A Limit Theorem for Financial Markets with Inert Investors

Authors:
Erhan Bayraktar;Ulrich Horst;Ronnie Sircar
Affiliations:
Department of Mathematics, University of Michigan, 530 Church Street, Ann Arbor, Michigan 48109;Department of Mathematics, University of British Columbia, 1984 Mathematics Road, Vancouver, British Columbia, Canada V6T 1Z2;Department of Operations Research and Financial Engineering, Princeton University, Princeton, New Jersey 08544
Venue:
Mathematics of Operations Research
Year:
2006

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Abstract

We study the effect of investor inertia on stock price fluctuations with a market microstructure model comprising many small investors who are inactive most of the time. It turns out that semi-Markov processes are tailor made for modelling inert investors. With a suitable scaling, we show that when the price is driven by the market imbalance, the log price process is approximated by a process with long-range dependence and non-Gaussian returns distributions, driven by a fractional Brownian motion. Consequently, investor inertia may lead to arbitrage opportunities for sophisticated market participants. The mathematical contributions are a functional central limit theorem for stationary semi-Markov processes and approximation results for stochastic integrals of continuous semimartingales with respect to fractional Brownian motion.