Change of measure and Girsanov theoremneumann.hec.ca/~p240/c80646en/12Girsanov_EN.pdf · Change of measure Radon-Nikodym th. Girsanov th. Example 1 Multidimensional References Girsanov

Girsanov

Change ofmeasure

Radon-Nikodymth.

Girsanov th.

Multidimensional

References

Change of measure and Girsanov theorem80-646-08

Stochastic calculus I

Geneviève Gauthier

HEC Montréal

Girsanov

Change ofmeasureExample 1

Radon-Nikodymth.

Girsanov th.

Multidimensional

References

An example I

Let (Ω,F , fFt : 0 t Tg ,P) be a ltered probabilityspace on whicha standard Brownian motion WP =

WPt : 0 t T

is

constructed.

The stochastic process S = fSt : 0 t Tg representsthe evolution of a risky security price and satises thestochastic di¤erential equation

dSt = µSt dt + σSt dWPt .

Lets also assume that the interest rate r is constant. Thediscount factor is therefore

β (t) = exp (rt)

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

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An example IILets set, for all 0 t T ,

Yt = βtSt

i.e. Yt represents the present value at time t of the riskysecurity.

Using Itôs lemma (more precisely the multiplication rule),we obtain

dYt = (µ r)Yt dt + σYt dWPt .

Indeed,

dYt = dβtSt= βt dSt + St dβt + d hβ,Sit= βt

µSt dt + σSt dWP

t

+ St (rβt dt)

= (µ r) βtSt dt + σβtSt dWPt .

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An example III

In its integral form, such a stochastic di¤erential equationbecomes

Yt = Y0 + (µ r)Z t

0Ys ds + σ

Z t

0Ys dWP

s .

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RefresherItô process

Let WP be a (fFtg ,P)Brownian motion.An Itô process is a process X = fXt : 0 t Tg takingits values in R such that:

Xt X0 +Z t

0Ks ds +

Z t

0Hs dWP

s

with K = fKt : 0 t Tg and H = fHt : 0 t Tg,processes adapted to the ltration fFtg,PhR T0 jKs j ds < ∞

i= 1

PhR T0 (Hs )

2 ds < ∞i= 1

Damien Lamberton and Bernard Lapeyre, Introduction au calculstochastique appliqué à la nance, Ellipses, page 53.

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Example (suite) I

Recall that WP is a (fFtg ,P)Brownian motion.

In a risk-neutral world (Ω,F , fFt : t 0g ,Q), thestochastic process Y = fYt : 0 t Tg should be a(fFtg ,Q)martingale.Thus, under the risk-neutral measure, the trend of Yshould be nil, i.e. we want the drift coe¢ cient to be 0.

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Example (suite) IILets set

WQt = W

Pt +

Z t

0γsds

and note that

1 WQ is not a Pmartingale (its expectation varies in time)and

2 dWQt = dW

Pt + γtdt. As a consequence

Yt = Y0 + (µ r)Z t0Ys ds + σ

Z t0Ys dWP

s

Yt = Y0 +Z t0(µ r σγs )Ys ds + σ

Z t0Ys dWQ

s .

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

µ r σγs = 0, γs =µ r

σ.

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Example (suite) III

Recall that

Yt = Y0 + σZ t

0Ys dWQ

s

Note that, under the measure P, the process WQ is not astandard Brownian motion since the law of WQ

t under the

measure P is N

µrσ t, t

.

The process Y will not be a (fFtg ,P)martingale sincethe stochastic integral is constructed with respect to WQ

which is not a (fFtg ,P)martingale.Indeed,

EPhWQt

i=

µ rσ

t

varies in time.

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Example (suite) IV

Recall that WP is a (fFtg ,P)Brownian motion,

Yt = Y0 + σZ t

0Ys dWQ

s

whereWQ (t) = WP (t) +

µ rσ

t.

So we want to nd the probability measure Q to be placedon the space (Ω,F , fFtg) such that WQ is aQstandard Brownian motion.By changing the probability on the set Ω, we transformthe drift coe¢ cient so that the trend becomes zero and weintegrate with respect to a (fFtg ,Q)martingale. As aresult, the process Y will be (fFtg ,Q)martingale.

