80-646-08StochasticcalculusIGeneviŁveGauthier ChangeofmeasureandGirsanovtheorem

Change of measure Radon- Nikodym th.

Girsanov th.

Multidimensional References

Change of measure and Girsanov theorem

80-646-08 Stochastic calculus I

Geneviève Gauthier

HEC Montréal

Change of measure

Example 1 Radon- Nikodym th.

Girsanov th.

Multidimensional References

An example I

Let ( _Ω , F ^, fF

^{: 0 t T} g ^, P ) be a …ltered probability space on which

a standard Brownian motion W

= W

: 0 t T is constructed.

The stochastic process S = f ^S

^{: 0 t T} g ^represents the evolution of a risky security price and satis…es the stochastic di¤erential equation

dS

= µS

dt + σS

dW

.

Let’s also assume that the interest rate r is constant. The discount factor is therefore

β ( t ) = exp ( rt )

which implies that d β ( t ) = r exp ( rt ) dt.

Change of measure

Example 1 Radon- Nikodym th.

Girsanov th.

Multidimensional References

An example II

Let’s set, for all 0 t T , Y

= β

S

i.e. Y

represents the present value at time t of the risky security.

Using Itô’s lemma (more precisely the multiplication rule), we obtain

dY

= ( µ r ) Y

dt + σY

dW

. Indeed,

dY

= d β

S

= β

dS

+ S

d β

+ d h ^{β, S} i

= β

µS

dt + σS

dW

+ S

( r β

dt )

= ( µ r ) β

S

dt + σβ

S

dW

.

Change of measure

Example 1 Radon- Nikodym th.

Girsanov th.

Multidimensional References

An example III

In its integral form, such a stochastic di¤erential equation becomes

Y

= Y

+ ( µ r )

Z

Y

ds + σ Z

0

Y

dW

.

Change of measure

Example 1 Radon- Nikodym th.

Girsanov th.

Multidimensional References

Refresher

Itô process

Let W

be a ( fF

g ^{, P} ) Brownian motion.

An Itô process is a process X = f ^X

^{: 0 t T} g ^taking its values in R such that:

X

X

+

Z

0

K

ds +

Z

0

H

dW

with K = f ^K

^{: 0 t T} g ^{and H} = f ^H

^{: 0 t T} g ^, processes adapted to the …ltration fF

g ^,

P h R

0

j ^K

j ^ds < _∞ ⁱ = 1 P h R

0

( H

)

ds < _∞ ⁱ = 1

Damien Lamberton and Bernard Lapeyre, Introduction au calcul

stochastique appliqué à la …nance, Ellipses, page 53.

Change of measure

Example 1 Radon- Nikodym th.

Girsanov th.

Multidimensional References

Example (suite) I

Recall that W

is a ( fF

g ^{, P} ) Brownian motion.

In a risk-neutral world ( _Ω , F ^, fF

^{: t 0} g ^{, Q} ) , the stochastic process Y = f ^Y

^{: 0 t T} g should be a ( fF

g ^{, Q} ) martingale.

Thus, under the risk-neutral measure, the trend of Y

should be nil, i.e. we want the drift coe¢ cient to be 0.

Change of measure

Example 1 Radon- Nikodym th.

Girsanov th.

Multidimensional References

Example (suite) II

Let’s set

W

= W

+

Z

0

γ

ds and note that

1

W

is not a P martingale (its expectation varies in time) and

2

dW

= dW

+ γ

dt . As a consequence

Y

= Y

+ ( µ r ) Z

0

Y

ds + σ Z

0

Y

dW

Y

= Y

+

Z

0

( µ r σγ

) Y

ds + σ Z

0

Y

dW

.

In order to get rid of the drift term, it is su¢ cient to set µ r σγ

= 0 , ^γ

= ^{µ r}

σ .

Change of measure

Example 1 Radon- Nikodym th.

Girsanov th.

Multidimensional References

Example (suite) III

Recall that

Y

= Y

+ σ Z

0

Y

dW

Note that, under the measure P , the process W

is not a standard Brownian motion since the law of W

under the measure P is N

t, t .

The process Y will not be a ( fF

g ^, P ) martingale since the stochastic integral is constructed with respect to W

which is not a ( fF

g ^{, P} ) martingale.

