a t a n e c e s s a r y a n d s u f f i c i e n t c o n d i t i o n f o r t h i s i s t h a t a t e v e r y n o d e i n t h e t r e e , t h e f o u r - p e r i o d b o n d d o e s n o t d o m i n a t e n o r i s i t d o m i n a t e d b y t h e m o n e y m a r k e t . W e w r o t e t h i s a s :

tsandtallfortstdtstrtstu );4,(),();4,( . ( 7 . 1 1 )

23

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

t h e r e e x i s t s a u n i q u e n u m b e r ),( tst s t r i c t l y

b e t w e e n 0 a n d 1 s u c h t h a t

tsandtallfortstdtsttstutsttstr );4,());(1();4,();();(

. ( 7 . 1 2 )

24

Expression (7.12), when rearranged, yields a formula for obtaining the ),( tst ’s.

This formula is

)];4,();4,(/[)];4,();([

),(

tstdtstutstdtstrtst

(7.13)

In computing the ),( tst ’s using expression (7.13), if

the computed ),( tst ’s are strictly between zero and

one for all times t and states ts, then the tree is

arbitrage-free.

25

We can now check to see if the tree is arbitrage-free using this new condition. We can compute the ),( tst ’s at every node. They

are:

5.0021622.1027250.1

021622.1024436.1);2(

5.0016857.1021529.1

016857.1019193.1);2(

5.0017958.1022828.1

017958.1020393.1);2(

5.0014006.1018056.1

014006.1016031.1);2(

5.0017851.1026961.1

017851.1022406.1);1(

5.0013754.1021455.1

013754.1017606.1);1(

5.0014400.1025602.1

014400.102.1)0(

dd

du

ud

uu

d

u

As this number is strictly between 0 and 1, the evolution in Figure 7.2 is arbitrage-free.

26

T h e o b s e r v a t i o n t h a t a l l t h e ),( tst ’ s a r e e q u a l t o

0 . 5 i s n o t a g e n e r a l r e s u l t , b u t j u s t a s p e c i a l c a s e . I t w o u l d n o t b e s a t i s f i e d f o r a n a r b i t r a r y t r e e . N o t i c e t h a t t h e s e ),( tst ’ s c a n b e i n t e r p r e t e d a s a

p s e u d o p r o b a b i l i t i e s o f t h e u p s t a t e o c c u r r i n g o v e r [ t , t + 1 ] . T h e y h a v e a n i m p o r t a n t u s a g e . T o s e e t h i s u s a g e , w e t r a n s f o r m e x p r e s s i o n ( 7 . 1 2 ) a g a i n . S i m p l e a l g e b r a s h o w s t h a t :

.

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

);4,(

tsandtallfortstr

dtstPtstutstPtsttstP

( 7 . 1 4 )

E x p r e s s i o n ( 7 . 1 4 ) g i v e s a p r e s e n t v a l u e f o r m u l a .

27

L e t t i n g )(~ tE d e n o t e t a k i n g a n e x p e c t e d v a l u e u s i n g

t h e p s e u d o p r o b a b i l i t i e s ),( tst , t h e n w e c a n

r e w r i t e e x p r e s s i o n ( 7 . 1 4 ) a s :

);(/))1;4,1((~

);4,( tstrtstPtEtstP . ( 7 . 1 5 )

I n t h i s f o r m w e s e e t h a t e x p r e s s i o n ( 7 . 1 5 ) i s a t r a n s f o r m a t i o n o f t h e p r e s e n t v a l u e f o r m o f t h e e x p e c t a t i o n s h y p o t h e s i s d i s c u s s e d i n c h a p t e r 5 . T h e d i f f e r e n c e h e r e i s t h a t t h e e x p e c t a t i o n s h y p o t h e s i s h o l d s i n t e r m s o f t h e p s e u d o p r o b a b i l i t i e s a n d n o t t h e a c t u a l p r o b a b i l i t i e s !

28

I n t e r m s o f F i g u r e 7 .2 , i t i s e a s y t o c h e c k t h a t t h e p r e s e n t v a l u e f o r m u l a e x p r e s s i o n ( 7 .1 5 ) i s s a t i s f i e d :

923845.002.1

937148.0)2/1(947497.0)2/1()0(/))1;4,1((0~

)4,0(

937148.0022406.1

953877.0)2/1(962414.0)2/1();1(/))2;4,2((1~

);4,1(

947497.0017606.1

960529.0)2/1(967826.0)2/1();1(/))2;4,2((1~

);4,1(

953877.0024436.1

974502.0)2/1(979870.0)2/1();2(/))3;4,3((2~

);4,2(

962414.0019193.1

978637.0)2/1(983134.0)2/1();2(/))3;4,3((2~

);4,2(

960529.0020393.1

977778.0)2/1(982456.0)2/1();2(/))3;4,3((2~

);4,2(

967826.0016031.1

981381.0)2/1(985301.0)2/1();2(/))3;4,3((2~

);4,2(

rsPEP

drsPEdP

ursPEuP

ddrsPEddP

dursPEduP

udrsPEudP

uursPEuuP

29

2 R i s k - N e u t r a l V a l u a t i o n I t i s a s u r p r i s i n g r e s u l t ( p r o v e n i n c h a p t e r 8 ) t h a t t h e p r e s e n t v a l u e f o r m u l a i n e x p r e s s i o n ( 7 . 1 5 ) u s i n g t h e p s e u d o - p r o b a b i l i t i e s a p p l i e s t o a n y r a n d o m c a s h f l o w w h o s e p a y o f f s d e p e n d o n t h e e v o l u t i o n o f t h e m o n e y m a r k e t a c c o u n t a n d t h e 4 - p e r i o d z e r o - c o u p o n b o n d . I n g e n e r a l , t h e p r e s e n t v a l u e f o r m u l a c a n b e w r i t t e n a s f o l l o w s . L e t )1( tsx r e p r e s e n t a r a n d o m c a s h f l o w r e c e i v e d a t t i m e

