We introduce models for financial markets and, in their context, the notions of portfolio rules and of arbitrage. The absence of arbitrage is a central requirement in the modern theories of mathematical economics and finance, as is the even stronger notion of equivalent martingale measure. We relate this to probabilistic concepts such as fair game, martingales, and coherence in the sense of de Finetti.

We also survey a newer, descriptive approach to finance, based on the existence of a growth-optimal portfolio (equivalently, of a portfolio with the so-called “numeraire” property). These equivalent notions proscribe only egregious forms of arbitrage, and lead to an entire theory for the subject which is flexible and simple, allows the outperformance of one portfolio by another, and is able to deal with an arbitrary number of assets. This part of the talk is based on a book in preparation, with Constantinos Kardaras.
Speaker(s): Ioannis Karatzas (Eugene Higgins Professor of Applied Probability Columbia University)