Sloan Research Grant 2020-2021:

How to Define and Detect Regime Shifts

The analysis of market mechanisms has a long history. In 1905, Louis Bachelier established that prices on financial markets follow a random walk: neither buyers nor sellers can systematically make a profit. It was in this vein that Paul Samuelson built a definition in response to an empirical study that revealed the random character of stock prices, providing the mathematical foundation for what has become known as the efficient-market hypothesis.

The early work of Benoît Mandelbrot showed that an efficient market situation with uncorrelated returns may not be observed and that long-range correlations and heavy-tailed return distributions may be typical. This analysis contrasts with the seminal work of Fischer Black and Myron Scholes. The strong impact of their work can partly be explained by the explicit approaches it provides for pricing and hedging. The Black-and-Scholes theory and associated pricing models assume a situation in which future returns are uncorrelated with respect to past information. Is it possible to reconcile such contradictory theories?

The project, supported by the Alfred P. Sloan Foundation, addresses the question of how deviations from an idealized efficient market situation can be understood from both economic and mathematical viewpoints. Appropriate tools are being developed for analysing historical data, with the aim of detecting such inefficiencies and developing market models that take into account such information. The project is interdisciplinary in nature. It is motivated and driven by data analysis and seeks to understand, from an economic perspective, the resulting observations. Sophisticated mathematical and probabilistic modelling is being developed to capture the essence of such markets. The data used to date come from the period following the establishment of the Black–Scholes framework; data from the pre-Black–Scholes period are also under examination. From a statistical viewpoint, modelling and analysis of locally stationary processes via time-frequency analysis are central ingredients. The modelling is carried out in terms of multifractal stochastic processes, where both a time-varying “memory effect” of returns and local market volatility can be incorporated. From an economic perspective, a method is being developed to understand what mathematicians call “intermittent” markets, mainly quiet periods that occasionally turn into periods of intense activity. The method focuses on characterizing such periods and on understanding how they can be correlated with specific economic conditions. How can these periods – for instance, in crude oil prices – be explained via collective behavior resulting from actions of small and large agents in the market place?