Statistics for Finance

Appendix B

Probability theory

In this appendix it is the intention to give a brief overview of probability theory. Some of the concepts introduced are widely used in the lecture notes. It is not necessary to understand all the technical details, but an intuitive understanding of the concepts introduced is important.

B.1 Measures and σ-algebras

Let Ω denote a finite sample space which contains all the elementary outcome ωi for i = 1,2,..., N. In a two period binomial model the elementary outcome is the state of the world at time t = 2, which determines the stock price at that time.

Definition B.1 (σ-algebra).

Let Ω be a set of points ω. A family ℱ of subset of Ω is called a σ-algebra if

• 1. Ø ∊ ℱ
• 2. A ∊ ℱ ⇒ Ac ∊ ℱ
• 3. An ∊ ℱ for $n=1,2,...\Rightarrow \underset{}{\overset{}{}}$