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Radon-Nikodym theorem IA way to construct new probability measures on themeasurable space (Ω,F ) when we already have aprobability measure P existing on that space is as follows:Let Y be a random variable constructed on the probabilityspace (Ω,F ,P) such that

8ω 2 Ω, Y (ω) 0 and EP [Y ] = 1.

For all event A 2 F , δA denotes the indicator function ofthat event:

δA (ω) =

1 if ω 2 A0 otherwise.

For all event A 2 F , lets set

Q (A) = EP [Y δA ] .

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

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Radon-Nikodym theorem II

Proof. We must verify that

(P1) Q (Ω) = 1,

(P2) 8A 2 F , 0 Q (A) 1,(P3) 8A1, A2, ...2 F such that Ai \ Aj = ∅ si i 6= j ,

QS

i1 Ai= ∑i1 Q (Ai ) .

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Radon-Nikodym theorem III

Verication of (P1). But, since for all ω, δΩ (ω) = 1and because we have assumed that EP [Y ] = 1,

Q (Ω) = EP [Y δΩ] = EP [Y ] = 1,

which establishes condition (P1).

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Radon-Nikodym theorem IV

Verication of (P2). The second condition is just as easyto prove: since Y is a positive random variable, Y δA is apositive random variable too, and Q (A) = EP [Y δA ] 0.Moreover,

Q (A) = EP [Y δA ]

EP [Y δΩ]

= EP [Y ]

= 1.

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Radon-Nikodym theorem VVerication of (P3). As we have established in anexercise in the rst chapter, 8A1, A2, ...2 F such thatAi \ Aj = ∅ if i 6= j ,

δSi1 Ai = ∑

i1δAi .

As a consequence,

Q

[i1Ai

!= EP

hY δS

i1 Ai

i= EP

"Y ∑i1

δAi

#= ∑

i1EP [Y δAi ]

= ∑i1

Q (Ai ) .

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Radon-Nikodym theorem VI

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

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

Girsanov

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Radon-Nikodym theorem VII

Question. Given two equivalent probability measures P

and Q, does there exist a non-negative valued randomvariable Y such that

Q (A) = EP [Y δA ] ?

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

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

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

The existence of such a variable is established in the nexttheorem which is a version of the famous Radon-Nikodymtheorem.

Girsanov

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Radon-Nikodym theorem VIII

TheoremRadon-Nikodym theorem. Given two equivalent probabilitymeasures P and Q constructed on the measurable space(Ω,F ), there exists a positive-valued random variable Y suchthat

Q (A) = EP [Y δA ] .

Such a random variable Y is often denoted by dQdP.

Such a theorem still does not tell us how to nd ourrisk-neutral measure. Actually, it is the next result thatwill provide us with the recipe to construct our measureand it involves the Radon-Nikodym derivative.

Girsanov

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Radon-Nikodym theorem IX

A few thoughts about the discrete case

Assume that Ω only contains a nite number of elements.Let Y = βTX be the present value of the attainablecontingent claim X . Si F0 = fΩ,∅g, then its price attime t = 0 is

EQ [Y ] = ∑ω2Ω

Y (ω)Q (ω)

= ∑ω2Ω

Y (ω)Q (ω)

P (ω)P (ω)

= EP

Y

Q

P

Girsanov

Change ofmeasure

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References

Radon-Nikodym theorem X

Consider the binomial market model: S (1) represents theevolution of the riskless asset and S (2) models a riskyasset. The unique risk-neutral measure is denoted by Q, Pbeing the realmeasure.

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. 1

4 0.085 0, 34

The Radon-Nykodym derivative is somewhat the memoryof the change of measure. For each path, it remembershow we have changed weights.

Girsanov

Change ofmeasure

Radon-Nikodymth.

Girsanov th.Example 1

Multidimensional

References

Girsanov theorem I

Lets focus on a bounded time interval: t 2 [0,T ].Let W = fWt : t 2 [0,T ]g represent a Brownian motionconstructed on a ltered probability space(Ω,F , fFtg ,P) such that the ltration fFtg is the onegenerated by the Brownian motion, plus it includes allzero-probability events, i.e. for all t 0,

Ft = σ (N and Ws : 0 s t) .

