probability for finance
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Probability for Finance
Patrick Roger
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Probability for Finance
Patrick Roger
Strasbourg University, EM Strasbourg Business School
May 2010
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Probability for Finance
© 2010 Patrick Roger & Ventus Publishing ApS
ISBN 978-87-7681-589-9
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Contents
Probability for Finance
Contents
Introduction
8
1.
1.1
1.1.1
1.1.2
1.1.3
1.2
1.2.1
1.2.2
1.2.3
1.3
1.3.1
1.3.2
1.3.3
1.3.4
1.3.5
Probability spaces and random variables
Measurable spaces and probability measures
σ algebra (or tribe) on a set Ω
Sub-tribes of A
Probability measures
Conditional probability and Bayes theorem
Independant events and independant tribes
Conditional probability measures
Bayes theorem
Random variables and probability distributions
Random variables and generated tribes
Independant random variables
Probability distributions and cumulative distributions
Discrete and continuous random variables
Transformations of random variables
10
10
11
13
16
18
19
21
24
25
25
29
30
34
35
2.
2.1
Moments of a random variable
Mathematical expectation
37
37
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Contents
Probability for Finance
2.1.1
2.1.2
2.1.3
2.2
2.2.1
2.2.2
2.3
2.3.1
2.3.2
2.3.3
2.3.4
2.4
2.4.1
2.4.2
2.5
2.5.1
2.5.2
Expectations of discrete and continous random variables
Expectation: the general case
Illustration: Jensen’s inequality and Saint-Peterburg paradox
Variance and higher moments
Second-order moments
Skewness and kurtosis
The vector space of random variables
Almost surely equal random variables
The space L1 (Ω, A, P)
The space L2 (Ω, A, P)
Covariance and correlation
Equivalent probabilities and Radon-Nikodym derivatives
Intuition
Radon Nikodym derivatives
Random vectors
Definitions
Application to portfolio choice
39
40
43
46
46
48
50
51
53
54
59
63
63
67
69
69
71
3.
3.1
3.1.1
3.1.2
3.1.3
Usual probability distributions in financial models
Discrete distributions
Bernoulli distribution
Binomial distribution
Poisson distribution
73
73
73
76
78
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Contents
Probability for Finance
3.2
3.2.1
3.2.2
3.2.3
3.3
3.3.1
3.3.2
3.3.3
Continuous distributions
Uniform distribution
Gaussian (normal) distribution
Log-normal distribution
Some other useful distributions
2
The X distribution
The Student-t distribution
The Fisher-Snedecor distribution
81
81
82
86
91
91
92
93
4.
4.1
4.1.1
4.1.2
4.1.3
4.1.4
4.1.5
4.2
4.2.1
4.2.2
4.3
4.3.1
4.4
4.4.1
Conditional expectations and Limit theorems
Conditional expectations
Introductive example
Conditional distributions
Conditional expectation with respect to an event
Conditional expectation with respect to a random variable
Conditional expectation with respect to a substribe
Geometric interpretation in L2 (Ω, A, P)
Introductive example
Conditional expectation as a projection in L2
Properties of conditional expectations
The Gaussian vector case
The law of large numbers and the central limit theorem
Stochastic Covergences
94
94
94
96
97
98
100
101
101
102
104
105
108
108
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Contents
Probability for Finance
4.4.2
4.4.3
Law of large numbers
Central limit theorem
109
112
Bibliography
114
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Introduction
Probability for Finance
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Probability for Finance
Introduction
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Probability spaces and random variables
Probability for Finance
t = 0 T = 1.
.
T
P
P
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Probability spaces and random variables
Probability for Finance
,
σ
σ
P()
σ A P()
∈ A
∀ B ∈ A, B c ∈ A B c B B c =
{ω ∈ /ω ∈
/ B} . A
(Bn , n ∈ N) A, +∞
n=1 Bn ∈ A.
A
(, A) A
T = 1 ω
A ω ∈ A A ω ∈
/ A.
, .
