In the book by \textit{J.-P. Fouqué}, \textit{G. Papanicolaou} and \textit{K. R. Sircar} [Derivatives in financial markets with stochastic volatility. Cambridge: University Press (2000; Zbl 0954.91025)] a class of models was considered where volatility is a mean-reverting diffusion with an intrinsic fast time scale. In the present paper a more general class of so-called multiscale stochastic volatility models is introduced. Here, the price of the underlying is the solution of the stochastic differential equation \[ dX_t= \mu X_t dt+ \sigma_t X_t dW^{(0)}_t, \] where the stochastic volatility is of the form \(\sigma_t= f(Y_t, Z_t)\) (\(f\) being a smooth positive function that is bounded and bounded away from zero). The first factor driving \(\sigma_t\) (a fast mean-reverting diffusion process) is given by \[ dY_t= {1\over\varepsilon} (m- Y_t)\,dt+ {\nu\sqrt{2}\over \sqrt{\varepsilon}}\,dW^{(1)}_t\qquad (\varepsilon> 0,\;\nu> 0) \] where \(W^{(0)}\), \(W^{(1)}\) are standard Brownian motions such that \(d\langle W^{(0)}, W^{(1)}\rangle_t= \rho_1 dt\). The correlation coefficient \(\rho_1\) is constant and satisfies \(|\rho_1| 0) \] (\(W^{(2)}\) being another standard Brownian motion) where the coefficients \(c(z)\), \(g(z)\) are smooth and at most linearly growing at infinity. The correlation structure is given by \[ \begin{pmatrix} W^{(0)}_t\\ W^{(1)}_t\\ W^{(2)}_t\end{pmatrix}= \begin{pmatrix} 1 & 0 & 0\\ \rho_1 & \sqrt{1- \rho^2_1} & 0\\ \rho_2 &\widetilde\rho_{12} & \sqrt{1- \rho^2_2}- \widetilde\rho^2_{12}\end{pmatrix}\,W_t \] (\(W\) being a standard three-dimensional Brownian motion). Here, \(\rho_1\), \(\rho_2\), \(\widetilde\rho_{12}\) are constants satisfying \(|\widetilde\rho_1|< 1\) and \(\rho^2_2+ \widetilde\rho^2_{12}< 1\). Combining a singular perturbation expansion with respect to the fast scale with a regular perturbation expansion with respect to the slow scale leads again to a leading order term which is the Black-Scholes price with a constant effective volatility. The first correction is made up of two parts which derive, respectively, from the fast and the slow factors and involves four parameters which still can be easily calibrated from the implied volatility surface. Using options data it is demonstrated that the addition of the slow factor to the model greatly improves the fit to the longer maturities.