The course gives a modern overview of the main concepts of mathematical finance using discrete-time stochastic models. The course focuses on the Cox-Ross-Rubinstein (binomial) model. Topics include no-arbitrage pricing of financial derivatives, replication, hedging, self-financed portfolios, risk-neutral valuation, stopping times, and portfolio optimization. As the main application, European and American options in discrete time are studied, and the numerical algorithms for their pricing are presented.