Exotic options are one of the subtypes of derivatives that were created by modifying the standard features of European and American options. There are numerous different types of exotic options. In this paper we deal with those that are most commonly used. The application of stochastic calculus in quantitative finance, and the development of the Black-Scholes-Merton formula have greatly contributed to the evolution of the markets for derivatives and for exotic options. In this paper, we explain the basics of stochastic calculus and derive the Black-Scholes-Merton formula. We also confirm that the mathematical theory fits in perfectly within financial modeling. When issuing options, we are faced with the problem of instrument pricing. Such pricing depends upon the price of the underlying asset on which the instrument is written. At the same time, we also discuss the problem of no arbitrage in financial markets. The aim of this diploma thesis is to describe in detail a method for the pricing of lookback and barrier options. Stock prices are modeled using the Geometric Brownian Motion model for stock prices. In both cases, the discounted prices of options are martingales with a risk-neutral probability. We calculate the formulas for the pricing of up-and-out barrier options, and for the floating strike lookback options using the expected value of option payoff.