Girsanov’s theorem and the risk-neutral measure

Girsanov’s theorem and the risk-neutral measure Please see Oksendal, 4th ed., pp 145–151.. Caveat: This theorem requires a technical condition on the size of... IP is a probability measu

Girsanov’s theorem and the risk-neutral measure

(Please see Oksendal, 4th ed., pp 145–151.)

Theorem 0.52 (Girsanov, One-dimensional) Let B ( t ) ; 0  t  T, be a Brownian motion on

a probability space ;F;P) Let F( t ) ; 0  t  T, be the accompanying filtration, and let

 ( t ) ; 0tT, be a process adapted to this filtration For0tT, define

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

;

and define a new probability measure by

f

IP ( A ) =Z

A Z ( T ) dIP; 8A2 F:

UnderfIP, the processBe( t ) ; 0tT, is a Brownian motion.

Caveat: This theorem requires a technical condition on the size of If

IE exp

(

1 2

Z T

0  2 ( u ) du

)

<1;

everything is OK

We make the following remarks:

Z ( t )is a matingale In fact,

dZ ( t ) =, ( t ) Z ( t ) dB ( t ) + 1 2  2 ( t ) Z ( t ) dB ( t ) dB ( t ),1 2  2 ( t ) Z ( t ) dt

=, ( t ) Z ( t ) dB ( t ) :

189

IP is a probability measure SinceZ (0) = 1, we haveIEZ ( t ) = 1for everyt0 In particular

f

Z ( T ) dIP = IEZ ( T ) = 1 ;

soIPfis a probability measure

f

IEin terms ofIE LetfIEdenote expectation underIPf IfXis a random variable, then

f

IEZ = IE [ Z ( T ) X ] :

To see this, consider first the caseX = 1A, whereA2 F We have

f

IEX = IPf( A ) =Z

A Z ( T ) dIP =Z

Z ( T ) 1A dIP = IE [ Z ( T ) X ] :

Now use Williams’ “standard machine”

f

IP andIP The intuition behind the formula

f

IP ( A ) =Z

A Z ( T ) dIP 8A2 F

is that we want to have

f

IP ( ! ) = Z ( T;! ) IP ( ! ) ;

but sinceIP ( ! ) = 0andfIP ( ! ) = 0, this doesn’t really tell us anything useful aboutIPf Thus,

we consider subsets of , rather than individual elements of

Distribution ofBe( T ) Ifis constant, then

Z ( T ) = expn

,B ( T ),1 2  2 To e

B ( T ) = T + B ( T ) :

UnderIP,B ( T )is normal with mean 0 and varianceT, soBe( T )is normal with meanTand varianceT:

IP ( Be( T )2d ~ b ) = 1p

2 T exp

(

,

(~ b,T ) 2

2 T

)

d ~ b:

Removal of Drift fromBe( T ) The change of measure fromIP tofIP removes the drift fromBe( T )

To see this, we compute

f

IE Be( T ) = IE [ Z ( T )( T + B ( T ))]

= IEh

expn ,B ( T ),1 2  2 To

( T + B ( T ))i

= 1p

2 T

Z 1

,1

( T + b )expf,b,

1

2  2 Tg exp

(

,

b 2

2 T

)

db

= 1p

2 T

Z 1

,1

( T + b )exp

(

,

( b + T ) 2

2 T

)

db ( y = T + b ) = 1p

2 T

Z 1

,1

y exp

(

,

y 2

2

)

dy (Substitutey = T + b)

= 0 :

We can also see thatIEfBe( T ) = 0by arguing directly from the density formula

IP n e

B ( t )2d ~ bo

= 1p

2 T exp

(

,

(~ b,T ) 2

2 T

)

d ~ b:

Because

Z ( T ) = expf,B ( T ),1 2  2 Tg

= expf, ( Be( T ),T ), 1 2  2 Tg

= expf, Be( T ) + 1 2  2 Tg;

we have

f

IPn e

B ( T )2d ~ bo

= IP n e

B ( T )2d ~ bo

expn , ~ b + 1 2  2 To

= 1p

2 T exp

(

,

(~ b,T ) 2

2 T , ~ b + 1 2  2 T

)

d ~ b:

