  In this paper we use Gaussian Process (GP) regression to propose a novel
approach for predicting volatility of financial returns by forecasting the
envelopes of the time series. We provide a direct comparison of their
performance to traditional approaches such as GARCH. We compare the forecasting
power of three approaches: GP regression on the absolute and squared returns;
regression on the envelope of the returns and the absolute returns; and
regression on the envelope of the negative and positive returns separately. We
use a maximum a posteriori estimate with a Gaussian prior to determine our
hyperparameters. We also test the effect of hyperparameter updating at each
forecasting step. We use our approaches to forecast out-of-sample volatility of
four currency pairs over a 2 year period, at half-hourly intervals. From three
kernels, we select the kernel giving the best performance for our data. We use
two published accuracy measures and four statistical loss functions to evaluate
the forecasting ability of GARCH vs GPs. In mean squared error the GP's perform
20% better than a random walk model, and 50% better than GARCH for the same
data.