t + 1 i n s t a t e 1ts . T h e n , t h e t i m e t v a l u e o f t h i s t i m e t + 1

c a s h f l o w i s :

);(/))1((~

Pr tstrtsxtEtes entValue . ( 7 . 1 6 )

I f 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 d t h e m a r k e t i s c o m p l e t e , t h e n e x p r e s s i o n ( 7 . 1 6 ) i s v a l i d .

30

T o i l l u s t r a t e t h i s 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 c o n s i d e r t h e 2 - a n d 3 - p e r i o d z e r o - c o u p o n b o n d s a s e x a m p l e s o f r a n d o m c a s h f l o w s . F o r t h e t w o - p e r i o d b o n d , t h e c a l c u l a t i o n s a r e a s f o l l o w s :

.961169.02.1978085.0)2/1(982699.0)2/1()0(1;2,10

~)2,0(

,978085.0022406.11)2/1(1)2/1();1(2;2,21~

);2,1(

,982699.0017606.11)2/1(1)2/1();1(2;2,21~

);2,1(

rsPEP

anddrsPEdP

ursPEuP

T h e s e v a l u e s g e n e r a t e t h e n o d e s o f t h e t r e e i n F i g u r e 7 . 4 .

31

To use this approach, we need to determine the portfolio that generates the synthetic two-period zero-coupon bond. This can be obtained using the procedure from Method 1. Alternatively, it can be obtained using only expression (7.16) and Figure 7.2. This alternative approach calculates the delta. The delta is defined as the change in the value of a derivative security with respect to the change in value of some underlying asset.

32

T h e d e l t a f o r t h e t w o - p e r i o d b o n d a t t i m e 1 , s t a t e u , d e n o t e dn 4 ( 1 ; u ) , i s :

0960529.0967826.0

11);4,2();4,2();2,2();2,2(

)2;4,2(

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

udPuuPudPuuP

sP

sPun .

T o g e t t h e n u m b e r o f u n i t s i n t h e m o n e y m a r k e t a c c o u n t , w eu s e :

963430.002.1

)947497.0(0982699.0)1(

);4,1();1(4);2,1();1(0

B

uPunuPun .

T h e n u m e r a t o r r e p r e s e n t s t h e c o s t o f c o n s t r u c t i o n ( P ( 1 , 2 ; u ) )l e s s t h e d o l l a r s i n v e s t e d i n t h e 4 - p e r i o d z e r o - c o u p o n b o n d( );4,1();1(4 uPun ) , o r t h e d o l l a r s i n t h e m o n e y m a r k e t a c c o u n t .

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

33

C o n t i n u i n g ,

0953877.0962414.0

11);4,2();4,2();2,2();2,2(

)2;4,2(

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

ddPduPddPduP

sP

sPdn

a n d

.958907.02.1)937148(.0978085.

)1();4,1();1(4;2,1);1(0

BdPdndPdn

F i n a l l y ,

445835.0937148.0947497.0978085.0982699.0

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

)1;4,1(

)1;2,1()0(4

dPuPdPuP

sP

sPn

a n d

.549286.1)923845(.445835.961169.

)0()4,0()0(42,0)0(0

BPnPn

34

1

Illustration of an arbitrage opportunity.

From the above calculations, the price of thetwo-period bond at time zero is P(0,2) = 0.961169.

Suppose that the actual price quoted in the marketwas 0.970000. An arbitrage opportunity exists.

To create the arbitrage opportunity, sell thetwo-period zero-coupon bond at time 0. This brings in+0.97000 dollar in cash.

Invest n0(0) = 0.549286 dollar of this in a moneymarket account and buy n4(0) = 0.445835 units ofP(0,4) = 0.923845, at a total cost of 0.961169 dollars. This leaves 0.97000 - 0.961169 = 0.008831 dollars inexcess cash available at time 0.

35

1

At time 1, if state u occurs, we rebalance our moneymarket account and four-period zero-coupon bondholdings to n0(1;u) and n4(1;u). No excess cash isgenerated. This follows from the self-financingconstraint.

At time 1, if state d occurs, we rebalance our moneymarket account and four-period zero-coupon bondholdings to n0(1;d) and n4(1;d). No excess cash isgenerated.

36

At time 2, our long position in the money marketaccount and four-period zero-coupon bond exactlyoffsets our short position in the two-period zero-coupon bond.

This completes the illustration for the two-periodzero-coupon bond.

There is an important observation to make. Forthe arbitrage opportunity to be executed, ingeneral, one must continue the strategy until thelast date relevant for the contingent claim.