Indeed,

E

h W

i

= ^{µ r}

σ t

varies in time.

Change of measure

Example 1 Radon- Nikodym th.

Girsanov th.

Multidimensional References

Example (suite) IV

Recall that W

is a ( fF

g ^{, P} ) Brownian motion, Y

= Y

+ σ

Z

0

Y

dW

where

W

( t ) = W

( t ) + ^{µ r} σ t.

So we want to …nd the probability measure Q to be placed on the space ( _Ω , F ^, fF

g ) such that W

is a

Q standard Brownian motion.

By changing the probability on the set Ω , we transform

the drift coe¢ cient so that the trend becomes zero and we

integrate with respect to a ( fF

g ^, Q ) martingale. As a

result, the process Y will be ( fF

g ^{, Q} ) martingale.

Change of measure Radon- Nikodym th.

Girsanov th.

Multidimensional References

Radon-Nikodym theorem I

A way to construct new probability measures on the measurable space ( Ω , F ) when we already have a

probability measure P existing on that space is as follows:

Let Y be a random variable constructed on the probability space ( Ω , F ^{, P} ) such that

8 ^ω 2 Ω , Y ( ω ) 0 and E

[ Y ] = 1.

For all event A 2 F ^, I

denotes the indicator function of that event:

I

( ω ) = ^{1 if ω} 2 ^A 0 otherwise.

For all event A 2 F , let’s set

Q ( A ) = E

[ Y I

] .

Then Q is a probability measure on ( Ω , F ) .

Change of measure Radon- Nikodym th.

Girsanov th.

Multidimensional References

Radon-Nikodym theorem II

Proof. We must verify that (P1) Q ( Ω ) = 1,

(P2) 8 ^A 2 F ^{, 0} Q ( A ) 1,

(P3) 8 ^A

^{, A}

^{, ...} 2 F ^{such that A}

\ ^A

= ∅ si i 6 = j,

Q ^S

A

= _∑

Q ( A

) .

Change of measure Radon- Nikodym th.

Girsanov th.

Multidimensional References

Radon-Nikodym theorem III

Veri…cation of (P1). But, since for all ω, I

( ω ) = 1 and because we have assumed that E

[ Y ] = 1,

Q ( Ω ) = E

[ Y I

] = E

[ Y ] = 1,

which establishes condition ( P 1 ) .

Change of measure Radon- Nikodym th.

Girsanov th.

Multidimensional References

Radon-Nikodym theorem IV

Veri…cation of (P2). The second condition is just as easy to prove: since Y is a positive random variable, Y I

is a positive random variable too, and Q ( A ) = E

[ Y I

] 0.

Moreover,

Q ( A ) = E

[ Y I

] E

[ Y I

]

= E

[ Y ]

= 1.

Change of measure Radon- Nikodym th.

Girsanov th.

Multidimensional References

Radon-Nikodym theorem V

Veri…cation of (P3). As we have established in an exercise in the …rst chapter, 8 ^A

^{, A}

^{, ...} 2 F ^{such that} A

\ ^A

= ∅ if i 6 = j ,

I

= ∑

i 1

I

.

As a consequence, Q ^[

i 1

A

!

= E

h

Y I

i

= E

"

Y ∑

i 1

I

#

= ∑

i 1

E

[ Y I

]

= ∑

i 1

Q ( A

) .

Change of measure Radon- Nikodym th.

Girsanov th.

Multidimensional References

Radon-Nikodym theorem VI

De…nition

Two probability measures P and Q constructed on the same measurable space ( Ω , F ) are said to be equivalent if they have the same set of impossible events, i.e.

P ( A ) = 0 , ^Q ( A ) = 0, A 2 F ^.

Change of measure Radon- Nikodym th.

Girsanov th.

Multidimensional References

Radon-Nikodym theorem VII

Question. Given two equivalent probability measures P and Q , does there exist a non-negative valued random variable Y such that

Q ( A ) = E

[ Y I

] ?

Note the di¤erence between such a problem and the result we have just proven.

In the latter, Y and P were given to us and we have constructed Q .

In this case, P and Q are given to us and we need to …nd Y , which is less easy.

The existence of such a variable is established in the next

theorem which is a version of the famous Radon-Nikodym

theorem.

Change of measure Radon- Nikodym th.

