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2007 Wolfe Lecture in Mathematics

Dr. Ioannis Karatzas
Columbia University
Thursday October 4, 2007
4:00pm, Herman Brown 227
Title: Volatility Stabilization, Diversity and Arbitrage in Stochastic Finance

Wolfe Lecture Poster Abstract: In this talk we start with an overview of the modern theory of portfolios, based on Stochastic Analysis. We introduce the notion of relative arbitrage and provide simple, easy-to-test criteria for the existence of such arbitrage in equity markets. These criteria postulate essentially that the excess growth rate of the market portfolio, a positive quantity that can be estimated or even computed from a given market structure, be “sufficiently large". We show that conditions which satisfy these criteria are manifestly present in the US equity market, and construct explicit portfolios under these conditions.  One such condition, market diversity, emerges when the volatility structure is bounded.

We then construct examples of abstract markets in which the criteria hold. We study in some detail a specific example of a non-diverse abstract market which is volatility-stabilized, in that the return from the market portfolio has constant drift and variance rates, while the smallest stocks are assigned the largest volatilities and individual stocks fluctuate widely. An interesting probabilistic structure emerges in which time changes, Bessel processes, and the asymptotic theory for planar Brownian motion, play crucial roles. Several open questions are raised for further study. (Joint work with E. Robert Fernholz.)


The Annual Wolfe Lectures are supported through a generous gift from the estate of Raquel A. Wolfe in loving memory of her husband, Dr. Alfred S. Wolfe, and his love for math which he instilled in both his children.

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