The next theorem will enable to construct our risk- neutralmeasures.

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Radon-Nikodymth.


Multidimensional

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Girsanov theorem II

TheoremCameron-Martin-Girsanov theorem. Letγ = fγt : t 2 [0,T ]g be a fFtgpredictable process suchthat

EP

exp

12

Z T

0γ2t dt

< ∞.

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

(CMG1) Q is equivalent to P

(CMG2) dQdP= exp

hR T0 γt dWt 1

2

R T0 γ2t dt

i(CMG3) The process fW =

nfWt : t 2 [0,T ]odened asfWt = Wt +

R t0 γs ds is a (fFtg ,Q)Brownian motion.

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

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Change ofmeasure

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Multidimensional

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Girsanov theorem III

The condition EPhexp

12

R T0 γ2t dt

i< ∞ is a su¢ cient

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

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Radon-Nikodymth.


Multidimensional

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Girsanov theorem IV

Consider the stochastic di¤erential equation

dXt = b (Xt , t) dt + a (Xt , t) dWt

where W represents a Brownian motion on the lteredprobability space (Ω,F , fFtg ,P).We assume that the drift and di¤usion coe¢ cients aresuch that there exists a unique solution to the equation,which we denote X .

We want to nd a probability measure Q, such that, onthe space (Ω,F , fFtg ,Q), the drift of X is eb (Xt , t)instead of b (Xt , t) .

Girsanov

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Girsanov theorem V

Lets go!

dXt = b (Xt , t) dt + a (Xt , t) dWt

= eb (Xt , t) dt + a (Xt , t) b (Xt , t) eb (Xt , t)a (Xt , t)

!dt

+a (Xt , t) dWt

provided that a (Xt , t) is di¤erent from 0.

= eb (Xt , t) dt + a (Xt , t) d Wt +Z t

0

b (Xs , s) eb (Xs , s)a (Xs , s)

ds

!= eb (Xt , t) dt + a (Xt , t) dfWt

where

fWt = Wt +Z t

0γs ds and γt =

b (Xt , t) eb (Xt , t)a (Xt , t)

.

Girsanov

Change ofmeasure

Radon-Nikodymth.


Multidimensional

References

Girsanov theorem VI

If EPhexp

12

R T0 γ2t dt

i< ∞ then by the

Radon-Nikodym and Cameron-Martin-Girsanov theorems,

Q (A) = EP

exp

Z T

0γt dWt

12

Z T

0γ2t dt

δA

, A 2 F

and fW =nfWt : t 2 [0,T ]

ois a (F,Q)Brownian

motion.

In practice, we dont need to determine the measure Q. Itis su¢ cient for us to know it exists, and to know thestochastic di¤erential equation of the process of intereston the space (Ω,F , fFtg ,Q) .

Girsanov

Change ofmeasure

Radon-Nikodymth.


Multidimensional

References

Example 1 IGirsanav theorem

Lets go back to the Black-Scholes market model. Thestochastic process Y = fYt : 0 t Tg constructed onthe space (Ω,F , fFtg ,P) used to construct theBrownian motion represents the evolution of the presentvalue of a risky security where

dYt = (µ r)Yt dt + σYt dWPt .

Girsanov

Change ofmeasure

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References

Example 1 IIGirsanav theorem

But, in a risk-neutral world (Ω,F , fFtg ,Q), the trend ofY should be zero, i.e. we want the drift coe¢ cient to bezero. Thus

dYt = (µ r)Yt dt + σYt dWPt

= σYtµ r

σdt + σYt dWP

t

= σYt dWPt +

µ rσ

t= σYt dW

Qt

where

WQt WP

t +µ r

σt = WP

t +Z t

0

µ rσ

ds.

Girsanov

Change ofmeasure

Radon-Nikodymth.


Multidimensional

References

Example 1 IIIGirsanav theorem

In the present case,

8s, γs =µ r

σ.

Girsanov

Change ofmeasure

Radon-Nikodymth.


