Local and Terminal Volatility of Equity in a Hybrid Model with Hull White Interest Rates

We consider a hybrid model for stocks and interest rates as it is proposed by GDV (Gesamtverband der Deutschen Versicherungswirtschaft) to assign market consistent values to the technical provisions of german life insurance companies. In this model, stock prices are modeled with a Black Scholes model with time-varying parameters, and interest rates are modeled with a Hull White short rate model.We find that local and terminal volatility of equity differ in this model and that the terminal volatility of equity is considerable higher than the local volatility. This raises the question, which of both volatilities is the right reference to calibrate the model. We argue that calibration should be done with respect to local volatility rather than terminal volatility, at least for the application we have in mind