Intended for graduate students in mathematics, this textbook is an introduction to probabilistic methods in finance that focuses on stochastic models in real time. It is based on courses taught by the authors at Humboldt U. and Technical U. in Germany. The core of the work is a dynamic arbitrage theory presented in the second section, but they first explain some of the main arguments in a more transparent one-period model. For the new edition they have simplified and clarified some of the material regarding robust representations of risk measures, arbitrage-free pricing of contingent claims, convergence to Black-Scholes prices, and stability under pasting with its connections to dynamically consistent coherent risk measures. They have also added several new sections discussing of law-invariant risk measures, concave distortions, and the relations between risk measures and Choquet integration.