Nikolay Nikolaev, Goldsmiths College, University of London

Dynamic Nonlinear GARCH Modelling with Neural Networks

NN-GARCH

Neural Network GARCH Modelling

The estimatiion of the changing variance (volatility) in time series of returns impacts the pricing of financial instruments, and it is a key concept in portfolio management, option pricing and financial market regulation. Popular tools for capturing the variance of the conditional distribution of returns are the Generalized Autoregressive Conditional Heteroscedastic (GARCH) models [Bollerslev, 1986], [Engle, 1982].

We elaborate dynamic neural network representations of the GARCH model (NN-GARCH) which allow us to use explicitly the time dependencies among the data in the estimation process (more precisely the gradient information is propagated through time). Our research maps GARCH to autoregressive with exogenous inputs (NARX) neural networks [Chen, Billings and Grant, 1990] in order to learn the time-dependent evolution of returns through specialized dynamic training algorithms, such as the Gauss-Newton algorithm with temporal derivatives, and the Broyden–Fletcher–Goldfarb–Shanno (BFGS) algorithm with temporal derivatives. These training algorithms are also modified to use not only Gaussian but also heavy-tailed return distributions.

Experimental investigations show that the dynamic NN-GARCH models have the capacity to capture simultaneously the main features of returns, namely their excess kurtosis, small autocorrelation of the squared returns and high persistence of the squared returns [Carnero, Pena and Ruiz, 2004]. Currently under investigation is a family of enhanced versions including NN-EGARCH, NN-TGARCH, and NN-PGARCH models

MDN-GARCH

Mixture Density Nonlinear GARCH Networks

Mixtures of nonlinear GARCH neural networks (MDN-GARCH) have been designed for accurate estimation of the volatility in non-stationary financial time series. The MDN-GARCH are mixtures of (normal) dynamic networks whose weighted average approximates the conditional distributions of returns. More precisely, pairs of expert networks specialize in modelling the moments of the conditional distributions of returns. Since dynamic networks are used, the time-dependent statistical features of returns, such as mean, variance, skewness and kurtosis, are predicted online with the arrival of the data.

We implemented first-order and second-order dynamic training algorithms for finding the mixture density network parameters. A distinguishing feature of MDN-GARCH networks is that all the parameters: mean level, persistence and moving average coefficients, are computed with temporal training algorithms (using analytical formula) which capture explicitly the dynamics of the data during training.

Preliminary experiments show that such nonlinear MDN-GARCH models produce results with better statistical characteristics (lower skewness and higher kurtosis) and economic performance (out-of-sample prediction of future directional changes) than conventional linear GARCH trained by MCMC sampling and maximum-likelihood estimation. The current research studies the relation between nonlinear MDN-GARCH and linear normal-mixture GARCH models (NM-GARCH) [Haas, Mittnik and Paoella, 2004], [Ausin and Galeano, 2007] as well as between MDN-GARCH and the similar Markov-switching GARCH models (MRS-GARCH) [Hamilton and Susmel, 1994], [Gray, 1996] which are popular contemporary tools for volatility inference.