Before revision:  For \rho < 0, our approach still works if the drift of the volatility takes a specific functional form and there is no local volatility component, and our results include the SABR model for \beta=1, \rho \le 0.
After revision:  If the correlation \rho < 0, our approach still works if the drift of the volatility takes a specific functional form and there is no local volatility component, and our results include the SABR model for \beta=1, \rho \le 0.