A limit theorem for financial markets with inert investors

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.