Girsanov th.

Multidimensional References

Radon-Nikodym theorem VIII

Theorem

Radon-Nikodym theorem . Given two equivalent probability measures P and Q constructed on the measurable space ( Ω , F ) , there exists a positive-valued random variable Y such that

Q ( A ) = E

[ Y I

] .

Such a random variable Y is often denoted by

.

Such a theorem still does not tell us how to …nd our

risk-neutral measure. Actually, it is the next result that

will provide us with the recipe to construct our measure

and it involves the Radon-Nikodym derivative.

Change of measure Radon- Nikodym th.

Girsanov th.

Multidimensional References

Radon-Nikodym theorem IX

A few thoughts about the discrete case

Assume that Ω only contains a …nite number of elements.

Let Y = β

X be the present value of the attainable contingent claim X . Si F

= f Ω , ∅ g , then its price at time t = 0 is

E

[ Y ] = ∑

ω2Ω

Y ( ω ) _Q ( ω )

= ∑

ω2Ω

Y ( ω ) ^Q ( ω ) P ( ω ) ^P ( ω )

= E

Y Q

P

Change of measure Radon- Nikodym th.

Girsanov th.

Multidimensional References

Radon-Nikodym theorem X

Consider the binomial market model: S

represents the evolution of the riskless asset and S

models a risky asset. The unique risk-neutral measure is denoted by Q , P being the ”real”measure.

ω S⁽¹⁾₀ (ω) S⁽²⁾₀ (ω)

! S⁽¹⁾₁ (ω) S⁽²⁾₁ (ω)

! S⁽¹⁾₂ (ω) S⁽²⁾₂ (ω)

!

P Q ^dQdP

ω1 (1;2)⁰ (1,1;2)⁰ (1,21;1)^{0 1}₄ 0,360 1,44

ω2 (1;2)⁰ (1,1;2)⁰ (1,21;3)^{0 1}₄ 0,540 2.16

ω3 (1;2)⁰ (1,1;4)⁰ (1,21;1)^{0 1}₄ 0,015 0,06

ω4 (1;2)⁰ (1,1;4)⁰ (1,21;5)⁰. ¹₄ 0.085 0,34

The Radon-Nykodym derivative is somewhat the memory

of the change of measure. For each path, it remembers

how we have changed weights.

Change of measure Radon- Nikodym th.

Girsanov th.

Example 1

Multidimensional References

Girsanov theorem I

Let’s focus on a bounded time interval: t 2 [ 0, T ] . Let W = f ^W

^{: t} 2 [ 0, T ] g represent a Brownian motion constructed on a …ltered probability space

( Ω , F ^, fF

g ^{, P} ) such that the …ltration fF

g ^{is the one} generated by the Brownian motion, plus it includes all zero-probability events, i.e. for all t 0,

F

= σ ( N ^{and W}

^{: 0 s t} ) .

The next theorem will enable to construct our risk- neutral

measures.

Change of measure Radon- Nikodym th.

Girsanov th.

Example 1

Multidimensional References

Girsanov theorem II

Theorem

Cameron-Martin-Girsanov theorem. Let

γ = f ^γ

: t 2 [ 0, T ] g ^{be a} fF

g predictable process such that

E

exp 1 2

Z

γ

dt < _∞ . There exists a measure Q on ( Ω , F ) such that (CMG1) Q is equivalent to P

(CMG2)

= exp h R

0

γ

dW

R

0

γ

dt i

(CMG3) The process f W = ⁿ W ^f

: t 2 [ 0, T ] ^o de…ned as f

W

= W

+ R

0

γ

ds is a ( fF

g ^{, Q} ) Brownian motion.

(ref. Baxter and Rennie, page 74; Lamberton and Lapeyre, page 84)

Change of measure Radon- Nikodym th.

Girsanov th.

Example 1

Multidimensional References

Girsanov theorem III

The condition E

h

exp

R

0

γ

dt i

< _∞ is a su¢ cient

but non-necessary condition. It is know as the Novikov

condition.

Change of measure Radon- Nikodym th.

Girsanov th.

Example 1

Multidimensional References

Girsanov theorem IV

Consider the stochastic di¤erential equation dX

= b ( X

, t ) dt + a ( X

, t ) dW

where W represents a Brownian motion on the …ltered probability space ( _Ω , F ^, fF

g ^, P ) .