= {ω1 , ω 2 , ω 3 , ω 4 } ,
A = {∅, } A′ =
{∅, {ω 1 , ω 2 } , {ω 3 , ω 4 } , } A = P(),
A
(Bn , n ∈ N) A, ∩+∞
n=1 Bn ∈
A A
∅ ∈ A.
σ
σ
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Probability spaces and random variables
Probability for Finance
Γ = {B1 , ..., BK }
Bi ∩ Bj = ∅ i = j
∪K
i=1 Bi = .
A
A.
A
Γ,
∅, Γ .
Bj
Bj ) Γ
Bj Γ Bj
∅
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Probability spaces and random variables
Probability for Finance
Γ = {B1 , ..., BK }
Γ, BΓ ,
Γ.
BΓ
BΓ ∅, ,
Γ.
BΓ 2K
A
T > 1
T
P) t < T,
P).
A′ P) A A′
A A
A′ .
, A′ ) A′
= {ω 1 , ω 2 , ω 3 , ω 4 } ,
A′ = {∅, {ω 1 , ω 2 } , {ω 3 , ω 4 } , } P).
∈ A′ B A′ B c A′ {ω 1 , ω 2 } =
{ω 3 , ω 4 }c . A′ A′
{ω 1 , ω 2 } ∪ {ω 3 , ω 4 } = .
A
A′ A′ ⊂ A Γ Γ′
Card( Card(
Card( < Card(P(.
P(
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Probability spaces and random variables
Probability for Finance
Γ Γ′
Γ′ Γ. Γ
Γ′ .
A′ A Γ A
Γ′ A′ .
2K
K K
A′ A;
u
d),
= {uu; ud; du; dd}
A′ = {∅; {uu; ud} ; {du; dd} ; }
P). {du; dd}
= {uu; ud}c
{uu; ud} {du; dd} = ∈ A.
1
ր
ց
ր
u
ց
d
ր
ց
uu = u2
ud
du
dd = d2
{uu; ud}
.
{uu; ud} .
ud du
ud
Γ′ Γ.
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Probability spaces and random variables
Probability for Finance
du.
R,
BR .
R R. BR
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Probability spaces and random variables
Probability for Finance
(, A)
A A [0; 1]
P () = 1
(Bn , n ∈ N) A
+∞ +∞
P
Bn =
P (Bn )
n=1
n=1
(, A, P )
∅
B B c ,
P (B) + P (B c ) = P () = 1
P (B c ) = 1−P (B).
B B c
σ
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Probability spaces and random variables
Probability for Finance
(, A, P )
P (∅) = 0
∀ (B1 , B2 ) ∈ A × A, B1 ⊆ B2 ⇒ P (B1 ) ≤ P (B2 )
(Bn , n ∈ N) Bn ⊂ Bn+1
A
lim P (Bn ) = P
Bn
n→+∞
n∈N
(Bn , n ∈ N) Bn ⊃ Bn+1
A
Bn
lim P (Bn ) = P
n→+∞
n∈N
∀ B ∈ A, P (B c ) = 1 − P (B)
∅ P ( ∅) = P () + P (∅) =
P () = 1. P (∅) = 0
B1 ⊆ B2 ⇒ P (B2 ) = P (B1 (B2 B1c )) = P (B1 ) + P (B2 B1c ) ≥
P (B1 )
n
(Bn , n ∈ N) un = P
p=1 Bp
P () = 1
(Bn , n ∈N)
P
n∈N Bn .
n
(Bn , n ∈ N) vn = P
B
p
p=1
P (∅) = 0
(Bn , n ∈N)
P
n∈N Bn .
P (B B c ) = P (B)
+ P (B c ) B
B c B B c = , P (B B c ) = P () = 1
P (B c ) = 1 − P (B)
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Probability spaces and random variables
Probability for Finance
Card() = N A = P() ;
A
1
∀ω ∈ , P (ω) =
N
[0; 1] × [0; 1]
R2 ; σ
A
, P (A) A P P () = 1;
P
[0; 1] × [0; 1]
B = [a; b] × [c; d] (d − c)(b − a) ≤ 1.
