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Best likelihood forecast of volatility in class of linear functions

An explicit analytical solution of the problem of constructing the best linear predictor of a stationary stochastic process with the autocorrelation function representing a superposition of several exponents is reported. The proposed method is applied further for volatility forecasting in financial time series. We account for the well-established deviations of financial time series from the Gaussian random walk, such as an approximate scaling and heavy tails of the return distributions, long-ranged volatility-volatility correlations (volatility clustering) and return-volatility correlations (leverage effect). Parameters of the predictor function are determined numerically by fitting the 100+ years of daily price returns of the Dow Jones 30 Industrial Average. Connection of the proposed method to the popular autoregressive conditional heteroscedasticity models is discussed.