On some stochastic models in finance

This thesis examines the core concepts and practical implications of key options pricing models, focusing on the Cox-Ross-Rubinstein (CRR) model and the Black-Scholes model. The CRR model provides a framework for valuing options in a discrete-time setting, describing a detailed analysis of stock price fluctuations and the influence of different parameters on option valuations. On the other hands, the Black-Scholes model offers a continuous-time perspective, deriving the formula through the principles of stochastic calculus and the assumption of market efficiency. The study explores the mathematical foundations of both models, starting with the significance of Brownian motion and the development of the BlackScholes partial differential equation. A thorough investigation of the underlying assumptions reveals critical insights into market dynamics, volatility, and interest rates. Additionally, the practical applications of these models in contemporary financial markets and the risk managment fields are discussed. Included in the appendix are two Python codes that enhance the empirical aspect of this research. The first code illustrates the pricing of options using a binomial model across multiple steps, proving how option prices converge as the number of steps increases. The second code applies the Black-Scholes model, showcasing its utility in calculating option values based on relevant market parameters.