B (d − c)(b − a).
(, A, P )
B ⊂
A
P (ω)
P ({ω})
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Probability spaces and random variables
Probability for Finance
B1 , B2 A P (B1
P (B1 ) × P (B2 ).
B2 ) =
B2 ∈ A P (B2 ) = 0 B1
B2 P (B1 |B2 ),
P (B1 B2 )
P (B1 |B2 ) =
P (B2 )
B2
B2 . B1
B2 ,
. B1 B2 = ∅, B1
B1
B1 B2
B2 B1 .
B1 B2
P (B1 B2 )
P (B1 ) × P (B2 )
P (B1 |B2 ) =
=
= P (B1 )
P (B2 )
P (B2 )
= [0; 1] × [0; 1]
(x, y) B1 = 0; 12 ×
1
; 1 B2 = 0; 13 × 0; 12 ;
3
1 2
1
× =
2 3
3
1 1
1
P (B2 ) =
× =
3 2
6
P (B1 ) =
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Probability spaces and random variables
Probability for Finance
B2 (x, y) ∈ B1 x ∈ 0; 13 y
1/3 13 ; 12 . (x, y) ∈ B2
y ≤ 12 . (x, y)
B1 y ≥ 13,
1/3
y ∈ 13 ; 12 .
y ∈ 0; 12
P (B1 |B2 ) = 13 B1 B2 = 0; 13 × 13 ; 12 ,
1
1 1
1
P (B1 B2 ) =
−0 ×
−
=
3
2 3
18
P (B1 |B2 ) =
1
18
1
6
=
1
= P (B1 )
3
B1 B2 .
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Probability spaces and random variables
Probability for Finance
B1 B2
B1 ,
B2 B1
σ
G G ′ A
∀B ∈ G, ∀B ′ ∈ G ′ , P (B ∩ B ′ ) = P (B) × P (B ′ )
G× G ′
P()
(, A).
∅.
B ∈ A AB
AB = A B A ∈ A ;
AB B. (B, AB )
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Probability spaces and random variables
Probability for Finance
B ∈ AB
B = B (Cn , n ∈ N)
Cn = An B
Cn =
An
An B =
B
n∈N
n∈N
n∈N
B ∈ AB
An ∈ A
n∈N An
C = A B ∈ AB CBc C B.
c
B = Ac B
Bc B
CBc = A B
= Ac B ∈ AB
A
n∈N
B ∈ P (B) = 0
P (. |B ), P (B1 |B ) B1 ,
(B, AB ) .
P (B |B ) = 1. (Cn , n ∈ N)
AB
P
P
B
B)
n∈N Cn
n∈N (Cn
P
Cn |B =
=
P (B)
P (B)
n∈N
n, Cn ⊂ B,
P
C
B)
n
n∈N
n∈N P (Cn )
n∈N P (Cn
=
=
=
P (Cn |B )
P (B)
P (B)
P (B)
n∈N
t B.
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Probability spaces and random variables
Probability for Finance
(B, AB , P (. |B )).
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Probability spaces and random variables
Probability for Finance
(B1 , B2 , ..., Bn ) C ∈ A,
P (C |Bj )P (Bj )
P (Bj |C ) = n
i=1 P (C |Bi )P (Bi )
Bj
n
C=
C
Bi
i=1
P (C) =
n
i=1
n
P C
Bi =
P (C |Bi )P (Bi )
i=1
P C
Bj = P (C |Bj )P (Bj ) = P (Bj |C )P (C)
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Probability spaces and random variables
Probability for Finance
P (C)
C
B1 B2 = B1c
P (B1 |C ) =
P (C |B1 )P (B1 )
P (C |B1 )P (B1 ) + P (C |B2 )P (B2 )
P (B1 ) = 10−4
P (C |B1 ) = 0.99
P (C |B2 ) = 0.01
P (B1 |C ) =
0.99 × 10−4
≃ 0.01
0.99 × 10−4 + 0.01 × (1 − 10−4 )
T = 1
R+
R
(S − S )/S ,
ln(S /S ), −∞.
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