A two-dimensional market model

To simplify notation, we omit the arguments whenever there is no ambiguity... The case =,1is analogous.. There are two cases:... This market admits arbitrage.. LetfIP be one of them.. Le

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A two-dimensional market model

LetB ( t ) = ( B 1 ( t ) ;B 2 ( t )) ; 0t T;be a two-dimensional Brownian motion on ;F;P) Let

F( t ) ; 0tT;be the filtration generated byB

In what follows, all processes can depend on tand !, but are adapted to F( t ) ; 0 t T To simplify notation, we omit the arguments whenever there is no ambiguity

Stocks:

dS 1 = S 1 [ 1 dt + 1 dB 1 ] ;

dS 2 = S 2

2 dt + 2 dB 1 +q

1, 2 2 dB 2

:

We assume 1 > 0 ; 2 > 0 ; ,11 :Note that

dS 1 dS 2 = S 21 21 dB 1 dB 1 = 21 S 21 dt;

dS 2 dS 2 = S 22 2 22 dB 1 dB 1 + S 22 (1, 2 ) 22 dB 2 dB 2

= 22 S 22 dt;

dS 1 dS 2 = S 1 1 S 2 2 dB 1 dB 1 = 1 2 S 1 S 2 dt:

In other words,

dS1

S1

has instantaneous variance 21,

dS2

S2

has instantaneous variance 22,

dS1

S1

and dS2

S2 have instantaneous covariance 1 2

Accumulation factor:

( t ) = expZ t

0 r du

:

The market price of risk equations are

1 1 = 1 ,r

2 1 +q

1, 2 2 2 = 2 ,r (MPR)

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The solution to these equations is

1 = 1,r

1 ;

2 = 1 ( 2,r ), 2 ( 1,r )

1 2p

1, 2 ;

provided,1 < < 1

Suppose,1 < < 1 Then (MPR) has a unique solution( 1 ; 2 ); we define

Z ( t ) = exp

,

Z t

0 1 dB 1,

Z t

0 2 dB 2,1 2Z t

0 ( 21 + 22 ) du

;

f

IP ( A ) =Z

A Z ( T ) dIP; 8A2 F:

f

IP is the unique risk-neutral measure Define

e

B 1 ( t ) =Z t

0 1 du + B 1 ( t ) ;

e

B 2 ( t ) =Z t

0 2 du + B 2 ( t ) :

Then

dS 1 = S 1

h

r dt + 1 d Be1

i

;

dS 2 = S 2

r dt + 2 d Be1 +q

1, 2 2 d Be2

:

We have changed the mean rates of return of the stock prices, but not the variances and covariances

dX = 1 dS 1 + 2 dS 2 + r ( X, 1 S 1, 2 S 2 ) dt

d

X

= 1 ( dX,rX dt )

= 1 1 ( dS 1,rS 1 dt ) + 1 2 ( dS 2,rS 2 dt )

= 1 1 S 1 1 d Be1 + 1 2 S 2

2 d Be1 +q

1, 2 2 d Be2

:

LetV beF( T )-measurable Define thefIP-martingale

Y ( t ) =fIE

V ( T )

F( t )

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The Martingale Representation Corollary implies

Y ( t ) = Y (0) +Z t

0 1 d Be1 +Z t

0 2 d Be2 :

We have

d

X

=

1

1 S 1 1 + 1 2 S 2 2

d Be1

+ 1 2 S 2

q

1, 2 2 d Be2 ;

dY = 1 d Be1 + 2 d Be2 :

We solve the equations

1

1 S 1 1 + 1 2 S 2 2 = 1

1

2 S 2

q

1, 2 2 = 2

for the hedging portfolio( 1 ; 2 ) With this choice of( 1 ; 2 )and setting

X (0) = Y (0) = IE Vf ( T ) ;

we haveX ( t ) = Y ( t ) ; 0tT;and in particular,

X ( T ) = V:

EveryF( T )-measurable random variable can be hedged; the market is complete.

The case =,1is analogous Assume that = 1 Then

dS 1 = S 1 [ 1 dt + 1 dB 1 ]

dS 2 = S 2 [ 2 dt + 2 dB 1 ]

The stocks are perfectly correlated

The market price of risk equations are

1 1 = 1,r

The process 2 is free There are two cases:

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Case I: 1, 1r

= 2, 2r :There is no solution to (MPR), and consequently, there is no risk-neutral measure This market admits arbitrage Indeed

d

X

= 1 1 ( dS 1,rS 1 dt ) + 1 2 ( dS 2,rS 2 dt )

= 1 1 S 1 [( 1,r ) dt + 1 dB 1 ] + 1 2 S 2 [( 2,r ) dt + 2 dB 1 ]

Suppose1

,r

1 > 2

,r

2 :Set

1 = 1 1 S 1 ; 2 =,

1

2 S 2 :

Then

d

X

= 1

1,r

1 dt + dB 1

,

1

2,r

2 dt + dB 1

= 1

1,r

2,r

2

Positive

dt

Case II: 1

,r

1 = 2

,r

2 :The market price of risk equations

1 1 = 1,r

2 1 = 2,r

have the solution

1 = 1,r

1 = 2,r

2 ;

2is free; there are infinitely many risk-neutral measures LetfIP be one of them

Hedging:

d

X

= 1 1 S 1 [( 1,r ) dt + 1 dB 1 ] + 1 2 S 2 [( 2,r ) dt + 2 dB 1 ]

= 1 1 S 1 1 [ 1 dt + dB 1 ] + 1 2 S 2 2 [ 1 dt + dB 1 ]

=

1

1 S 1 1 + 1 2 S 2 2

d Be1 :

Notice thatBe2does not appear

LetV be anF( T )-measurable random variable IfV depends onB 2, then it can probably not

be hedged For example, if

V = h ( S 1 ( T ) ;S 2 ( T )) ;

and 1 or 2 depend onB 2, then there is trouble

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More precisely, we define thefIP-martingale

Y ( t ) = IEf

V ( T )

F( t )

We can write

Y ( t ) = Y (0) +Z t

0 1 d Be1 +Z t

0 2 d Be2 ;

so

dY = 1 d Be1 + 2 d Be2 :

To getd

X

to matchdY, we must have

2 = 0 :