Multidimensional

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Example 1 IVGirsanav theorem

Recall the Cameron-Martin-Girsanov theorem. Letγ = fγt : t 2 [0,T ]g be a fFtgpredictable process suchthat

EP

exp

12

Z T

0γ2t dt

< ∞.



(CMG2) dQdP= exp

hR T0 γt dWt 1

2

R T0 γ2t dt

i(CMG3) The process fW =

nfWt : t 2 [0,T ]odened asfWt = Wt +

R t0 γs ds is a (fFtg ,Q)Brownian motion.

Girsanov

Change ofmeasure

Radon-Nikodymth.


Multidimensional

References

Example 1 VGirsanav theorem

Lets verify that the condition on the process γ is indeedsatised:

EP

exp

12

Z T

0γ2t dt

= EP

"exp

12

Z T

0

µ r

σ

2dt

!#

= exp

12

µ r

σ

2T

!< ∞.

Girsanov

Change ofmeasure

Radon-Nikodymth.


Multidimensional

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Example 1 VIGirsanav theorem

Lets apply Girsanov theorem :

dQ

dP= exp

Z T

0γt dW

Pt

12

Z T

0γ2t dt

= exp

"Z T

0

µ rσ

dWPt

12

Z T

0

µ r

σ

2dt

#

= exp

"µ r

σWPT

12

µ r

σ

2T

#.

This implies that

Q [A] = EP

"exp

µ r

σWPT

12

µ r

σ

2T

!δA

#, A 2 F .

Girsanov

Change ofmeasure

Radon-Nikodymth.


Multidimensional

References

Example 1 VIIGirsanav theorem

Moreover, under the measure Q, the evolution of thepresent value of the risky security satises the equation

dYt = σYt dWQt

where WQ is a QBrownian motion.We can also deduce the stochastic di¤erential equationsatised by the evolution of the risky security price S :

dSt = µSt dt + σSt dWPt

= µSt dt + σSt dWQt

µ rσ

t

puisque WQt WP

t +µ r

σt

= µSt dt + σSt dWQt σSt

µ rσ

dt

= rSt dt + σSt dWQt .

Girsanov

Change ofmeasure

Radon-Nikodymth.


Multidimensional

References

Example 1 VIIIGirsanav theorem

Girsanov

Change ofmeasure

Radon-Nikodymth.


Multidimensional

References

Example 1 IXGirsanav theorem

Note that we dont really need to calculate Q, we simplyneed to know that it exists then toestablish what is theequation satised by the processus of interest, i.e. theevolution of the risky security price. Indeed, on(Ω,F , fFtg ,Q),

dSt = rSt dt + σSt dWQt

where fW is a QBrownian motion.But the unique solution to that equation is

St = S0 expr σ2

2

t + σWQ

t

.

Girsanov

Change ofmeasure

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Multidimensional

References

Example 1 XGirsanav theorem

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

EQ [exp (rT )max (ST K ; 0)]

= EQ

exp (rT )max

S0 exp

r σ2

2

T + σW Q

T

K ; 0

= EQ

max

S0 exp

σ2

2T + σW Q

T

K exp (rT ) ; 0

=

Z ∞

∞max

S0 exp

σ2

2T + σz

KerT ; 0

fZ (z) dz .

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

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Example 1 XIGirsanav theorem

Since

S0 expσ2

2T + σz

> KerT

, σ2

2T + σz > ln

KerT

S0= lnK rT lnS0

, z > lnS0 lnK +

r σ2

2

T

σ d2

pT

Girsanov

Change ofmeasure

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Multidimensional

References

Example 1 XIIGirsanav theorem

Z ∞

∞max

S0 exp

σ2

2T + σz

KerT ; 0

fZ (z) dz

=Z ∞

d2pT

S0 exp

σ2

2T + σz

KerT

fZ (z) dz

=Z ∞

d2pTS0 exp

σ2

2T + σz

fZ (z) dz

Z ∞

d2pTKerT fZ (z) dz

= S0Z ∞

d2pT

1p2π

1pTexp

z

2 2Tσz + σ2T 2

2T

dz

KerTZ ∞

d2pT

1p2π

1pTexp

z

2

2T

dz

Girsanov

Change ofmeasure

Radon-Nikodymth.


