Change of measure and Girsanov theoremneumann.hec.ca/~p240/c80646en/12Girsanov_EN.pdf · Change of...
Transcript of Change of measure and Girsanov theoremneumann.hec.ca/~p240/c80646en/12Girsanov_EN.pdf · Change of...
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.
Girsanov
Change ofmeasureExample 1
Radon-Nikodymth.
Girsanov th.
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References
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 .
Girsanov
Change ofmeasureExample 1
Radon-Nikodymth.
Girsanov th.
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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 .
Girsanov
Change ofmeasureExample 1
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Girsanov th.
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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.
Girsanov
Change ofmeasureExample 1
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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.
Girsanov
Change ofmeasureExample 1
Radon-Nikodymth.
Girsanov th.
Multidimensional
References
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.
Girsanov
Change ofmeasureExample 1
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Girsanov th.
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References
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.
Girsanov
Change ofmeasureExample 1
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Girsanov th.
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References
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.
Girsanov
Change ofmeasure
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References
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 ).
Girsanov
Change ofmeasure
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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 ) .
Girsanov
Change ofmeasure
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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).
Girsanov
Change ofmeasure
Radon-Nikodymth.
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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.
Girsanov
Change ofmeasure
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References
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 ) .
Girsanov
Change ofmeasure
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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
Change ofmeasure
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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
Change ofmeasure
Radon-Nikodymth.
Girsanov th.
Multidimensional
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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
Change ofmeasure
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Girsanov th.
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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
Radon-Nikodymth.
Girsanov th.
Multidimensional
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.
Girsanov
Change ofmeasure
Radon-Nikodymth.
Girsanov th.Example 1
Multidimensional
References
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)
Girsanov
Change ofmeasure
Radon-Nikodymth.
Girsanov th.Example 1
Multidimensional
References
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.
Girsanov
Change ofmeasure
Radon-Nikodymth.
Girsanov th.Example 1
Multidimensional
References
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
Change ofmeasure
Radon-Nikodymth.
Girsanov th.Example 1
Multidimensional
References
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.
Girsanov th.Example 1
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.
Girsanov th.Example 1
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
Radon-Nikodymth.
Girsanov th.Example 1
Multidimensional
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.
Girsanov th.Example 1
Multidimensional
References
Example 1 IIIGirsanav theorem
In the present case,
8s, γs =µ r
σ.
Girsanov
Change ofmeasure
Radon-Nikodymth.
Girsanov th.Example 1
Multidimensional
References
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
< ∞.
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.
Girsanov
Change ofmeasure
Radon-Nikodymth.
Girsanov th.Example 1
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.
Girsanov th.Example 1
Multidimensional
References
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.
Girsanov th.Example 1
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.
Girsanov th.Example 1
Multidimensional
References
Example 1 VIIIGirsanav theorem
Girsanov
Change ofmeasure
Radon-Nikodymth.
Girsanov th.Example 1
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
Radon-Nikodymth.
Girsanov th.Example 1
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.
Girsanov
Change ofmeasure
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Girsanov th.Example 1
Multidimensional
References
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
Radon-Nikodymth.
Girsanov th.Example 1
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.
Girsanov th.Example 1
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
Change ofmeasure
Radon-Nikodymth.
Girsanov th.Example 1
Multidimensional
References
Example 1 XIVGirsanav theorem
where N () is the cumulative distribution function of astandard normal random variable.
Girsanov
Change ofmeasure
Radon-Nikodymth.
Girsanov th.Example 1
Multidimensional
References
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.
Girsanov th.
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.
MultidimensionalExample 2GirsanovExample 2(continued)Example 3
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.
MultidimensionalExample 2GirsanovExample 2(continued)Example 3
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.
MultidimensionalExample 2GirsanovExample 2(continued)Example 3
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.
MultidimensionalExample 2GirsanovExample 2(continued)Example 3
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.
Girsanov th.
MultidimensionalExample 2GirsanovExample 2(continued)Example 3
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.
MultidimensionalExample 2GirsanovExample 2(continued)Example 3
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.
MultidimensionalExample 2GirsanovExample 2(continued)Example 3
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.
MultidimensionalExample 2GirsanovExample 2(continued)Example 3
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.
MultidimensionalExample 2GirsanovExample 2(continued)Example 3
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.
MultidimensionalExample 2GirsanovExample 2(continued)Example 3
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.
MultidimensionalExample 2GirsanovExample 2(continued)Example 3
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.
MultidimensionalExample 2GirsanovExample 2(continued)Example 3
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
< ∞.
There exists a measure Q on (Ω,F ) such that
(CMG1) Q is equivalent to P
Girsanov
Change ofmeasure
Radon-Nikodymth.
Girsanov th.
MultidimensionalExample 2GirsanovExample 2(continued)Example 3
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.
MultidimensionalExample 2GirsanovExample 2(continued)Example 3
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.
MultidimensionalExample 2GirsanovExample 2(continued)Example 3
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
dCt = µCCt dt + σCCt dWPt
= 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.
MultidimensionalExample 2GirsanovExample 2(continued)Example 3
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
ρ (µC + v r )σCp1 ρ2
!t
!
= rCtSt dt + (σS ρ+ σC )CtSt dWQt + σS
q1 ρ2CtSt dfW Q
t
Girsanov
Change ofmeasure
Radon-Nikodymth.
Girsanov th.
MultidimensionalExample 2GirsanovExample 2(continued)Example 3
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.
MultidimensionalExample 2GirsanovExample 2(continued)Example 3
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.
MultidimensionalExample 2GirsanovExample 2(continued)Example 3
References
Example 2 (continued) VI
To summarize,
dCt = µCCt dt + σCCt dWPt
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.
MultidimensionalExample 2GirsanovExample 2(continued)Example 3
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
ρ (µC + v r )σCp1 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.
MultidimensionalExample 2GirsanovExample 2(continued)Example 3
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.
MultidimensionalExample 2GirsanovExample 2(continued)Example 3
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.
MultidimensionalExample 2GirsanovExample 2(continued)Example 3
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.
MultidimensionalExample 2GirsanovExample 2(continued)Example 3
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.
MultidimensionalExample 2GirsanovExample 2(continued)Example 3
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.
MultidimensionalExample 2GirsanovExample 2(continued)Example 3
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.