Stochastic finance and financial engineering have been rapidly expanding fields of science over the past four decades, mainly due to the success of sophisticated quantitative methodologies in helping professionals manage financial risks. In recent years, we have witnessed a tremendous acceleration in research efforts aimed at better comprehending, modeling and hedging this kind of risk.These two volumes aim to provide a foundation course on applied stochastic finance. They are designed for three groups of readers: firstly, students of various backgrounds seeking a core knowledge on the subject of stochastic finance; secondly financial analysts and practitioners in the investment, banking and insurance industries; and finally other professionals who are interested in learning advanced mathematical and stochastic methods, which are basic knowledge in many areas, through finance.In Volume 2 we study continuous time models by presenting the necessary material from continuous martingales, measure theory and stochastic differential equations as models for various assets, such as the Wiener process, Brownian motion, etc. We then build, with many examples and intuitive explanations, the necessary stochastic analysis background i.e. Itô's lemma, stochastic integration, Girsanovís theorem, etc. The book then guides the reader into the pricing of vanilla options in continuous time i.e. the continuous time models of Black and Scholes, followed by interest rate models and the models of Heath-Jarrow-Morton and the forward Libor model. The final part of the book presents the pricing of credit derivatives.