= 1p

2 T exp

(

,

~

b 2

2 T

)

d ~ b:

UnderfIP, Be( T )is normal with mean zero and varianceT UnderIP,Be( T )is normal with

meanT and varianceT

Means change, variances don’t When we use the Girsanov Theorem to change the probability

measure, means change but variances do not Martingales may be destroyed or created Volatilities, quadratic variations and cross variations are unaffected Check:

d B de Be = (  ( t ) dt + dB ( t )) 2 = dB:dB = dt:

17.1 Conditional expectations under f

I P

Lemma 1.53 Let0tT IfXisF( t )-measurable, then

f

IEX = IE [ X:Z ( t )] :

Proof:

f

IEX = IE [ X:Z ( T )] = IE [ IE [ X:Z ( T )jF( t )] ]

= IE [ X IE [ Z ( T )jF( t )] ]

= IE [ X:Z ( t )]

becauseZ ( t ) ; 0 t T, is a martingale underIP

Lemma 1.54 (Baye’s Rule) IfXisF( t )-measurable and0stT, then

f

IE [ XjF( s )] = 1 Z ( s ) IE [ XZ ( t )jF( s )] : (1.1)

Proof: It is clear that 1

Z(s) IE [ XZ ( t )jF( s )]isF( s )-measurable We check the partial averaging property ForA2 F( s ), we have

Z

A 1

Z ( s ) IE [ XZ ( t )jF( s )] dfIP =fIE

1A Z 1 ( s ) IE [ XZ ( t )jF( s )]

= IE [ 1A IE [ XZ ( t )jF( s )]] (Lemma 1.53)

= IE [ IE [ 1A XZ ( t )jF( s )]] (Taking in what is known)

= IE [ 1A XZ ( t )]

=fIE [ 1A X ] (Lemma 1.53 again)

=Z

A X dfIP:

Although we have proved Lemmas 1.53 and 1.54, we have not proved Girsanov’s Theorem We will not prove it completely, but here is the beginning of the proof

Lemma 1.55 Using the notation of Girsanov’s Theorem, we have the martingale property

f

IE [ Be( t )jF( s )] = Be( s ) ; 0stT:

Proof: We first check thatBe( t ) Z ( t )is a martingale underIP Recall

d Be( t ) =  ( t ) dt + dB ( t ) ;

dZ ( t ) =, ( t ) Z ( t ) dB ( t ) :

Therefore,

d ( BZe ) = B dZe + Z d Be+ d B dZe

=, e

BZ dB + Z dt + Z dB,Z dt

= (, e

BZ + Z ) dB:

Next we use Bayes’ Rule For0stT,

f

IE [ Be( t )jF( s )] = 1 Z ( s ) IE [ Be( t ) Z ( t )jF( s )]

= 1 Z ( s ) Be( s ) Z ( s )

= Be( s ) :

Definition 17.1 (Equivalent measures) Two measures on the same probability space which have

the same measure-zero sets are said to be equivalent.

The probability measures IP and fIP of the Girsanov Theorem are equivalent Recall that IPf is defined by

f

IP ( A ) =Z

Z ( T ) dIP; A2 F:

IfIP ( A ) = 0, then

R

A Z ( T ) dIP = 0 :BecauseZ ( T ) > 0for every!, we can invert the definition

offIP to obtain

IP ( A ) =Z

A 1

Z ( T ) dfIP; A2 F:

IffIP ( A ) = 0, then

R

A 1 Z(T) dIP = 0 :

17.2 Risk-neutral measure

As usual we are given the Brownian motion: B ( t ) ; 0  t T, with filtrationF( t ) ; 0  t T, defined on a probability space ;F;P) We can then define the following

Stock price:

dS ( t ) =  ( t ) S ( t ) dt +  ( t ) S ( t ) dB ( t ) :

