figure 6.1: an example of a zero-coupon bond price curve

1

6 Trading Strategies, Arbitrage Opportunities, and Complete Markets A Motivation These are the basic concepts needed to develop the pricing and hedging methodology for fixed income securities and interest rate options. B Trading Strategies Intuitively, a trading strategy is a dynamic investment portfolio involving the traded zero-coupon bonds. Portfolio rebalancings can occur within the investment horizon and they are based on the information available at the time that the portfolio is rebalanced.

2

EXAMPLE: A TRADING STRATEGY

A trading strategy is a complete (state and timecontingent) listing of the holdings of each tradedsecurity for each time and state in a tree.

A trading strategy imposes no restrictions onthese holdings.

3Figure 6.1: An Example of a Zero-coupon Bond Price Curve

210

1

976147.

953877.

1

978085.

957211.

937148.

1

981169.

962414.

1

980392.

961169.

942322.

923845.

)0,0(

)1,0(

)2,0(

)3,0(

)4,0(

1

980015.

960529.

1

982699.

965127.

947497.

1

984222.

967826.

time

P

P

P

P

P

4Figure 6.2: An Example of a Zero-coupon Bond Price Curve Evolution and a Trading Strategy (n0(t), n4(t), n3(t), n2(t)).

210time(0,0,0,0)

1976147.953877.

1.042854)-.5962084 (0,0,0,

1978085.957211.937148. (0,0,0,0)

1.02 1981169.962414.

1.0428541980392.961169.942322.923845.

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

1B(0) 1980015.960529.

1.037958

(0,0,3.5,0)

1982699.965127.947497. (0,0,0,0)1.02 1

984222.967826.

1.037958

)2,5.2,0,1()0(n),0(n),0(n),0(n 2340

r(0) = 1.02

1.017606

1.022406

5

A t t i m e 0 , t h e h o l d i n g s v e c t o r i s :

))0(2),0(3),0(4),0(0( nnnn = ( - 1 , 0 , 2 . 5 , - 2 ) .

)0(0n i s 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 h e l d a t t i m e 0 .)0(4n i s t h e n u m b e r o f u n i t s o f t h e 4 - p e r i o d z e r o -

c o u p o n b o n d h e l d a t t i m e 0 .)0(3n i s t h e n u m b e r o f u n i t s o f t h e 3 - p e r i o d z e r o -

c o u p o n b o n d h e l d a t t i m e 0 .)0(2n i s t h e n u m b e r o f u n i t s o f t h e 2 - p e r i o d z e r o -

c o u p o n b o n d h e l d a t t i m e 0 .

T h e 1 - p e r i o d z e r o - c o u p o n b o n d i s n o t i n c l u d e db e c a u s e a p o s i t i o n i n t h i s b o n d i s i m p l i c i t l yi n c o r p o r a t e d i n t o t h e p o s i t i o n i n t h e m o n e ym a r k e t a c c o u n t .

6

C a s h f l o w s e n t e r i n g a n d l e a v i n g a t i m e p e r i o d a r ei m p o r t a n t .

A t t i m e 0 , t h e i n i t i a l c a s h f l o w f r o m f o r m i n g t h i st r a d i n g s t r a t e g y i s : – c o s t =

) ]2,0()0(2)3,0()0(3)4,0()0(4)0()0([ PnPnPnBon =

]961169) .0(2942322) .0(3923845) .0(41)0([ nnnon =

961169) .2(942322) .5.2(923845) .0(1)1( = + . 5 6 6 5 3 3 .

A p o s i t i v e c a s h f l o w i m p l i e s t h a t t h e v a l u e o f t h i sp o s i t i o n i s - . 5 6 6 5 3 3 d o l l a r s , i t i s a l i a b i l i t y .

7

Other trading strategies may have an initial cashflow that is negative or zero.

When the initial cash inflow is zero, the tradingstrategy is called a zero investment tradingstrategy

8

T h e s e h o l d i n g s a r e f o r m e d a t t i m e 0 a n d h e l du n t i l t i m e 1 .

A t t i m e 1 , t w o o u t c o m e s a r e p o s s i b l e - e i t h e r u po r d o w n o c c u r s .

I f u p o c c u r s , t h e t r a d i n g s t r a t e g y e n t e r s t i m e 1w i t h a v a l u e o f :

);2,1()0(2);3,1()0(3);4,1()0(4)1()0( uPnuPnuPnBon =

9 8 2 6 9 9).2(9 6 5 1 2 7).5.2(9 4 7 4 9 7).0(0 2.1)1( = - . 5 7 2 5 8 0 5 .

T h i s p o s i t i o n h a s l o s t v a l u e , m o v i n g f r o m -. 5 6 6 5 3 3 t o - . 5 7 2 5 8 0 5 .

9

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

T h e n e w h o l d i n g s a t t i m e 1 i n t h e u p - s t a t e a r e :) );1(2) ,;1(3) ,;1(4) ,;1(0( unununun = ( 0 , 0 , 3 . 5 , 0 ) .

