Application of GARCH Models Generalized with Machine Learning and Quantum Statistical Physics Methodology in Exotic Financial Data and Beyond

With the recent development of econophysics, information theory, and correlated system mechanics, many random and stochastic processes in complex systems can be much better described, such as the financial market, the infectious disease dynamics, the social network construction, and others. The generalized autoregressive conditional heteroskedasticity (GARCH) model is a well-established method in analyzing and predicting the volatility of a fluctuating system. However, within the traditional GARCH scenario, the parameter fitting and optimization is mostly limited to the linear space.

In the traditional GARCH model regime, the absolute magnitudes of multiple historical volatilities and increments are taken as the elements of the input vector for a target value prediction. In our present work, we will further use the concepts in quantum statistical physics, such as the von Neumann entropy, the canonical ensemble, and the quantum fluctuation to build and extract new dimensions of data as the elements of our input vector. Further, to conquer the limitation of linear-only predictions, we will introduce statistically significant non-linearity by implementing multiple machine learning methods that have been developed in the recent decade, such as the random forest machine, the neural network and the deep learning method. The prediction and the description of volatility in time series can be notably enhanced using our newly improved methodology.

Our new approach is then tested in simulated data using Monte Carlo and bootstrap methods, as well as real data from the secondary financial markets, the disease infection records, and climate change patterns.