Probability for finance

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
algebra (or tribe) on a set
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

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13
16
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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 ( , A, P)
The space L2 ( , A, P)
Covariance and correlation
Equivalent probabilities and Radon-Nikodym derivatives
Intuition
Radon Nikodym derivatives
Random vectors
Deinitions
Application to portfolio choice

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40
43

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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 inancial models
Discrete distributions
Bernoulli distribution
Binomial distribution
Poisson distribution

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73
73

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78

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