This research studies option pricing using the Black-Scholes model and its relationship with market volatility. The study begins by establishing the mathematical foundations necessary for understanding financial models, including random number generation, Brownian motion, and stochastic differential equations. Pseudo-random numbers are introduced through the Linear Congruential Generator, while exponential and normal random variables are used to simulate financial processes. The Itô formula and Itô integral provide the analytical framework for handling stochastic systems influenced by uncertainty and random fluctuations. Building on these foundations, the research presents the Black-Scholes model as a central tool in financial mathematics for pricing European call and put options. The model is derived through geometric Brownian motion and expressed in both stochastic and partial differential equation forms. Key variables including stock price, strike price, risk-free interest rate, time to expiration, and volatility are examined within the pricing formulas. A significant focus is placed on volatility as a critical measure of market risk. The study distinguishes between implied volatility, derived from current derivative prices, and historical volatility, estimated from past price data. The difference between low and high volatility and their applications in investment decisions and risk management are also discussed. The limitations of the Black-Scholes assumption of constant volatility are highlighted, as real markets exhibit dynamic volatility behavior. Extensions of the classical framework incorporating transaction costs and nonlinear volatility terms are reviewed to address these limitations. Finally, the research presents a practical case study applying the Black-Scholes model to a real-world hedging problem for a bakery in Sulaymaniyah, demonstrating how the model can be used as an effective financial planning tool in emerging markets such as Kurdistan and Iraq.
