The purpose of this paper is to present a European call and put options in discrete time on the financial market. The first chapter defines the basic terms related to the financial market and financial instruments, which are the main trading tools on the financial market. Furthermore, the basic division of financial instruments into basic and derivative financial instruments is explained. The second chapter deals with the general characteristics of the most flexible type of derivatives, which are options. The following chapter gives a brief overview of different types of options, which are classified into six groups according to certain characteristics. The following options are clarified: call, put, European, American, commodity, financial, covered, uncovered, short-term, long-term, standard and exotic. The fourth chapter presents the concept of discrete time in more detail and describes basic concepts of discrete probability space. The basic assumptions of mathematical models of financial markets, single period and multiperiod financial market models are also presented in this chapter. The last chapter is devoted to a European call and put options, their values at maturity and examples of successful and unsuccessful European call options. The single period binary model of the financial market, the prices of risk-free financial instruments and the prices of financial instruments with risk are also explained. Finally, theorem when the single period binary model does not allow arbitrage and theorem on the non-arbitrage price of ECO and EPO on the binary model are presented, as well as proofs and examples related to them.