We assume that the drift and di¤usion coe¢ cients are such that there exists a unique solution to the equation, which we denote X .

We want to …nd a probability measure Q , such that, on

the space ( _Ω , F ^, fF

g ^{, Q} ) , the drift of X is e b ( X

, t )

instead of b ( X

, t ) .

Change of measure Radon- Nikodym th.

Girsanov th.

Example 1

Multidimensional References

Girsanov theorem V

Let’s go!

dX

= b ( X

, t ) dt + a ( X

, t ) dW

= e b ( X

, t ) dt + a ( X

, t ) ^b ( X

, t ) e b ( X

, t ) a ( X

, t )

! dt + a ( X

, t ) dW

provided that a ( X

, t ) is di¤erent from 0.

= e b ( X

, t ) dt + a ( X

, t ) d W

+

0

b ( X

, s ) e b ( X

, s ) a ( X

, s ) ^ds

!

= e b ( X

, t ) dt + a ( X

, t ) d W f

where f

W

= W

+

Z

0

γ

ds and γ

= ^b ( X

, t ) ^e b ( X

, t )

a ( X

, t ) ^.

Change of measure Radon- Nikodym th.

Girsanov th.

Example 1

Multidimensional References

Girsanov theorem VI

If E

h

exp

R

0

γ

dt i

< ∞ then by the

Radon-Nikodym and Cameron-Martin-Girsanov theorems, Q ( A ) = E

exp

Z

0

γ

dW

1 2

Z

0

γ

dt I

, A 2 F and W f = ^{n f} W

: t 2 [ 0, T ] ^o is a ( F , Q ) Brownian

motion.

In practice, we don’t need to determine the measure Q . It

is su¢ cient for us to know it exists, and to know the

stochastic di¤erential equation of the process of interest

on the space ( Ω , F ^, fF

g ^{, Q} ) .

Change of measure Radon- Nikodym th.

Girsanov th.

Example 1

Multidimensional References

Example 1 I

Girsanav theorem

Let’s go back to the Black-Scholes market model. The stochastic process Y = f ^Y

^{: 0 t T} g constructed on the space ( _Ω , F ^, fF

g ^, P ) used to construct the

Brownian motion represents the evolution of the present value of a risky security where

dY

= ( µ r ) Y

dt + σY

dW

.

Change of measure Radon- Nikodym th.

Girsanov th.

Example 1

Multidimensional References

Example 1 II

Girsanav theorem

But, in a risk-neutral world ( Ω , F ^, fF

g ^{, Q} ) , the trend of Y should be zero, i.e. we want the drift coe¢ cient to be zero. Thus

dY

= ( µ r ) Y

dt + σY

dW

= σY

µ r

σ dt + σY

dW

= σY

d W

+ ^{µ r}

σ t = σY

dW

where

W

W

+ ^{µ r}

σ t = W

+

Z

µ r

σ ds.

Change of measure Radon- Nikodym th.

Girsanov th.

Example 1

Multidimensional References

Example 1 III

Girsanav theorem

In the present case,

8 ^{s, γ}

= ^{µ r}

σ .

Change of measure Radon- Nikodym th.

Girsanov th.

Example 1

Multidimensional References

Example 1 IV

Girsanav theorem

Recall the Cameron-Martin-Girsanov theorem. Let γ = f ^γ

: t 2 [ 0, T ] g ^{be a} fF

g predictable process such that

E

exp 1 2

Z

0

γ

dt < _∞ . There exists a measure Q on ( Ω , F ) such that (CMG1) Q is equivalent to P

(CMG2)

= exp h R

0

γ

dW

R

0

γ

dt i

(CMG3) The process f W = ⁿ W f

: t 2 [ 0, T ] ^o de…ned as W f

= W

+ R

0

γ

ds is a ( fF

g ^{, Q} ) Brownian motion.

Change of measure Radon- Nikodym th.

Girsanov th.

Example 1

Multidimensional References

Example 1 V

Girsanav theorem

Let’s verify that the condition on the process γ is indeed satis…ed:

E

exp 1 2

Z

0

γ

dt

= E

"

exp 1 2

Z

µ r σ

2

dt

!#

= exp 1 2

µ r σ

2

T

!

