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Probability Theory
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02 — Article
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Quiz
Quiz: Lebesgue Integration and Expectation
Module 2 of 5 · Medium
Quick Quiz
1.
Let
Ω
=
{
A
,
B
,
C
}
\Omega=\{A,B,C\}
Ω
=
{
A
,
B
,
C
}
with
P
(
A
)
=
1
/
2
\mathbb{P}(A)=1/2
P
(
A
)
=
1/2
,
P
(
B
)
=
1
/
4
\mathbb{P}(B)=1/4
P
(
B
)
=
1/4
,
P
(
C
)
=
1
/
4
\mathbb{P}(C)=1/4
P
(
C
)
=
1/4
. Define
X
(
A
)
=
4
X(A)=4
X
(
A
)
=
4
,
X
(
B
)
=
−
2
X(B)=-2
X
(
B
)
=
−
2
,
X
(
C
)
=
0
X(C)=0
X
(
C
)
=
0
. What is
E
[
X
]
\mathbb{E}[X]
E
[
X
]
?
2/3
7/4
3/2
1
2.
Let
φ
(
x
)
=
(
x
−
K
)
+
\varphi(x)=(x-K)^+
φ
(
x
)
=
(
x
−
K
)
+
for a fixed
K
>
0
K>0
K
>
0
. Under
Q
\mathbb{Q}
Q
with
E
Q
[
S
T
]
=
S
0
e
r
T
\mathbb{E}^\mathbb{Q}[S_T]=S_0e^{rT}
E
Q
[
S
T
]
=
S
0
e
r
T
, Jensen's inequality implies which of the following?
E
Q
[
(
S
T
−
K
)
+
]
=
(
E
Q
[
S
T
]
−
K
)
+
\mathbb{E}^\mathbb{Q}[(S_T-K)^+]=(\mathbb{E}^\mathbb{Q}[S_T]-K)^+
E
Q
[(
S
T
−
K
)
+
]
=
(
E
Q
[
S
T
]
−
K
)
+
e
−
r
T
E
Q
[
(
S
T
−
K
)
+
]
≥
(
S
0
−
K
e
−
r
T
)
+
e^{-rT}\mathbb{E}^\mathbb{Q}[(S_T-K)^+]\ge(S_0-Ke^{-rT})^+
e
−
r
T
E
Q
[(
S
T
−
K
)
+
]
≥
(
S
0
−
K
e
−
r
T
)
+
e
−
r
T
E
Q
[
(
S
T
−
K
)
+
]
≤
(
S
0
−
K
e
−
r
T
)
+
e^{-rT}\mathbb{E}^\mathbb{Q}[(S_T-K)^+]\le(S_0-Ke^{-rT})^+
e
−
r
T
E
Q
[(
S
T
−
K
)
+
]
≤
(
S
0
−
K
e
−
r
T
)
+
E
Q
[
(
S
T
−
K
)
+
]
≥
0
\mathbb{E}^\mathbb{Q}[(S_T-K)^+]\ge0
E
Q
[(
S
T
−
K
)
+
]
≥
0
only if
S
0
>
K
S_0>K
S
0
>
K
3.
The DCT justifies computing
Δ
=
∂
S
V
0
\Delta=\partial_S V_0
Δ
=
∂
S
V
0
by differentiating under
V
0
=
e
−
r
T
E
Q
[
(
S
T
−
K
)
+
]
V_0=e^{-rT}\mathbb{E}^\mathbb{Q}[(S_T-K)^+]
V
0
=
e
−
r
T
E
Q
[(
S
T
−
K
)
+
]
. What is the dominating function that makes it applicable?
No dominating function exists; DCT cannot apply to option payoffs.
g
=
K
g=K
g
=
K
, since the strike is fixed and the payoff cannot exceed
K
K
K
.
g
=
S
T
g=S_T
g
=
S
T
, since the payoff is bounded above by the stock price.
g
=
1
g=1
g
=
1
, since
∣
∂
S
(
S
T
−
K
)
+
∣
=
1
S
T
>
K
≤
1
|\partial_S(S_T-K)^+|=\mathbf{1}_{S_T>K}\le1
∣
∂
S
(
S
T
−
K
)
+
∣
=
1
S
T
>
K
≤
1
.
4.