Multidimensional

References

Example 1 XIIIGirsanav theorem

= S0Z ∞

d2pT

1p2π

1pTexp

(z σT )2

2T

!dz

KerTZ ∞

d2pT

1p2π

1pTexp

z

2

2T

dz

Lets set u =z σTp

Tand v =

zpT

= S0Z ∞

d2σpT

1p2π

expu

2

2

du

KerTZ ∞

d2

1p2π

expv

2

2

dv

= S01N

d2 σ

pTKerT (1N (d2))

Girsanov

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Radon-Nikodymth.


Multidimensional

References

Example 1 XIVGirsanav theorem

where N () is the cumulative distribution function of astandard normal random variable.

Girsanov

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Radon-Nikodymth.


Multidimensional

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Example 1 XVGirsanav theorem

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

S01N

d2 σ

pTKerT (1N (d2))

= S0Nd2 + σ

pTKerT (N (d2))

Girsanov

Change ofmeasure

Radon-Nikodymth.

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MultidimensionalExample 2GirsanovExample 2(continued)Example 3

References

Example 2 I

Lets assume that WP and fWP represent two standardBrownian motions constructed on the ltered probabilityspace (Ω,F , fFtg ,P).Note that

neBPt : t 0

owhere

eBt ρWPt +

q1 ρ2fWP

t

is a standard Brownian motion such that

CorrPWPt , eBP

t

= ρ.

Exercise: prove it.

Girsanov

Change ofmeasure

Radon-Nikodymth.

Girsanov th.


References

Example 2 II

The instantaneous exchange rate

dCt = µCCt dt + σCCt dWPt

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

Suppose also that the stochastic di¤erential equation

dSt = µSSt dt + σSSt d eBPt

= µSSt dt + σSρSt dWPt + σS

q1 ρ2St dfWP

t

models the evolution of a foreign risky asset price.

Girsanov

Change ofmeasure

Radon-Nikodymth.

Girsanov th.


References

Example 2 III

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

βt = exp (rt) .

and the value in foreign currency of an initial investmentequal to one foreign currency unit is Bt = exp (vt) .

Girsanov

Change ofmeasure

Radon-Nikodymth.

Girsanov th.


References

Example 2 IV

Lets put ourselves in the shoes of a Canadian investor.

CtSt gives us the Canadian dollar value of a risky asse attime t,CtBt gives us the Canadian dollar value, at time t, of oneforeign currency unit invested in a foreign bank accountUt = βtCtSt gives us the Canadian dollar present value ofthe risky asset at time t.Vt = βtCtBt gives us the Canadian dollar present value,at time t, of one foreign currency unit invested in a foreignbank account.

We wish to nd a measure Q, such that the stochasticprocesses U and V are (fFtg ,Q)martingales.

Girsanov

Change ofmeasure

Radon-Nikodymth.

Girsanov th.


References

Example 2 VRecall:

dCt = µCCt dt + σCCt dWPt ,

dBt = vBt dt

dβt = rβt dt

First, lets determine the stochastic di¤erential equationsatised by the Canadian dollar present value of oneforeign currency unit invested in a foreign bank accountV = βCB under the measure P. Itôs lemma allows us towrite

dCtBt = Ct dBt + Bt dCt + d hB,C it= Ct (vBt dt) + Bt

µCCt dt + σCCt dW

Pt

= (v + µC )CtBt dt + σCCtBt dW

Pt .

Girsanov

Change ofmeasure

Radon-Nikodymth.

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References

Example 2 VI

Thus,

dVt = dβtCtBt= βt dCtBt + CtBt dβt + d hβ,CBit= βt

(v + µC )CtBt dt + σCCtBt dW

Pt

+CtBt (rβt dt)

= (µC + v r) βtCtBt dt + σC βtCtBt dWPt

= (µC + v r)Vt dt + σCVt dWPt .

Girsanov

Change ofmeasure

Radon-Nikodymth.

Girsanov th.


References

Example 2 VII

Recall:

dSt = µSSt dt + σSρSt dWPt + σS

q1 ρ2St dfWP

t ,

dCt = µCCt dt + σCCt dWPt ,

dBt = vBt dt,

dβt = rβt dt.

