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Probability Theory
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04 — Article
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Quiz
Quiz: Filtrations, Adapted Processes, and Martingales
Module 4 of 5 · Hard
Quick Quiz
1.
A filtration
(
F
t
)
t
≥
0
(\mathcal{F}_t)_{t\ge0}
(
F
t
)
t
≥
0
on
(
Ω
,
F
,
P
)
(\Omega,\mathcal{F},\mathbb{P})
(
Ω
,
F
,
P
)
is defined by which core property?
F
s
=
F
t
\mathcal{F}_s=\mathcal{F}_t
F
s
=
F
t
for all
s
,
t
s,t
s
,
t
(all σ-algebras equal).
F
s
⊆
F
t
\mathcal{F}_s\subseteq\mathcal{F}_t
F
s
⊆
F
t
for all
s
≤
t
s\le t
s
≤
t
(increasing family of sub-σ-algebras).
F
s
⊇
F
t
\mathcal{F}_s\supseteq\mathcal{F}_t
F
s
⊇
F
t
for all
s
≤
t
s\le t
s
≤
t
(decreasing family).
F
t
=
F
\mathcal{F}_t=\mathcal{F}
F
t
=
F
for all
t
t
t
(each equals the full σ-algebra).
2.
For
Ω
=
{
H
H
,
H
T
,
T
H
,
T
T
}
\Omega=\{HH,HT,TH,TT\}
Ω
=
{
H
H
,
H
T
,
T
H
,
T
T
}
with
F
1
=
σ
(
{
H
H
,
H
T
}
,
{
T
H
,
T
T
}
)
\mathcal{F}_1=\sigma(\{HH,HT\},\{TH,TT\})
F
1
=
σ
({
H
H
,
H
T
}
,
{
T
H
,
T
T
})
, how many events does
F
1
\mathcal{F}_1
F
1
contain?
8
16
4
2
3.
A process
(
X
t
)
t
≥
0
(X_t)_{t\ge0}
(
X
t
)
t
≥
0
is adapted to
(
F
t
)
(\mathcal{F}_t)
(
F
t
)
if and only if:
X
t
X_t
X
t
is constant on each
F
t
\mathcal{F}_t
F
t
-atom for every
t
≥
0
t\ge0
t
≥
0
.
X
t
X_t
X
t
is independent of
F
t
\mathcal{F}_t
F
t
for every
t
≥
0
t\ge0
t
≥
0
.
X
t
X_t
X
t
is
F
\mathcal{F}
F
-measurable for every
t
≥
0
t\ge0
t
≥
0
.
X
t
X_t
X
t
is
F
t
\mathcal{F}_t
F
t
-measurable for every
t
≥
0
t\ge0
t
≥
0
.
4.
Which of the following is a stopping time with respect to the natural filtration
(
F
t
B
)
(\mathcal{F}_t^B)
(
F
t
B
)
of a Brownian motion
(
B
t
)
(B_t)
(
B
t
)
?
τ
=
arg
max
0
≤
t
≤
T
B
t
\tau=\arg\max_{0\le t\le T}B_t
τ
=
ar
g
max
0
≤
t
≤
T
B
t
(time of the maximum).
τ
=
inf
{
t
≥
0
:
B
t
=
a
}
\tau=\inf\{t\ge0:B_t=a\}
τ
=
in
f
{
t
≥
0
:
B
t
=
a
}
for a constant
a
>
0
a>0
a
>
0
(first hitting time).
τ
=
inf
{
t
≥
0
:
B
t
+
1
=
a
}
\tau=\inf\{t\ge0:B_{t+1}=a\}
τ
=
in
f
{
t
≥
0
:
B
t
+
1
=
a
}
(uses the future path).
τ
=
T
−
inf
{
t
≥
0
:
B
t
=
a
}
\tau=T-\inf\{t\ge0:B_t=a\}
τ
=
T
−
in
f
{
t
≥
0
:
B
t
=
a
}
(time-reversed hitting time).
5.
