There are some standard models for forecasting volatility (e.g., GARCH) and for forecasting returns (e.g., factor models). What kind of standard models exist for forecasting future correlation between securities?
One common "model" is to assume the correlation to be constant, such as in a CCC-MVGARCH model. If you want a review of different multivariate GARCH models, you could look at:
Silvennoinen and Täräsvirta 2009, Multivariate Garch models, in Handbook of financial time series.