< _∞ .

Change of measure Radon- Nikodym th.

Girsanov th.

Example 1

Multidimensional References

Example 1 VI

Girsanav theorem

Let’s apply Girsanov theorem : d Q

d P = exp

Z

0

γ

dW

1 2

Z

γ

dt

= exp

" _Z

T 0

µ r

σ dW

1 2

Z

0

µ r σ

2

dt

#

= exp

"

µ r

σ W

1 2

µ r σ

2

T

# .

This implies that Q [ A ] = E

"

exp µ r

σ W

1 2

µ r σ

2

T

! I

#

, A 2 F ^.

Change of measure Radon- Nikodym th.

Girsanov th.

Example 1

Multidimensional References

Example 1 VII

Girsanav theorem

Moreover, under the measure Q , the evolution of the present value of the risky security satis…es the equation

dY

= σY

dW

where W

is a Q Brownian motion.

We can also deduce the stochastic di¤erential equation satis…ed by the evolution of the risky security price S :

dS

= µS

dt + σS

dW

= µS

dt + σS

d W

µ r σ t puisque W

W

+ ^{µ r}

σ t

= µS

dt + σS

dW

σS

µ r σ dt

= rS

dt + σS

dW

.

Change of measure Radon- Nikodym th.

Girsanov th.

Example 1

Multidimensional References

Example 1 VIII

Girsanav theorem

Change of measure Radon- Nikodym th.

Girsanov th.

Example 1

Multidimensional References

Example 1 IX

Girsanav theorem

Note that we don’t really need to calculate Q , we simply need to know that it exists then toestablish what is the equation satis…ed by the processus of interest, i.e. the evolution of the risky security price. Indeed, on ( Ω , F ^, fF

g ^{, Q} ) ,

dS

= rS

dt + σS

dW

where W f is a Q Brownian motion.

But the unique solution to that equation is S

= S

exp r σ

2 t + σW

.

Change of measure Radon- Nikodym th.

Girsanov th.

Example 1

Multidimensional References

Example 1 X

Girsanav theorem

Since the price of a call option, the strike price of which is K and the maturity of which is T , is given by

E

[ exp ( rT ) max ( S

K ; 0 )]

= E

exp ( rT ) max S

exp r σ

2 T + σW

K ; 0

= E

max S

exp σ

2 T + σW

K exp ( rT ) ; 0

=

∞

max S

exp σ

2 T + σz Ke

; 0 f

( z ) dz.

where f

( ) represents the probability density function of a normal random variable with zero expectation and variance T . The rest of the calculation is a pure application of the

properties of the normal law.

Change of measure Radon- Nikodym th.

Girsanov th.

Example 1

Multidimensional References

Example 1 XI

Girsanav theorem

Since

S

exp σ

2 T + σz > Ke

, ^σ

2

2 T + σz > ln Ke

S

= ln K rT ln S

, ^z >

ln S

ln K + r

T

σ d

p

T

Change of measure Radon- Nikodym th.

Girsanov th.

Example 1

Multidimensional References

Example 1 XII

Girsanav theorem

Z

∞

max S

exp σ

2 T + σz Ke

; 0 f

( z ) dz

=

Z

d2p T

S

exp σ

2 T + σz Ke

f

( z ) dz

=

Z

d2

pT

S

exp σ

2 T + σz f

( z ) dz Z

d2

pT

Ke

f

( z ) dz

= S

Z

d2p T

p 1 2π

p 1

T exp z

2T σz + σ

T

2T dz

Ke

Z

d2

pT

p 1 2π

p 1

T exp z

2T dz

Change of measure Radon- Nikodym th.

Girsanov th.

Example 1

Multidimensional References

Example 1 XIII

Girsanav theorem

= S

Z

d2

pT

p 1 2π

p 1

T exp ( z σT )

2T

! dz

Ke

Z

d₂p T

p 1 2π

p 1

T exp z

2T dz Let’s set u = ^z p ^σT

T and v = p ^z T

= S

Z

d2 σp T

p 1

2π exp u

2 du Ke

Z

p 1

2π exp v

2 dv

= S

1 N d

σ p

T Ke

( 1 N ( d

))

Change of measure Radon- Nikodym th.

Girsanov th.

