Review of 642:621 Mathematical Finance I

A. Probability spaces, random variables. (Shreve, Chapter 1) In this course, risk-neutral pricing theory is formulated in the language ofmeasuretheoretic probability. To the aspiring quant, measure-theoretic probability might at first appear mysterious, overly abstract, and irrelevant to quantitative analysis. However, it provides a general and powerful way to express both the conceptual ideas and the basic formulas of risk-neutral pricing, and it is the framework employed in the research literature of mathematical finance. For this reason, you should aim to acquire a confidant, intuitive understanding of measure-theoretic language and be able to translate it into practical computational formulae. You do not need to learn advanced, measure-theoretic techniques or proofs. The basis of measure-theoretic probability is the probability space. This is a triple (Ω,F ,P), where Ω is a set called the outcome space, F is a collection of subsets of Ω, and P is a measure that assigns to each A ∈ F a number P(A) in [0, 1]. It provides a general template for probability modeling. Given a random phenomenon to model, the elements of Ω represent all its possible outcomes; P(A), where A is a subset of Ω, represents the probability of A, or, more precisely, the probability that the outcome will belong to the set A; and F is the collection of subsets to which P assigns probabilities. The elements of F are referred to as events. In order that (Ω,F ,P) qualify as a probability space, F and P must satisfy a few basic conditions, called the axioms of probability, which are consistent with intuitive notions of what probability means and with logical coherence. These are: (i) F is σ-algebra, meaning it is a nonempty collection that is closed under the set operations of taking complements and of forming countable intersections or unions; (ii) P(Ω) = 1; and (iii) P is countably additive, namely, if A1, A2, . . . is a sequence of disjoint events, then