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

);2,1();1(2);3,1();1(3);4,1();1(4)1();1( uPunuPunuPunBuon =

982699) .0(965127) .5.3(947497) .0(02.1)0( = + 3 . 3 7 7 9 4 4 5 .

T h i s p o r t f o l i o w a s r e b a l a n c e d f r o m - . 5 7 2 5 8 0 5 d o l l a r s t o+ 3 . 3 7 7 9 4 4 5 d o l l a r s .

T h i s c a n o n l y o c c u r i f t h e r e w a s c a s h i n p u t t o t h e t r a d i n gs t r a t e g y a t t i m e 1 i n s t a t e u .

T h e c a s h i n f l o w t o t h i s t r a d i n g s t r a t e g y w a s e q u a l t o( . 5 7 2 5 8 0 5 + 3 . 3 7 7 9 4 4 5 ) = 3 . 9 5 0 5 2 5 d o l l a r s .

10

I f t h e r e b a l a n c e d v a l u e o f t h i s t r a d i n g s t r a t e g y a tt i m e 1 s t a t e u h a d b e e n e q u a l t o t h e e n t e r i n g v a l u eo f t h e p o r t f o l i o , t h e n t h e s t r a t e g y w o u l d b e c a l l e ds e l f - f i n a n c i n g .

A s e l f - f i n a n c i n g r e b a l a n c i n g i s i l l u s t r a t e d a t t i m e 1i n t h e d o w n - s t a t e .

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

);2,1()0(2);3,1()0(3);4,1()0(4)1()0( dPndPndPnBon =

9 7 8 0 8 5).2(9 5 7 2 11).5.2(9 3 7 1 4 8).0(0 2.1)1( =

- . 5 8 3 1 4 2 5 .

11

T h e r e b a l a n c e d p o r t f o l i o i s));1(2),;1(3),;1(4),;1(0( dndndndn = ( 0 , 0 , 0 , - . 5 9 6 2 0 8 4 ) .

T h e v a l u e o f t h i s r e b a l a n c e d p o r t f o l i o i s :

);2,1();1(2);3,1();1(3);4,1();1(4)1();1( dPdndPdndPdnBdon =

.9 7 8 0 8 5(.5 9 6 2 0 8 4 )(0 ).9 5 7 2 11(0 ).9 3 7 1 4 8(0 )1 .0 2

= - . 5 8 3 1 4 2 5 .

A s t h e s e t w o v a l u e s a r e e q u a l , t h i s r e b a l a n c i n g i s s e l f -f i n a n c i n g .

12

F o r t h i s t r a d i n g s t r a t e g y , a l l h o l d i n g s a r e l i q u i d a t e d a t t i m e 2 . T h i s i s i n d i c a t e d b y t h e h o l d i n g s v e c t o r a t t i m e 2 i n s t a t e u u , u d , d u , a n d d d h a v i n g o n l y z e r o e n t r i e s , i . e . ( 0 : 0 , 0 , 0 ) . T h e l i q u i d a t e d p o r t f o l i o v a l u e s a t t i m e 2 i n t h e v a r i o u s s t a t e s a r e :

( t i m e 2 s t a t e u u )

);2,2();1(2);3,2();1(3);4,2();1(4);2();1( uuPunuuPunuuPunuBuon =

( 0 ) 1 . 0 3 7 9 5 8 + ( 0 ) . 9 6 7 8 2 6 + ( 3 . 5 ) . 9 8 4 2 2 2 + ( 0 ) 1 = 3 . 4 4 4 7 7 7

( t i m e 2 s t a t e u d )

);2,2();1(2);3,2();1(3);4,2();1(4);2();1( udPunudPunudPunuBuon =

( 0 ) 1 . 0 3 7 9 5 8 + ( 0 ) . 9 6 0 5 2 9 + ( 3 . 5 ) . 9 8 0 0 1 5 + ( 0 ) 1 = 3 . 4 3 0 0 5 3

13

( t i m e 2 s t a t e d u ));2,2();1(2);3,2();1(3);4,2();1(4);2();1( d uPdnd uPdnd uPdndBdon =

1(. 5 9 6 2 0 8 4 )(0 ).9 8 116 9(0 ).9 6 2 4 1 44(0 )1 .0 4 2 8 5 = - . 5 9 6 2 0 8 4

( t i m e 2 s t a t e d d ));2,2();1(2);3,2();1(3);4,2();1(4);2();1( d dPdnd dPdnd dPdndBdon =

1(. 5 9 6 2 0 8 4 )(0 ).9 7 6 1 4 7(0 ).9 5 3 8 7 74(0 )1 .0 4 2 8 5 = - . 5 9 6 2 0 8 4 .

T h e t r a d i n g s t r a t e g y h a s a n e g a t i v e v a l u e a t l i q u i d a t i o n a n dt h u s a n e g a t i v e c a s h f l o w .

14

C Arbitrage Opportunities

An arbitrage opportunity is a trading strategy that haszero initial investment and generates positive cash flows(with positive probability) at no risk of a loss.