Girsanov

Change ofmeasure

Radon-Nikodymth.

Girsanov th.


References

Example 2 VIII

Second, lets determine the stochastic di¤erential equationsatised by U = βCS under the measure P. Itôs lemmaallows us to write

dCtSt= Ct dSt + St dCt + d hS ,C it= Ct

µSSt dt + σSρSt dWP

t + σS

q1 ρ2St dfWP

t

+St

µCCt dt + σCCt dW

Pt

+ σSρStσCCt dt

= (µS + µC + σSσC ρ)CtSt dt

+ (σSρ+ σC )CtSt dWPt + σ

q1 ρ2CtSt dfWP

t .

Girsanov

Change ofmeasure

Radon-Nikodymth.

Girsanov th.


References

Example 2 IX

Applying Itôs lemma again, we obtain

dUt = dβtCtSt= βt dCtSt + CtSt dβt + d hβ,CSit

= βt

(µS + µC + σSσC ρ)CtSt dt

+ (σS ρ+ σC )CtSt dW Pt + σS

p1 ρ2CtSt dfW P

t

+CtSt (rβt dt)

= (µS + µC + σSσC ρ r ) βtCtSt dt

+ (σS ρ+ σC ) βtCtSt dWPt + σS

q1 ρ2βtCtSt dfW P

t

= (µS + µC + σSσC ρ r )Ut dt

+ (σS ρ+ σC )Ut dWPt + σS

q1 ρ2Ut dfW P

t .

Girsanov

Change ofmeasure

Radon-Nikodymth.

Girsanov th.


References

Example 2 XWe have

dVt = (µC + v r)Vt dt + σCVt dWPt ,

dUt = (µS + µC + σSσC ρ r)Ut dt

+ (σSρ+ σC )Ut dWPt + σS

q1 ρ2Ut dfWP

t

which allows us to write

dVt = (µC + v r σC γt )Vt dt

+σCVt dW Pt +

Z t

0γsds

dUt =

µS + µC + σSσC ρ r

(σS ρ+ σC ) γt σSp1 ρ2eγt

Ut dt

+ (σS ρ+ σC )Ut dW Pt +

Z t

0γsds

+σS

q1 ρ2Ut d

fW Pt +

Z t

0eγsds .

Girsanov

Change ofmeasure

Radon-Nikodymth.

Girsanov th.


References

Example 2 XI

So, we want to solve the linear system

µC + v r σC γt = 0

µS + µC + σSσC ρ r (σS ρ+ σC ) γt σS

q1 ρ2eγt = 0

the unknowns of which are γt and eγt . In matrix form, wewrite

σC 0σS ρ+ σC σS

p1 ρ2

γteγt

=

µC + v r

µS + µC + σSσC ρ r

.

The solution is

γt =µC + v r

σCeγt =µS v + σSσC ρ

σSp1 ρ2

ρ (µC + v r)σCp1 ρ2

Girsanov

Change ofmeasure

Radon-Nikodymth.

Girsanov th.


References

Example 2 XII

So, lets set WQt = W

Pt +

R t0 γs ds andfWQ

t = fWPt +

R t0 eγs ds where

γs =µC + v r

σC

eγs =µS v + σSσC ρ

σSp1 ρ2

ρ (µC + v r )σCp1 ρ2

.

We can then write

dVt = σCVt dWQt

dUt = (σSρ+ σC )Ut dWQt + σS

q1 ρ2Ut dfWQ

t .

Is it possible to nd a measure Q such that WQ and fWQ

are (fFtg ,Q)Brownian motions simultaneously?

Girsanov

Change ofmeasure

Radon-Nikodymth.

Girsanov th.


References

Girsanav theorem I

Let W =W (1), ...,W (n)

be a Brownian motion with

dimension n, i.e. its components are independent standardBrownian motions on the ltered probability space(Ω,F , fFtg ,P)

TheoremCameron-Martin-Girsanov theorem. For all i 2 f1, ..., ng ,γ(i ) =

γ(i )t : 0 t T

is a fFtgpredictable process such

that

EP

exp

12

Z T

0

γ(i )t

2dt

< ∞.



