This textbook provides an easily understandable introduction to modern financial mathematics and explains fundamental mathematical concepts of option pricing, portfolio optimization, and risk management. This includes pricing through arbitrage considerations, pricing American options by solving optimal stopping problems, determining optimal consumption and investment strategies, and expected value-variance portfolios. Current concepts of risk measurement such as Value at Risk and Expected Shortfall are also presented. A basic understanding of stochastic processes and optimization is sufficient for grasping the content, and numerous exercises with detailed solutions, along with three appendices, facilitate self-study.