This book builds on previous and current research by the four authors, including results introduced in the book Derivatives in Financial Markets with Stochastic Volatility. Recent research demonstrates that the introduction of two time scales in volatility, a fast and a slow, is needed and efficient for capturing the main features of the observed term structure of implied volatility. For practitioners, the modeling of the implied volatility consistent with no-arbitrage is crucial. The authors present an approach to this problem which consists in combining singular and regular perturbation techniques. The book will serve a dual purpose: present ‘off the shelf’ formulas and calibration tools for practitioners, and introduce, explain and develop the mathematical framework to handle the multi-scale asymptotics. Detailed presentation of the analysis as well as a thorough insight into the modeling approach makes this an excellent text for a second level graduate course in financial and applied mathematics.