Example 1

Multidimensional References

Example 1 XIV

Girsanav theorem

where N ( ) is the cumulative distribution function of a

standard normal random variable.

Change of measure Radon- Nikodym th.

Girsanov th.

Example 1

Multidimensional References

Example 1 XV

Girsanav theorem

But the symmetry of N implies that 1 N ( x ) = N ( x ) . Then

S

1 N d

σ p

T Ke

( 1 N ( d

))

= S

N d

+ σ p

T Ke

( N ( d

))

Change of measure Radon- Nikodym th.

Girsanov th.

Multidimensional Example 2 Girsanov Example 2 (continued) Example 3 References

Example 2 I

Let’s assume that W

and W f

represent two standard Brownian motions constructed on the …ltered probability space ( _Ω , F ^, fF

g ^{, P} ) .

Note that n e

B

: t 0 o where B e

ρW

+

q

1 ρ

W f

is a standard Brownian motion such that

Corr

W

, B e

= ρ.

Exercise: prove it.

Change of measure Radon- Nikodym th.

Girsanov th.

Multidimensional Example 2 Girsanov Example 2 (continued) Example 3 References

Example 2 II

The instantaneous exchange rate

dC

= µ

C

dt + σ

C

dW

enables us to model the number of Canadian dollars per unit of foreign currency at any time.

Suppose also that the stochastic di¤erential equation dS

= µ

S

dt + σ

S

d B e

= µ

S

dt + σ

ρS

dW

+ σ

q

1 ρ

S

d W f

models the evolution of a foreign risky asset price.

Change of measure Radon- Nikodym th.

Girsanov th.

Multidimensional Example 2 Girsanov Example 2 (continued) Example 3 References

Example 2 III

Lastly, the Canadian instantaneous interest rate r and the foreign instantaneous interest rate v are assumed to be constant. As a consequence, the discount factor is

β

= exp ( rt ) .

and the value in foreign currency of an initial investment

equal to one foreign currency unit is B

= exp ( vt ) .

Change of measure Radon- Nikodym th.

Girsanov th.

Multidimensional Example 2 Girsanov Example 2 (continued) Example 3 References

Example 2 IV

Let’s put ourselves in the shoes of a Canadian investor.

C

S

gives us the Canadian dollar value of a risky asse at time t ,

C

B

gives us the Canadian dollar value, at time t , of one foreign currency unit invested in a foreign bank account U

= β

C

S

gives us the Canadian dollar present value of the risky asset at time t.

V

= β

C

B

gives us the Canadian dollar present value, at time t, of one foreign currency unit invested in a foreign bank account.

We wish to …nd a measure Q , such that the stochastic

processes U and V are ( fF

g ^{, Q} ) martingales.

Change of measure Radon- Nikodym th.

Girsanov th.

Multidimensional Example 2 Girsanov Example 2 (continued) Example 3 References

Example 2 V

Recall:

dC

= µ

C

dt + σ

C

dW

, dB

= vB

dt

d β

= r β

dt

First, let’s determine the stochastic di¤erential equation satis…ed by the Canadian dollar present value of one foreign currency unit invested in a foreign bank account V = βCB under the measure P . Itô’s lemma allows us to write

dC

B

= C

dB

+ B

dC

+ d h ^{B, C} i

= C

( vB

dt ) + B

µ

C

dt + σ

C

dW

= ( v + µ

) C

B

dt + σ

C

B

dW

.

Change of measure Radon- Nikodym th.

Girsanov th.

Multidimensional Example 2 Girsanov Example 2 (continued) Example 3 References

Example 2 VI

Thus,

dV

= d β

C

B

= β

dC

B

+ C

B

d β

+ d h ^{β, CB} i

= β

( v + µ

) C

B

dt + σ

C

B

dW

+ C

B

( r β

dt )

= ( µ

+ v r ) β

C

B

dt + σ

β

C

B

dW

= ( µ

+ v r ) V

dt + σ

V

dW

.

Change of measure Radon- Nikodym th.

Girsanov th.

Multidimensional Example 2 Girsanov Example 2 (continued) Example 3 References

Example 2 VII

Recall:

dS

= µ

S

dt + σ

ρS

dW

+ σ

q

1 ρ

S

d W f

, dC

= µ

C

dt + σ

C

dW

,

dB

= vB

dt,

d β

= r β

dt .

