Ioannis Karatzas
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MATHEMATICS COLLOQUIUM
Speaker: Ioannis Karatzas Abstract. We introduce models for financial markets and, in their context, the notions of portfolio rules and of arbitrage. The normative assumption of "absence of arbitrage" is central in the modern theories of mathematical economics and finance. We relate it to probabilistic concepts such as fair game, martingale, coherence in the sense of deFinetti, and equivalent martingale measure. We also survey recent work in the context of the Stochastic Portfolio Theory pioneered by E.R. Fernholz. This theory provides descriptive conditions under which opportunities for relative arbitrage, or outperformance, do exist, then constructs simple portfolios that implement them. Finally we explain how, even in the presence of such arbitrage, most of the standard mathematical theory of finance still functions, though in somewhat modified form. |