Probability for Finance

This book is a technical support for students in finance. It reviews the main probabilistic tools used in financial models in a pedagogical way, starting from simplke concepts like random variables and tribes and going to more sophiticated ones like conditional expectations and limit theorems. Many illustrations are given, taken from the financial literature. The book is also a prerequisite for “Stochastic Processes for Finance” published in the same collection.

Introduction

1. Probability spaces and random variables
1.1 Measurable spaces and probability measures
1.1.1 σ algebra (or tribe) on a set Ω
1.1.2 Sub-tribes of A
1.1.3 Probability measures
1.2 Conditional probability and Bayes theorem
1.2.1 Independant events and independant tribes
1.2.2 Conditional probability measures
1.2.3 Bayes theorem
1.3 Random variables and probability distributions
1.3.1 Random variables and generated tribes
1.3.2 Independant random variables
1.3.3 Probability distributions and cumulative distributions
1.3.4 Discrete and continuous random variables
1.3.5 Transformations of random variables

2. Moments of a random variable
2.1 Mathematical expectation
2.1.1 Expectations of discrete and continous random variables
2.1.2 Expectation: the general case
2.1.3 Illustration: Jensen’s inequality and Saint-Peterburg paradox
2.2 Variance and higher moments
2.2.1 Second-order moments
2.2.2 Skewness and kurtosis
2.3 The vector space of random variables
2.3.1 Almost surely equal random variables
2.3.2 The space L1 (Ω, A, P)
2.3.3 The space L2 (Ω, A, P)
2.3.4 Covariance and correlation
2.4 Equivalent probabilities and Radon-Nikodym derivatives
2.4.1 Intuition
2.4.2 Radon Nikodym derivatives
2.5 Random vectors
2.5.1 Definitions
2.5.2 Application to portfolio choice

3. Usual probability distributions in financial models
3.1 Discrete distributions
3.1.1 Bernoulli distribution
3.1.2 Binomial distribution
3.1.3 Poisson distribution
3.2 Continuous distributions
3.2.1 Uniform distribution
3.2.2 Gaussian (normal) distribution
3.2.3 Log-normal distribution
3.3 Some other useful distributions
3.3.1 The X 2 distribution
3.3.2 The Student-t distribution
3.3.3 The Fisher-Snedecor distribution

4. Conditional expectations and Limit theorems
4.1 Conditional expectations
4.1.1 Introductive example
4.1.2 Conditional distributions
4.1.3 Conditional expectation with respect to an event
4.1.4 Conditional expectation with respect to a random variable
4.1.5 Conditional expectation with respect to a substribe
4.2 Geometric interpretation in L2 (Ω, A, P)
4.2.1 Introductive example
4.2.2 Conditional expectation as a projection in L2
4.3 Properties of conditional expectations
4.3.1 The Gaussian vector case
4.4 The law of large numbers and the central limit theorem
4.4.1 Stochastic Covergences
4.4.2 Law of large numbers
4.4.3 Central limit theorem

Bibliography