Fundamental concepts such as missing arbitrage, fair price, complete market, and martingale are defined using a market with a risk-free bond and a stock. Subsequently, the transition to the continuous-time model derives the Black-Scholes formula for options and introduces the factors for practical implementation. Methods of stochastic analysis, such as the Ito formula, are derived, and the classical approach according to Black-Scholes is presented using stochastic differential equations. The approach through martingale theory is a subject necessary for the valuation of complex options (American and exotic). The final chapter examines the fundamentals of interest rate term structure models. In all sections, numerical methods are provided, which are implemented with programs for practical illustration.