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Girsanov's Theorem and Equivalent Martingale Measures
Hard
·
25 min read
Stochastic Calculus
Girsanov's Theorem
Risk-Neutral Pricing
Change of Measure
1
Article
2
Quiz
Quick Quiz
1.
Two probability measures
P
\mathbb{P}
P
and
Q
\mathbb{Q}
Q
on the same space are equivalent if:
P
(
A
)
=
0
⟺
Q
(
A
)
=
0
\mathbb{P}(A)=0\iff\mathbb{Q}(A)=0
P
(
A
)
=
0
⟺
Q
(
A
)
=
0
for every event
A
A
A
P
\mathbb{P}
P
and
Q
\mathbb{Q}
Q
are defined on the same filtration
Q
(
A
)
≤
P
(
A
)
\mathbb{Q}(A)\le\mathbb{P}(A)
Q
(
A
)
≤
P
(
A
)
for every event
A
A
A
E
P
[
X
]
=
E
Q
[
X
]
\mathbb{E}^{\mathbb{P}}[X]=\mathbb{E}^{\mathbb{Q}}[X]
E
P
[
X
]
=
E
Q
[
X
]
for every
X
X
X
2.
The Girsanov density
Z
t
=
exp
(
−
∫
0
t
θ
s
d
W
s
−
1
2
∫
0
t
θ
s
2
d
s
)
Z_t=\exp\!\left(-\int_0^t\theta_s\,dW_s-\tfrac12\int_0^t\theta_s^2\,ds\right)
Z
t
=
exp
(
−
∫
0
t
θ
s
d
W
s
−
2
1
∫
0
t
θ
s
2
d
s
)
satisfies which SDE?
d
Z
t
=
−
θ
t
Z
t
d
t
dZ_t=-\theta_t Z_t\,dt
d
Z
t
=
−
θ
t
Z
t
d
t
d
Z
t
=
−
θ
t
Z
t
d
W
t
dZ_t=-\theta_t Z_t\,dW_t
d
Z
t
=
−
θ
t
Z
t
d
W
t
d
Z
t
=
θ
t
Z
t
d
t
−
Z
t
d
W
t
dZ_t=\theta_t Z_t\,dt-Z_t\,dW_t
d
Z
t
=
θ
t
Z
t
d
t
−
Z
t
d
W
t
d
Z
t
=
θ
t
Z
t
d
W
t
dZ_t=\theta_t Z_t\,dW_t
d
Z
t
=
θ
t
Z
t
d
W
t
3.
Under
P
\mathbb{P}
P
,
d
S
t
=
μ
S
t
d
t
+
σ
S
t
d
W
t
P
dS_t=\mu S_t\,dt+\sigma S_t\,dW_t^{\mathbb{P}}
d
S
t
=
μ
S
t
d
t
+
σ
S
t
d
W
t
P
. After Girsanov with market price of risk
θ
=
(
μ
−
r
)
/
σ
\theta=(\mu-r)/\sigma
θ
=
(
μ
−
r
)
/
σ
, the dynamics under
Q
\mathbb{Q}
Q
are:
d
S
t
=
0
⋅
d
t
+
σ
S
t
d
W
~
t
dS_t=0\cdot dt+\sigma S_t\,d\widetilde{W}_t
d
S
t
=
0
⋅
d
t
+
σ
S
t
d
W
t
d
S
t
=
(
μ
−
r
)
S
t
d
t
+
σ
S
t
d
W
~
t
dS_t=(\mu-r)S_t\,dt+\sigma S_t\,d\widetilde{W}_t
d
S
t
=
(
μ
−
r
)
S
t
d
t
+
σ
S
t
d
W
t
d
S
t
=
μ
S
t
d
t
+
σ
S
t
d
W
~
t
dS_t=\mu S_t\,dt+\sigma S_t\,d\widetilde{W}_t
d
S
t
=
μ
S
t
d
t
+
σ
S
t
d
W
t
d
S
t
=
r
S
t
d
t
+
σ
S
t
d
W
~
t
dS_t=r S_t\,dt+\sigma S_t\,d\widetilde{W}_t
d
S
t
=
r
S
t
d
t
+
σ
S
t
d
W
t
4.
In a complete Black-Scholes market, the equivalent martingale measure is unique.
True
False
5.
The Novikov condition
E
P
[
exp
(
1
2
∫
0
T
θ
t
2
d
t
)
]
<
∞
\mathbb{E}^{\mathbb{P}}\!\left[\exp\!\left(\tfrac12\int_0^T\theta_t^2\,dt\right)\right]<\infty
E
P
[
exp
(
2
1
∫
0
T
θ
t
2
d
t
)
]
<
∞
guarantees that:
The market price of risk process
θ
t
\theta_t
θ
t
stays bounded
The underlying stock price remains strictly positive
Z
t
Z_t
Z
t
is a true martingale, not just a local martingale
The measure
Q
\mathbb{Q}
Q
assigns positive probability to every event
6.
Under the
T
T
T
-forward measure
Q
T
\mathbb{Q}^T
Q
T
(numeraire = zero-coupon bond
P
(
t
,
T
)
P(t,T)
P
(
t
,
T
)
), which process is a
Q
T
\mathbb{Q}^T
Q
T
-martingale?
The bond-deflated stock
S
t
/
P
(
t
,
T
)
S_t/P(t,T)
S
t
/
P
(
t
,
T
)
The discounted stock price
e
−
r
t
S
t
e^{-rt}S_t
e
−
r
t
S
t
The short rate
r
t
r_t
r
t
The stock price
S
t
S_t
S
t
itself
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