Bayesian analysis of chaos: The joint return-volatility dynamical system

We use a novel Bayesian inference procedure for the Lyapunov exponent in the dynamical system of returns and their unobserved volatility. In the dynamical system, computation of largest Lyapunov exponent by tradi- tional methods is impossible as the stochastic nature has to be taken explicitly into account due to unobserved volatility. We apply the new techniques to daily stock return data for a group of six world countries, namely USA, UK, Switzerland, Netherlands, Germany and France, from 2003 to 2014 by means of Sequential Monte Carlo for Bayesian inference. The evidence points to the direction that there is indeed noisy chaos both before and after the recent financial crisis. However, when a much simpler model is examined where the interaction between returns and volatility is not taken into consideration jointly, the hypothesis of chaotic dynamics does not receive much support by the data (“neglected chaos”).