The processes ( t ) and  ( t ) are adapted to the filtration The stock price model is completely general, subject only to the condition that the paths of the process are continuous

Interest rate:r ( t ) ; 0tT The processr ( t )is adapted

Wealth of an agent, starting withX (0) = x We can write the wealth process differential in several ways:

dX ( t ) =
( t ) dS ( t )

Capital gains from Stock

+ r ( t )[ X ( t ),
( t ) S ( t )] dt

Interest earnings

= r ( t ) X ( t ) dt +
( t )[ dS ( t ),rS ( t ) dt ]

= r ( t ) X ( t ) dt +
( t )(  ( t ),r ( t ))

Risk premium

S ( t ) dt +
( t )  ( t ) S ( t ) dB ( t )

= r ( t ) X ( t ) dt +
( t )  ( t ) S ( t )

2

6

6

4

 ( t ),r ( t )

 ( t )

Market price of risk=(t)

dt + dB ( t )

3

7

7

5

Discounted processes:

d

e,

Rt

0r(u) du S ( t )

= e,

Rt

0r(u) du [,r ( t ) S ( t ) dt + dS ( t )]

d

e,

Rt

0r(u) du X ( t )

= e,

Rt

0r(u) du [,r ( t ) X ( t ) dt + dX ( t )]

=
( t ) d

e,

Rt

0r(u) du S ( t )

:

Notation:

( t ) = eRt

0r(u) du ;  1 ( t ) = e,

Rt

0r(u) du ;

d ( t ) = r ( t ) ( t ) dt; d

1 ( t )



=,

r ( t ) ( t ) dt:

The discounted formulas are

d

S ( t ) ( t )



= 1 ( t ) [,r ( t ) S ( t ) dt + dS ( t )]

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

= 1 ( t )  ( t ) S ( t )[  ( t ) dt + dB ( t )] ;

d

X ( t ) ( t )



=
( t ) d

S ( t ) ( t )



=
( t ) ( t )  ( t ) S ( t ) [  ( t ) dt + dB ( t )] :

Changing the measure Define

e

B ( t ) =Z t

0  ( u ) du + B ( t ) :

Then

d

S ( t ) ( t )



= 1 ( t )  ( t ) S ( t ) d Be( t ) ;

d

X ( t ) ( t )



=
( t ) ( t )  ( t ) S ( t ) d Be( t ) :

UnderfIP, S(t)

(t)and X(t)

(t) are martingales

Definition 17.2 (Risk-neutral measure) A risk-neutral measure (sometimes called a martingale

measure) is any probability measure, equivalent to the market measureIP, which makes all dis-counted asset prices martingales

For the market model considered here,

f

IP ( A ) =Z

A Z ( T ) dIP; A2 F;

where

Z ( t ) = exp

,

Z t

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

0  2 ( u ) du

;

is the unique risk-neutral measure Note that because ( t ) = (t) ,(t) r(t) ;we must assume that ( t )6=

0

Risk-neutral valuation Consider a contingent claim paying anF( T )-measurable random variable

V at timeT

Example 17.1

V = (S(T),K)+; European call

V = (K,S(T))+; European put

V = T 1Z

T

0

S(u) du,K

! +

V = max

0tT

S(t); Look back

If there is a hedging portfolio, i.e., a process
( t ) ; 0tT, whose corresponding wealth process satisfiesX ( T ) = V, then

X (0) =fIE

V ( T )



:

This is because X(t)

(t) is a martingale underIPf, so

X (0) = X (0)

(0) =fIE

X ( T ) ( T )



=fIE

V ( T )



:

(t)and X(t)

(t) are martingales

Definition 17.2 (Risk-neutral measure) A risk-neutral measure (sometimes called a martingale

measure) ... :

Changing the measure Define

e

B ( t ) =Z t

0  ( u ) du + B ( t ) :

Then

d... measure (sometimes called a martingale

measure) is any probability measure, equivalent to the market measure< /i>IP, which makes all dis-counted asset prices martingales