This paper proposes a fast, efficient and robust way to compute the prices of compound options such as the popular call-on-call options within the context of multiscale stochastic volatility models. Recent empirical studies indicate the existence of at least two characteristic time scales for volatility factors including one highly persistent factor and one quickly mean-reverting factor. Here we introduce one relatively slow time scale and another relatively fast time scale, with respect to typical time to maturities, into our multiscale stochastic volatility models. Using a combination of singular and regular perturbation techniques we approximate the price of a compound option by the price under constant volatility of the corresponding option corrected in order to take into account the effects of stochastic volatility. We provide formulas for these corrections, which involve universal parameters calibrated to the term structure of implied volatility. Our method is not model sensitive, and the calibration and computational efforts are drastically reduced compared with solving fully specified models.