Martingale Representation Theorem

Martingale Representation Theorem18.1 Martingale Representation Theorem See Oksendal, 4th ed., Theorem 4.11, p.50.. In the context of Girsanov’s Theorem, suppse thatF t ; 0 t T;is the f

Martingale Representation Theorem

18.1 Martingale Representation Theorem

See Oksendal, 4th ed., Theorem 4.11, p.50

Theorem 1.56 LetB ( t ) ; 0tT;be a Brownian motion on ;F;P) LetF( t ) ; 0tT, be the filtration generated by this Brownian motion LetX ( t ) ; 0tT, be a martingale (underIP) relative to this filtration Then there is an adapted process ( t ) ; 0tT, such that

X ( t ) = X (0) +Z t

0  ( u ) dB ( u ) ; 0tT:

In particular, the paths ofXare continuous.

Remark 18.1 We already know that ifX ( t )is a process satisfying

dX ( t ) =  ( t ) dB ( t ) ;

thenX ( t )is a martingale Now we see that ifX ( t )is a martingale adapted to the filtration generated

by the Brownian motionB ( t ), i.e, the Brownian motion is the only source of randomness inX, then

dX ( t ) =  ( t ) dB ( t )

for some ( t )

18.2 A hedging application

Homework Problem 4.5 In the context of Girsanov’s Theorem, suppse thatF( t ) ; 0 t T;is the filtration generated by the Brownian motionB (underIP) Suppose thatY is afIP-martingale Then there is an adapted process ( t ) ; 0tT, such that

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

0 ( u ) d Be( u ) ; 0tT:

197

dS ( t ) =  ( t ) S ( t ) dt +  ( t ) S ( t ) dB ( t ) ; ( t ) = expZ t

0 r ( u ) du

;

 ( t ) =  ( t ),r ( t )

 ( t ) ;

e

B ( t ) =Z t

0  ( u ) du + B ( t ) ;

Z ( t ) = exp

,

Z t

0  ( u ) dB ( u ),

1 2

Z t

0  2 ( u ) du

;

f

IP ( A ) =Z

A Z ( T ) dIP; 8A2 F:

Then

d

S ( t )

( t )



= S ( t )

( t )  ( t ) d Be( t ) :

Let
( t ) ; 0tT;be a portfolio process The corresponding wealth processX ( t )satisfies

d

X ( t )

( t )



=
( t )  ( t ) S ( t )

( t ) d Be( t ) ;

i.e.,

X ( t )

( t ) = X (0) +Z t

0
( u )  ( u ) S ( u )

( u ) d Be( u ) ; 0tT:

LetV be anF( T )-measurable random variable, representing the payoff of a contingent claim at timeT We want to chooseX (0)and
( t ) ; 0tT, so that

X ( T ) = V:

Define thefIP-martingale

Y ( t ) =fIE

V ( T )

F( t )

; 0tT:

According to Homework Problem 4.5, there is an adapted process ( t ) ; 0tT, such that

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

0 ( u ) d Be( u ) ; 0tT:

SetX (0) = Y (0) =fIEh

V (T)

i

and choose
( u )so that

( u )  ( u ) S ( u )

( u ) = ( u ) :

X ( t )

( t ) = Y ( t ) = IEf 

V ( T )

F( t )



; 0tT:

In particular,

X ( T )

( T ) =fIE

V ( T )

F( T )

( T ) ;

so

X ( T ) = V:

The Martingale Representation Theorem guarantees the existence of a hedging portfolio, although

it does not tell us how to compute it It also justifies the risk-neutral pricing formula

X ( t ) = ( t )fIE

V ( T )

F( t )

= ( t )

Z ( t ) IE

Z ( T )

( T ) V

F( t )

= 1  ( t ) IE

 ( T ) V

F( t )

; 0tT;

where

 ( t ) = Z ( t )

( t )

= exp

,

Z t

0  ( u ) dB ( u ),

Z t

0 ( r ( u ) + 1 2  2 ( u )) du

18.3 d-dimensional Girsanov Theorem

Theorem 3.57 (d-dimensional Girsanov)  B ( t ) = ( B 1 ( t ) ;::: ;B d ( t )) ; 0  t  T, a d -dimensional Brownian motion on ;F;P);

 F( t ) ; 0tT;the accompanying filtration, perhaps larger than the one generated byB;

  ( t ) = (  1 ( t ) ;::: ; d ( t )) ; 0tT,d-dimensional adapted process.