The Monotone Convergence Theorem requires a non-decreasing sequence
(
f
n
)
(f_n)
(
f
n
)
. Which counterexample demonstrates that MCT fails without monotonicity?
f
n
=
1
[
0
,
1
/
n
]
f_n=\mathbf{1}_{[0,1/n]}
f
n
=
1
[
0
,
1/
n
]
:
f
n
→
0
f_n\to0
f
n
→
0
and
∫
f
n
d
λ
→
0
\int f_n\,d\lambda\to0
∫
f
n
d
λ
→
0
.
f
n
=
x
/
n
f_n=x/n
f
n
=
x
/
n
on
[
0
,
1
]
[0,1]
[
0
,
1
]
:
f
n
→
0
f_n\to0
f
n
→
0
and
∫
f
n
d
λ
→
0
\int f_n\,d\lambda\to0
∫
f
n
d
λ
→
0
.
f
n
=
1
[
n
,
n
+
1
]
f_n=\mathbf{1}_{[n,n+1]}
f
n
=
1
[
n
,
n
+
1
]
on
R
\mathbb{R}
R
with Lebesgue measure:
f
n
→
0
f_n\to0
f
n
→
0
but
∫
f
n
d
λ
=
1
\int f_n\,d\lambda=1
∫
f
n
d
λ
=
1
.
f
n
=
n
⋅
1
[
0
,
1
/
n
]
f_n=n\cdot\mathbf{1}_{[0,1/n]}
f
n
=
n
⋅
1
[
0
,
1/
n
]
:
f
n
→
0
f_n\to0
f
n
→
0
a.e. but
∫
f
n
d
λ
=
1
\int f_n\,d\lambda=1
∫
f
n
d
λ
=
1
— a DCT failure (no dominating function), not an MCT one.
5.
For
Ω
=
{
1
,
2
,
3
,
4
}
\Omega=\{1,2,3,4\}
Ω
=
{
1
,
2
,
3
,
4
}
uniform with
X
(
k
)
=
k
−
5
/
2
X(k)=k-5/2
X
(
k
)
=
k
−
5/2
, compute
Var
(
X
)
=
E
[
X
2
]
−
(
E
[
X
]
)
2
\text{Var}(X)=\mathbb{E}[X^2]-(\mathbb{E}[X])^2
Var
(
X
)
=
E
[
X
2
]
−
(
E
[
X
]
)
2
.
5/2
3/2
1
5/4
6.
A portfolio's daily P&L
X
X
X
is non-negative with
E
[
X
]
=
100
\mathbb{E}[X]=100
E
[
X
]
=
100
. Using Markov's inequality, what is the tightest model-free upper bound on the probability of a daily gain exceeding 500?
Markov's inequality cannot bound gains — only losses.
1/25
1/5
1/2500
7.
Which statement correctly distinguishes
X
∈
L
1
X\in L^1
X
∈
L
1
from
X
∈
L
2
X\in L^2
X
∈
L
2
on a probability space, and states which the Itô integral requires?
L
1
L^1
L
1
:
E
[
∣
X
∣
]
<
∞
\mathbb{E}[|X|]<\infty
E
[
∣
X
∣
]
<
∞
;
L
2
L^2
L
2
:
E
[
X
2
]
<
∞
\mathbb{E}[X^2]<\infty
E
[
X
2
]
<
∞
; the Itô integral requires
L
2
L^2
L
2
so the Itô isometry holds.
L
1
L^1
L
1
:
E
[
∣
X
∣
]
<
∞
\mathbb{E}[|X|]<\infty
E
[
∣
X
∣
]
<
∞
;
L
2
L^2
L
2
:
E
[
X
2
]
<
∞
\mathbb{E}[X^2]<\infty
E
[
X
2
]
<
∞
; the Itô integral requires only
L
1
L^1
L
1
.
L
1
L^1
L
1
and
L
2
L^2
L
2
coincide on finite probability spaces, so the distinction is purely theoretical.
L
1
L^1
L
1
:
E
[
X
]
<
∞
\mathbb{E}[X]<\infty
E
[
X
]
<
∞
;
L
2
L^2
L
2
:
E
[
X
2
]
<
∞
\mathbb{E}[X^2]<\infty
E
[
X
2
]
<
∞
; they are equivalent on bounded spaces.
8.
You price an Asian option
Φ
=
(
S
ˉ
T
−
K
)
+
\Phi=(\bar S_T-K)^+
Φ
=
(
S
ˉ
T
−
K
)
+
by Monte Carlo, producing estimates
V
^
N
\hat V_N
V
^
N
as the number of paths
N
N
N
grows. Which theorem guarantees
V
^
N
→
V
0
\hat V_N\to V_0
V
^
N
→
V
0
almost surely?
The Monotone Convergence Theorem, applied to the increasing sequence of sample averages.
Jensen's inequality with
φ
(
x
)
=
(
x
−
K
)
+
\varphi(x)=(x-K)^+
φ
(
x
)
=
(
x
−
K
)
+
.
The Dominated Convergence Theorem, with dominating function the maximum payoff.
The strong law of large numbers — a consequence of Lebesgue integration theory.
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