Revisiting the finite mixture of Gaussian distributions with application to futures markets

We present a new estimation method for Gaussian mixture modeling, namely, the kurtosis-controlled expectation-maximization (EM) algorithm, which overcomes the limitations of the usual estimation techniques via kurtosis control and kernel splitting. Our simulation study shows that the dynamic allocation of kernels according to the value of the total kurtosis measure makes the proposed kurtosis-controlled EM algorithm an efficient method for Gaussian mixture density estimation. This algorithm yielded considerable improvements over the classical EM algorithm. We then used the discrete Gaussian mixture framework to account for the observed thick-tailed distributions of futures returns and applied the kurtosis-controlled EM algorithm to estimate the distributions of real (agricultural, metal, and energy) and financial (stock index and currency) futures returns. We proved that this framework is perfectly adapted to capturing the departures from normality of the observed return distributions. Unlike in previous studies, we found that a two-component Gaussian mixture is too poor a model to accurately capture the distributional properties of returns. Similar results have been obtained for stocks, indexes, currencies, interest rates, and commodities. This has important implications in many financial studies using Gaussian mixtures to incorporate the thickness of the tails of the distributions in the computation of the value at risk or to infer implied risk-neutral densities from option prices, to name but a few.