For0tT;define

e

B j ( t ) =Z t

0  j ( u ) du + B j ( t ) ; j = 1 ;::: ;d;

Z ( t ) = exp

 ,

Z t

0  ( u ) : dB ( u ),1 2Z t

0 jj ( u )jj

2 du

;

f

IP ( A ) =Z

A Z ( T ) dIP:

Then, underIPf, the process

e

B ( t ) = ( Be1 ( t ) ;::: ; Bed ( t )) ; 0tT;

is ad-dimensional Brownian motion.

18.4 d-dimensional Martingale Representation Theorem

Theorem 4.58  B ( t ) = ( B 1 ( t ) ;::: ;B d ( t )) ; 0 t T;ad-dimensional Brownian motion

on ;F;P);

 F( t ) ; 0tT;the filtration generated by the Brownian motionB.

If X ( t ) ; 0  t  T, is a martingale (under IP) relative to F( t ) ; 0  t  T, then there is a

d-dimensional adpated process ( t ) = (  1 ( t ) ;::: ; d ( t )), such that

X ( t ) = X (0)+Z t

0  ( u ) : dB ( u ) ; 0tT:

Corollary 4.59 If we have ad-dimensional adapted process ( t ) = (  1 ( t ) ;::: ; d ( t )) ;then we can defineB;Ze andfIP as in Girsanov’s Theorem IfY ( t ) ; 0tT, is a martingale underIPfrelative

toF( t ) ; 0tT, then there is ad-dimensional adpated process ( t ) = ( 1 ( t ) d ( t ))such that

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

0 ( u ) : d Be( u ) ; 0tT:

18.5 Multi-dimensional market model

Let B ( t ) = ( B 1 ( t ) ;::: ;B d ( t )) ; 0  t  T, be a d-dimensional Brownian motion on some

;F;P), and letF( t ) ; 0  t  T, be the filtration generated byB Then we can define the following:

Stocks

dS i ( t ) =  i ( t ) S i ( t ) dt + S i ( t )Xd

j=1  ij ( t ) dB j ( t ) ; i = 1 ;::: ;m

Accumulation factor

( t ) = expZ t

0 r ( u ) du

:

Here, i ( t ) ; ij ( t )andr ( t )are adpated processes

S i ( t )

( t )



= (  i ( t ),r ( t ))

| {z }

Risk Premium

S i ( t )

( t ) dt + S i ( t )

( t )

d

X

j=1  ij ( t ) dB j ( t )

?

= S i ( t )

( t )

d

X

j=1  ij ( t )[  j ( t ) + dB j ( t )]

| {z }

d Bej(t)

(5.1)

For 5.1 to be satisfied, we need to choose 1 ( t ) ;::: ; d ( t ), so that

d

X

j=1  ij ( t )  j ( t ) =  i ( t ),r ( t ) ; i = 1 ;::: ;m: (MPR)

Market price of risk The market price of risk is an adapted process  ( t ) = (  1 ( t ) ;::: ; d ( t ))

satisfying the system of equations (MPR) above There are three cases to consider:

Case I: (Unique Solution) For Lebesgue-almost every t and IP-almost every !, (MPR) has a

unique solution ( t ) Using ( t )in thed-dimensional Girsanov Theorem, we define a unique

risk-neutral probability measureIPf UnderfIP, every discounted stock price is a martingale Consequently, the discounted wealth process corresponding to any portfolio process is afIP -martingale, and this implies that the market admits no arbitrage Finally, the Martingale Representation Theorem can be used to show that every contingent claim can be hedged; the

market is said to be complete.

Case II: (No solution.) If (MPR) has no solution, then there is no risk-neutral probability measure

and the market admits arbitrage.

Case III: (Multiple solutions) If (MPR) has multiple solutions, then there are multiple risk-neutral

probability measures The market admits no arbitrage, but there are contingent claims which

cannot be hedged; the market is said to be incomplete.

Theorem 5.60 (Fundamental Theorem of Asset Pricing) Part I (Harrison and Pliska,

Martin-gales and Stochastic integrals in the theory of continuous trading, Stochastic Proc and Applications

11 (1981), pp 215-260.):

If a market has a risk-neutral probability measure, then it admits no arbitrage.

Part II (Harrison and Pliska, A stochastic calculus model of continuous trading: complete markets,

Stochastic Proc and Applications 15 (1983), pp 313-316):

The risk-neutral measure is unique if and only if every contingent claim can be hedged.

18.4 d-dimensional Martingale Representation Theorem< /b>

Theorem 4.58  B ( t ) = ( B ( t ) ;::: ;B d (... portfolio process is afIP -martingale, and this implies that the market admits no arbitrage Finally, the Martingale Representation Theorem can be used to show that every contingent... )

( T ) ;

so

X ( T ) = V:

The Martingale Representation Theorem guarantees the existence of a hedging portfolio, although

it does not