This book introduces Uncertain Volatility Models in mathematical finance and their computer implementation for portfolios of vanilla, barrier and American options in equity and FX markets. Uncertain Volatility Models place subjective constraints such as upper and lower bounds on volatility and evaluate option portfolios under worst- and best-case scenarios. This book is for graduate students, researchers and practitioners who wish to study advanced aspects of volatility risk in portfolios of vanilla and exotic options. The accompanying CD contains the source code of a C++ implementation of the algorithms presented in the book.