The main idea behind this work was to model the price of an option, a financial instrument that often serves investors as a method of risk protection in the market. The goal was to establish a model using partial differential equations and then, if possible, derive a closed form solution for the option price at time t, with underlying asset price being S. In the first chapter, we defined some of the key concepts, both financial and mathematical, that were necessary throughout the paper. We learned more about options, introduced concepts such as arbitrage and Brownian motion, and, in the end, finalized the chapter with an introduction to partial differential equations. The second chapter was dedicated specifically to European options. After introducing few basic assumptions, we derived one of the most famous equations in the world of finance, the Black-Scholes equation, which we then solved to obtain the formula for the price of an European option: the Black-Scholes formula. Following that, we made a few changes to the basic assumptions to obtain a more general model, applicable in the real world, and concluded with deriving models for several types of exotic options. The work was finalized with the third chapter, where we exclusively dealt with American options. As the concept of payout for American options is slightly more complicated, a closed-form solution for the option price generally does not exist. We still managed to derive a specific formula for perpetual American options, which are somewhat simpler, while for the general case, we developed a model and proved a formula for the decomposition of the American option price that consists of the European option price and an additional premium.