Estimating Conditional Volatility with Neural Networks

It is well known that one of the obstacles to e ective forecasting of exchange rates is het eroscedasticity input dependent conditional variance The autoregressive conditional het eroscedastic ARCH model and its variants have been used to estimate a time dependent variance for many nancial time series However such models are essentially linear in form and we can ask whether a non linear model for variance can improve forecasting results just as non linear models such as neural networks for the mean have done In this paper we consider two neural network models for variance estimation Mixture Density Networks combine a Multi Layer Perceptron MLP and a mixture model to estimate the conditional data density They are trained using a maximum likelihood approach However it is known that maximum likelihood estimates are biased and lead to a systematic under estimate of variance More recently a Bayesian approach approach to parameter estimation in such models has been developed that shows promise in removing the maximum likelihood bias However up to now this model has not been used for time series prediction Here we compare these algorithms with two other models to provide benchmark results a linear ARIMA model and a conventional neural network trained with a sum of squares error function In both these cases the model estimates the conditional mean of the time series with a constant variance noise model This comparison is carried out on daily exchange rate data for ve currencies In this paper we are concerned with models that predict the conditional variance for the next time step The conditional variance can be used to provide error bars also known as prediction intervals in the regression literature around the conditional mean When the size of the error bars increases then the value of the next forecast is less certain This is less useful for options pricing than longer term variance forecasts but the information can be incorporated into trading rules For example the size of error bars is a measure of how likely the predicted price movement is likely to be accurate We are interested in comparing the generalisation performance of di erent models and use log likelihood on out of sample data with one step ahead prediction to compare results The rest of this paper is organised as follows In section the various di erent models that we employ are described and contrasted Section describes the methodology that