Change of measure Radon- Nikodym th.

Girsanov th.

Multidimensional Example 2 Girsanov Example 2 (continued) Example 3 References

Example 2 VIII

Second, let’s determine the stochastic di¤erential equation satis…ed by U = βCS under the measure P . Itô’s lemma allows us to write

dC

S

= C

dS

+ S

dC

+ d h ^{S , C} i

= C

µ

S

dt + σ

ρS

dW

+ σ

q

1 ρ

S

d W f

+ S

µ

C

dt + σ

C

dW

+ σ

ρS

σ

C

dt

= ( µ

+ µ

+ σ

σ

ρ ) C

S

dt + ( σ

ρ + σ

) C

S

dW

+ σ

q

1 ρ

C

S

d W f

.

Change of measure Radon- Nikodym th.

Girsanov th.

Multidimensional Example 2 Girsanov Example 2 (continued) Example 3 References

Example 2 IX

Applying Itô’s lemma again, we obtain

dU

= d β

C

S

= β

dC

S

+ C

S

d β

+ d h ^{β, CS} i

= β

( µ

+ µ

+ σ

σ

ρ ) C

S

dt + ( σ

ρ + σ

) C

S

dW

+ σ

p

1 ρ

C

S

d W f

+ C

S

( r β

dt )

= ( µ

+ µ

+ σ

σ

ρ r ) β

C

S

dt + ( σ

ρ + σ

) β

C

S

dW

+ σ

q

1 ρ

β

C

S

d W f

= ( µ

+ µ

+ σ

σ

ρ r ) U

dt + ( σ

ρ + σ

) U

dW

+ σ

q

1 ρ

U

d f W

.

Change of measure Radon- Nikodym th.

Girsanov th.

Multidimensional Example 2 Girsanov Example 2 (continued) Example 3 References

Example 2 X

We have

dV

= ( µ

+ v r ) V

dt + σ

V

dW

, dU

= ( µ

+ µ

+ σ

σ

ρ r ) U

dt

+ ( σ

ρ + σ

) U

dW

+ σ

q

1 ρ

U

d W f

which allows us to write

dV

= ( µ

+ v r σ

γ

) V

dt + σ

V

d W

+

Z _t 0

γ

ds dU

= ^µ

+ µ

+ σ

σ

ρ r

( σ

ρ + σ

) γ

σ

p

1 ρ

e γ

U

dt + ( σ

ρ + σ

) U

d W

+

Z_t 0

γ

ds + σ

q

1 ρ

U

d W f

+

0

e γ

ds .

Change of measure Radon- Nikodym th.

Girsanov th.

Multidimensional Example 2 Girsanov Example 2 (continued) Example 3 References

Example 2 XI

So, we want to solve the linear system

µ

+ v r σ

γ

= 0 µ

+ µ

+ σ

σ

ρ r ( σ

ρ + σ

) γ

σ

q

1 ρ

γ e

= 0

the unknowns of which are γ

and γ e

. In matrix form, we write

σ

0

σ

ρ + σ

σ

p 1 ρ

γ

e

γ

= ^µ

+ v r

µ

+ µ

+ σ

σ

ρ r .

The solution is

γ

= ^µ

+ v r σ

e

γ

= ^µ

^v + σ

σ

ρ σ

p 1 ρ

ρ ( µ

+ v r ) σ

p 1 ρ

Change of measure Radon- Nikodym th.

Girsanov th.

Multidimensional Example 2 Girsanov Example 2 (continued) Example 3 References

Example 2 XII

So, let’s set W

= W

+ R

0

γ

ds and W f

= ^f W

+ R

0

γ e

ds where

γ

= ^µ

+ v r σ

e

γ

= ^µ

^v + σ

σ

ρ σ

p 1 ρ

ρ ( µ

+ v r ) σ

p 1 ρ

.

We can then write dV

= σ

V

dW

dU

= ( σ

ρ + σ

) U

dW

+ σ

q

1 ρ

U

d W f

.

Is it possible to …nd a measure Q such that W

and W f

are ( fF

g ^, Q ) Brownian motions simultaneously?

Change of measure Radon- Nikodym th.

Girsanov th.