Girsanov

Change ofmeasure

Radon-Nikodymth.

Girsanov th.


References

Girsanav theorem II

(CMG2) dQdP =

exp∑n

i=1

R T0 γ

(i )t dW (i )

t 12

R T0 ∑n

i=1

γ(i )t

2dt

(CMG3) For all i 2 f1, ..., ng , the processfW (i ) =fW (i )

t : 0 t Tdened asfW (i )

t = W (i )t +

R t0 γ

(i )s ds is a (fFtg ,Q)Brownian

motion.

(ref. Baxter and Rennie, page 186)

Girsanov

Change ofmeasure

Radon-Nikodymth.

Girsanov th.


References

Example 2 (continued) I

Since the functions γ and eγ are constant, the Novikovcondition is satised and Girsanov theorem(multidimensional version) allows to conclude there existsa martingale measure Q such that WQ and fWQ are(fFtg ,Q)Brownian motions.

Girsanov

Change ofmeasure

Radon-Nikodymth.

Girsanov th.


References

Example 2 (continued) II

An interesting fact, on the space (Ω,F , fFtg ,Q), wehave the stochastic di¤erential equation satised by theinstantaneous exchange rate


= µCCt dt + σCCt dWQt

µC + v rσC

t

= (r v)Ct dt + σCCt dWQt .

The di¤erence between the domestic and foreigninstantaneous interest rates can be recognized in the driftcoe¢ cient.

Girsanov

Change ofmeasure

Radon-Nikodymth.

Girsanov th.


References

Example 2 (continued) III

Again on the space (Ω,F , fFtg ,Q), the stochasticdi¤erential equation for the evolution of the risky assetCanadian dollar price is

dCtSt= (µS + µC + σSσC ρ)CtSt dt

+ (σS ρ+ σC )CtSt dWPt + σS

q1 ρ2CtSt dfW P

t

= (µS + µC + σSσC ρ)CtSt dt

+ (σS ρ+ σC )CtSt dW Qt

µC + v rσC

t

+σS

q1 ρ2CtSt d

fW Qt

µS v + σSσC ρ

σSp1 ρ2


!t

!

= rCtSt dt + (σS ρ+ σC )CtSt dWQt + σS

q1 ρ2CtSt dfW Q

t

Girsanov

Change ofmeasure

Radon-Nikodymth.

Girsanov th.


References

Example 2 (continued) IV

where the last equality is obtained by simplifying the driftcoe¢ cient

(µS + µC + σSσC ρ) (σSρ+ σC )µC + v r

σC

σS

q1 ρ2

µS v + σSσC ρ

σSp1 ρ2

ρ (µC + v r)σCp1 ρ2

!.

Girsanov

Change ofmeasure

Radon-Nikodymth.

Girsanov th.


References

Example 2 (continued) V

Again under the risk-neutral measure Q, the Canadiandollar value of one foreign currency invested in a foreignbank account satises

dCtBt= (v + µC )CtBt dt + σCCtBt dW

Pt

= (v + µC )CtBt dt + σCCtBt dW Qt

µC + v rσC

t

=

v + µC σC

µC + v rσC

CtBt dt + σCCtBt dW

Qt

= rCtBt dt + σCCtBt dWQt .

Girsanov

Change ofmeasure

Radon-Nikodymth.

Girsanov th.


References

Example 2 (continued) VI

To summarize,


dCt = (r v )Ct dt + σCCt dWQt

dCtSt = (µS + µC + σSσC ρ)CtSt dt

+ (σS ρ+ σC )CtSt dWPt + σS

q1 ρ2CtSt dfW P

t

dCtSt = rCtSt dt

+ (σS ρ+ σC )CtSt dWQt + σS

q1 ρ2CtSt dfW Q

t

dCtBt = (v + µC )CtBt dt + σCCtBt dWPt

dCtBt = rCtBt dt + σCCtBt dWQt

Girsanov

Change ofmeasure

Radon-Nikodymth.

Girsanov th.


