Stochastic Volatility Model with Time-dependent Skew

A formula is derived for the 'effective' skew in a stochastic volatility model with a time-dependent local volatility function. The formula relates the total amount of skew generated by the model over a given time period to the time-dependent slope of the instantaneous local volatility function. A new 'effective' volatility approximation is also derived. The utility of the formulas is demonstrated by building a forward Libor model that can be calibrated to swaption smiles that vary across the swaption grid.