A Langevin Approach to Stock Market Fluctuations and Crashes

TL;DR

This paper introduces a nonlinear Langevin model for stock market fluctuations and crashes, highlighting feedback effects, risk aversion, and the rare, exponential probability of crashes, with predictions on crash dynamics and correlations.

Contribution

It presents a novel nonlinear Langevin equation model capturing feedback and risk aversion effects, explaining crashes as rare activated events with specific crash shape predictions.

Findings

Crashes are rare, exponentially small probability events.

Price fall during crashes is predicted to be logarithmic.

Normal market behavior shows non-trivial correlations across time scales.

Abstract

We propose a non linear Langevin equation as a model for stock market fluctuations and crashes. This equation is based on an identification of the different processes influencing the demand and supply, and their mathematical transcription. We emphasize the importance of feedback effects of price variations onto themselves. Risk aversion, in particular, leads to an up-down symmetry breaking term which is responsible for crashes, where `panic' is self reinforcing. It is also responsible for the sudden collapse of speculative bubbles. Interestingly, these crashes appear as rare, `activated' events, and have an exponentially small probability of occurence. We predict that the shape of the falldown of the price during a crash should be logarithmic. The normal regime, where the stock price exhibits behavior similar to that of a random walk, however reveals non trivial correlations on different time scales, in particular on the time scale over which operators perceive a change of trend.