References

Example 2 (continued) VII

For the foreign currency securities, we have the equation,under probability Q, which characterizes the evolution ofthe risky asset foreign currency price:

dSt

= µSSt dt + σS ρSt dW Pt + σS

q1 ρ2St dfW P

t

= dSt = µSSt dt + σS ρSt dW Qt

µC + v rσC

t

+σS

q1 ρ2St d

fW Qt

µS v + σSσC ρ

σSp1 ρ2


!t

!

= (v σSσC ρ) St dt + σS ρSt dWQt + σS

q1 ρ2St dfW Q

t

and the equation of the evolution of a foreign currencybank account

dBt = vBt dt.

Girsanov

Change ofmeasure

Radon-Nikodymth.

Girsanov th.


References

Example 3 INon-uniqueness of the martingale measure

Let WP and fWP be two independent standard Brownianmotions constructed on the ltered probability space(Ω,F , fFtg ,P).Assume the risky asset price evolves according to the SDE

dSt = µSt dt + σSt dWPt + σSt dfWP

t

and that the instantaneous interest rate r is constant.

Girsanov

Change ofmeasure

Radon-Nikodymth.

Girsanov th.


References

Example 3 IINon-uniqueness of the martingale measure

Lets set, for all 0 t T ,

Yt = βtSt

i.e. Yt represents the present value at time t of the riskysecurity. Using Itôs lemma (more precisely themultiplication rule), we obtain

dYt = (µ r)Yt dt + σYt dWPt + σYt dfWP

t .

Indeed,

dYt= dβtSt= βt dSt + St dβt + d hβ,Sit= βt

µSt dt + σSt dWP

t + σSt dfWPt

+ St (rβt dt)

= (µ r) βtSt dt + σβtSt dWPt + σβtSt dfWP

t .

Girsanov

Change ofmeasure

Radon-Nikodymth.

Girsanov th.


References

Example 3 IIINon-uniqueness of the martingale measure

dYt= (µ r)Yt dt + σYt dWP

t + σYt dfWPt

= (µ r σγt σeγt )Yt dt+σYt d

WPt +

Z t

0γsds

+ σYt d

fWPt +

Z t

0eγsds .

Lets force the drift coe¢ cient to cancel out:

µ r σγt σeγt = 0, eγt = µ rσ

γt .

So there is an innity of solutions.

Girsanov

Change ofmeasure

Radon-Nikodymth.

Girsanov th.


References

Example 3 IVNon-uniqueness of the martingale measure

Recall: eγt = µ rσ

γt .

If we decide that the process fγt : 0 t Tg does notdepend on time, then the same will be true for eγ. Since γand eγ are deterministic and constant, the Novikovcondition is satised. As a consequence, for all γ 2 R,there exists a martingale measure Qγ such that

W γt = WP

t + γt

and fW γt = fWP

t + eγt = fWPt +

µ r

σ γ

t

are (fFtg ,Qγ)Brownian motions.

Girsanov

Change ofmeasure

Radon-Nikodymth.

Girsanov th.


References

Example 3 VNon-uniqueness of the martingale measure

For all γ 2 R, the process Y ,

dYt = σYt dWγt + σYt dfW γ

t

is a (fFtg ,Qγ)martingale.

Girsanov

Change ofmeasure

Radon-Nikodymth.

Girsanov th.


References

Example 3 VINon-uniqueness of the martingale measure

Consequences

The market is incompleteSome contingent claims cannot be replicated. In such acase, the expectation, under a risk-neutral measure, of thepresent value of the contingent claim will give us A price,but not THE price.How to determine whether a contingent claim isattainable? The answer can be found in the next series ofslides!

Girsanov

Change ofmeasure

Radon-Nikodymth.

Girsanov th.

Multidimensional

References

References

Martin Baxter and Andrew Rennie (1996). FinancialCalculus, an introduction to derivative pricing, Cambridgeuniversity press.

Christophe Bisière (1997). La structure par terme des tauxdintérêt, Presses universitaires de France.

Damien Lamberton and Bernard Lapeyre (1991).Introduction au calcul stochastique appliqué à la nance,Ellipses.

Change of measure and Girsanov theoremneumann.hec.ca/~p240/c80646en/12Girsanov_EN.pdf · Change of measure Radon-Nikodym th. Girsanov th. Example 1 Multidimensional References Girsanov

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