This book presents a factor-based model of the stochastic evolution of the implied volatility surface. The model allows for the integrated and consistent pricing and hedging, risk management, and trading of equity index derivatives as well as volatility derivatives. In the first part, the book develops a unifying theory for the analysis of contingent claims under both the real-world measure and the risk-neutral measure in an environment of stochastic implied volatility. On the basis of transaction data, the second part of the book provides extensive statistical analyses on the dynamics of the implied volatility surface of German DAX options and proposes a four-factor model to describe its evolution. The model is validated and tested on market data. The final part deals with potential applications of the model in the fields of exotic option pricing, value at risk, and volatility trading.