EXAMPLE: AN ARBITRAGE OPPORTUNITY

15

Figure 6.3: An Example of a Zero-coupon Bond Price Curve Evolution with an Arbitrage Opportunity

10time

1978085.957211.

02.1

02.1)0(r

1980392.960000.942322.

1

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

)0(B1

982695.965127.

02.1

16

I claim that there is a mispricing implicit in thetime 0 market prices of these zero-coupon bonds.

The 2-period zero-coupon bond is undervalued,and it should sell for .961169 dollars.

The trader should sell .58287 units of the 3-periodzero-coupon bond, sell .4119177 units of themoney market account, and buy 1 unit of the 2-period zero-coupon bond. The trader should holdthis portfolio until time 1, then liquidate hisposition.

This is called a buy and hold trading strategy.

17

The initial cash flow from this position is:

+ (.58287)P(0,3) - (1)P(0,2) + (.4119177)B(0) =

+ (.58287).942322 - (1).96000 + (.4119177)1 = +.001169.

A positive cash flow yields a negative value (a liability).

We next determine the time 1 cash flows from liquidating thisposition.

18

The time 1 cash flow in the up-state is:

- (.58287)P(1,3;u) + (1)P(1,2;u) - (.4119177)B(1) =

- (.58287).965127 + (1).982699 - (.4119177)1.02 = 0.

The time 1 cash flow in the down-state is:

- (.58287)P(1,3;d) + (1)P(1,2;d) - (.4119177)B(1) =

- (.58287).957211 + (1).978085 - (.4119177)1.02 = 0.

19

Surprisingly, there is no additional cash flowfrom liquidating this portfolio at time 1.

This portfolio generates +.001169 dollars at time0 and has no further liability. It creates cashfrom nothing! It is a money pump.

The trading strategy employed in this example isan arbitrage opportunity.

20

The subsequent pricing theory is based on thesimple notion that we would not expect to seemany arbitrage opportunities in well-functioningmarkets. Why?

Because bright investors would hold thesearbitrage opportunities, becoming wealthy in theprocess. They would desire to hold as many ofthem as possible.

The process of arbitrageurs taking advantage ofthese arbitrage opportunities would causeequilibrium prices (supply and demand) to changeuntil these arbitrage opportunities are eliminated.

21

D Complete Markets

A complete market is one in which any cash flowpattern desired can be obtained via a tradingstrategy.

EXAMPLE: A COMPLETE MARKET

22

Figure 6.4: An Example of a Zero-coupon Bond Price Curve Evolution in a Complete Market

10time

1978085.

02.1

02.1)0(r

1980392.961169.

1

)0,0(P)1,0(P)2,0(P

)0(B1

982699.02.1

23

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

T o p r o v e t h i s , s u p p o s e a t t i m e 1 w e d e s i r e t oc o n s t r u c t a p o r t f o l i o w h o s e v a l u e i s ),( dxux i n t h e

u p a n d d o w n s t a t e s , r e s p e c t i v e l y w h e r e dxux .

F o r c o n v e n i e n c e , l e t u s c a l l t h i s c a s h f l o w t h ev a l u e o f a n ( a r b i t r a r y ) t r a d e d i n t e r e s t r a t e o p t i o n .

T h e i d e a i s t o f o r m a t r a d i n g s t r a t e g y ))0(2),0(0( nn

i n t h e m o n e y m a r k e t a c c o u n t a n d 2 - p e r i o d z e r o -c o u p o n b o n d a t t i m e 0 s u c h t h a t t h e l i q u i d a t i o nv a l u e a t t i m e 1 m a t c h e s t h e c a s h f l o w ),( dxux .

24

The initial cost of this trading strategy is

)2,0()0(2)0()0(0 PnBn = 961169).0(21)0(0 nn .

(7.14) The objective is to choose ))0(2),0(0( nn such that

);2,1()0(2)1()0(0 uPnBn = ux and

(7.15) );2,1()0(2)1()0(0 dPnBn =

dx.

25

Substitution of the prices from Figure 6.4 yields

982699).0(202.1)0(0 nn = ux and

(6.16) 978085).0(202.1)0(0 nn = dx.

26

T h e s o l u t i o n i s

)005614(.02.1

)978085(.)982699(.)0(0

uxdxn

( 6 . 1 7 )

005614.)0(2

dxuxn

.

E x p r e s s i o n ( 6 . 1 7 ) g i v e s t h e t r a d i n g s t r a t e g y t h a t r e p l i c a t e s t h e c a s h f l o w ),( dxux .

T h i s t r a d i n g s t r a t e g y i s c a l l e d t h e s y n t h e t i c i n t e r e s t r a t e o p t i o n .

27

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 t r a d i n g s t r a t e g y i s o b t a i n e d b y s u b s t i t u t i n g e x p r e s s i o n ( 6 . 1 7 ) i n t o e x p r e s s i o n ( 6 . 1 4 ) , i . e .

)961169(.005614.

)1()005614(.02.1

)978085(.)982699(. dxuxuxdx

.

( 6 . 1 8 )

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