An adapted integrable process
(
M
t
)
(M_t)
(
M
t
)
is a martingale if and only if:
E
[
M
t
∣
F
s
]
=
M
s
\mathbb{E}[M_t\mid\mathcal{F}_s]=M_s
E
[
M
t
∣
F
s
]
=
M
s
a.s. for all
0
≤
s
≤
t
0\le s\le t
0
≤
s
≤
t
.
E
[
M
t
]
=
0
\mathbb{E}[M_t]=0
E
[
M
t
]
=
0
for all
t
≥
0
t\ge0
t
≥
0
.
M
t
−
M
s
M_t-M_s
M
t
−
M
s
is independent of
F
s
\mathcal{F}_s
F
s
for all
s
≤
t
s\le t
s
≤
t
.
E
[
M
t
∣
F
s
]
≥
M
s
\mathbb{E}[M_t\mid\mathcal{F}_s]\ge M_s
E
[
M
t
∣
F
s
]
≥
M
s
a.s. for all
0
≤
s
≤
t
0\le s\le t
0
≤
s
≤
t
.
6.
Why is the discounted stock price
S
~
t
=
e
−
r
t
S
t
\tilde S_t=e^{-rt}S_t
S
~
t
=
e
−
r
t
S
t
a martingale under
Q
\mathbb{Q}
Q
but not under
P
\mathbb{P}
P
?
Under
Q
\mathbb{Q}
Q
the volatility
σ
=
0
\sigma=0
σ
=
0
; under
P
\mathbb{P}
P
volatility makes
S
t
S_t
S
t
a submartingale.
Under
P
\mathbb{P}
P
,
S
t
=
0
S_t=0
S
t
=
0
, so the discounted price is trivially a martingale.
The Girsanov theorem adds a drift
μ
\mu
μ
under
Q
\mathbb{Q}
Q
that exactly cancels
e
−
r
t
e^{-rt}
e
−
r
t
.
Under
Q
\mathbb{Q}
Q
the drift of
d
(
e
−
r
t
S
t
)
d(e^{-rt}S_t)
d
(
e
−
r
t
S
t
)
is zero; under
P
\mathbb{P}
P
it is
(
μ
−
r
)
e
−
r
t
S
t
d
t
≠
0
(\mu-r)e^{-rt}S_t\,dt\ne0
(
μ
−
r
)
e
−
r
t
S
t
d
t
=
0
when
μ
≠
r
\mu\ne r
μ
=
r
.
7.
Doob's Optional Stopping Theorem gives
E
[
M
τ
]
=
E
[
M
0
]
\mathbb{E}[M_\tau]=\mathbb{E}[M_0]
E
[
M
τ
]
=
E
[
M
0
]
. Which integrability condition is sufficient when
τ
\tau
τ
may be unbounded?
τ
\tau
τ
has finite variance:
Var
(
τ
)
<
∞
\text{Var}(\tau)<\infty
Var
(
τ
)
<
∞
.
τ
\tau
τ
is a.s. finite, with no further condition needed.
M
n
M_n
M
n
is non-negative for all
n
n
n
.
The stopped process
(
M
τ
∧
n
)
(M_{\tau\wedge n})
(
M
τ
∧
n
)
is uniformly integrable.
8.
A rates quant checks that the discounted NPV of a knock-out barrier swaption is a
Q
\mathbb{Q}
Q
-martingale, yet her OST calculation gives
E
[
NPV
τ
]
≠
E
[
NPV
0
]
\mathbb{E}[\text{NPV}_\tau]\ne\mathbb{E}[\text{NPV}_0]
E
[
NPV
τ
]
=
E
[
NPV
0
]
. What is the most likely explanation?
The stopping time
τ
\tau
τ
is unbounded and the stopped process is not uniformly integrable.
The discounted NPV is a submartingale, not a martingale, under
Q
\mathbb{Q}
Q
.
The optional stopping theorem does not apply to interest-rate models.
The martingale property fails under the swap measure but not under the risk-neutral measure.
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