This textbook provides the necessary techniques from financial mathematics and stochastic analysis for the valuation of more complex financial products and strategies. The author discusses how to make use of mathematical methods to analyse volatilities in capital markets. Furthermore, he illustrates how to apply and extend the Black-Scholes theory to several fields in finance. In the final section of the book, the author introduces the readers to the fundamentals of stochastic analysis and presents examples of applications. This book builds on the previous volume of the author's trilogy on quantitative finance. The aim of the second volume is to present and discuss more complex and advanced techniques of modern financial mathematics in a way that is intuitive and easy to follow. As in the previous volume, the author provides financial mathematicians with insights into practical requirements when applying financial mathematical techniques in the real world.