Multidimensional Example 2 Girsanov Example 2 (continued) Example 3 References

Girsanav theorem I

Let W = W

, ..., W

be a Brownian motion with dimension n, i.e. its components are independent standard Brownian motions on the …ltered probability space

( Ω , F ^, fF

g ^{, P} ) Theorem

Cameron-Martin-Girsanov theorem. For all i 2 f ^{1, ..., n} g ^, γ

= γ

: 0 t T is a fF

g predictable process such that

E

exp 1 2

Z

0

γ

2

dt < ∞ .

There exists a measure Q on ( _Ω , F ) such that

(CMG1) Q is equivalent to P

Change of measure Radon- Nikodym th.

Girsanov th.

Multidimensional Example 2 Girsanov Example 2 (continued) Example 3 References

Girsanav theorem II

(CMG2)

= exp ∑

R

0

γ

dW

R

0

∑

γ

2

dt (CMG3) For all i 2 f ^{1, ..., n} g ^, the process

f

W

= W f

: 0 t T de…ned as f

W

= W

+ R

0

γ

ds is a ( fF

g ^, Q ) Brownian motion.

(ref. Baxter and Rennie, page 186)

Change of measure Radon- Nikodym th.

Girsanov th.

Multidimensional Example 2 Girsanov Example 2 (continued) Example 3 References

Example 2 (continued) I

Since the functions γ and γ e are constant, the Novikov condition is satis…ed and Girsanov theorem

(multidimensional version) allows to conclude there exists

a martingale measure Q such that W

and W f

are

( fF

g ^{, Q} ) Brownian motions.

Change of measure Radon- Nikodym th.

Girsanov th.

Multidimensional Example 2 Girsanov Example 2 (continued) Example 3 References

Example 2 (continued) II

An interesting fact, on the space ( Ω , F ^, fF

g ^{, Q} ) , we have the stochastic di¤erential equation satis…ed by the instantaneous exchange rate

dC

= µ

C

dt + σ

C

dW

= µ

C

dt + σ

C

d W

µ

+ v r σ

t

= ( r v ) C

dt + σ

C

dW

.

The di¤erence between the domestic and foreign

instantaneous interest rates can be recognized in the drift

coe¢ cient.

Change of measure Radon- Nikodym th.

Girsanov th.

Multidimensional Example 2 Girsanov Example 2 (continued) Example 3 References

Example 2 (continued) III

Again on the space ( Ω , F ^, fF

g ^{, Q} ) , the stochastic di¤erential equation for the evolution of the risky asset Canadian dollar price is

dCtSt

= (µ_S+µ_C+σSσCρ)CtSt dt + (σSρ+σC)CtSt dW_t^P+σS

q

1 ρ²CtSt dWf_t^P

= (µ_S+µ_C+σSσCρ)CtSt dt

+ (σSρ+σC)CtSt d W_t^Q µ_C+v r σC

t

+σS

q

1 ρ²CtSt d Wf_t^Q µ_S v+_σSσCρ σS

p1 ρ²

ρ(µ_C+v r) σC

p1 ρ²

! t

!

= rCtSt dt+ (σSρ+σC)CtSt dW_t^Q+σS

q

1 ρ²CtSt dWf_t^Q

Change of measure Radon- Nikodym th.

Girsanov th.

Multidimensional Example 2 Girsanov Example 2 (continued) Example 3 References

Example 2 (continued) IV

where the last equality is obtained by simplifying the drift coe¢ cient

( µ

+ µ

+ σ

σ

ρ ) ( σ

ρ + σ

) ^µ

+ v r σ

σ

q

1 ρ

µ

v + σ

σ

ρ σ

p 1 ρ

ρ ( µ

+ v r ) σ

p 1 ρ

!

.

Change of measure Radon- Nikodym th.

Girsanov th.

Multidimensional Example 2 Girsanov Example 2 (continued) Example 3 References

Example 2 (continued) V

Again under the risk-neutral measure Q , the Canadian dollar value of one foreign currency invested in a foreign bank account satis…es

dC

B

= ( v + µ

) C

B

dt + σ

C

B

dW

= ( v + µ

) C

B

dt + σ

C

B

d W

µ

+ v r σ

t

= v + µ

σ

µ

+ v r

σ

C

B

dt + σ

C

B

dW

= rC

B

dt + σ

C

B

dW

.