ON THE STUDY OF OPTION PRICING: FOCUS ON CALL OPTION

This study explores the methodology underlying the pricing of call options using continuous-time models. Beginning with the Wiener process as a foundational model, the research extends into stochastic calculus and the derivation of the Black-Scholes equation. The analysis delves into the fundamental properties of Brownian motion, the normal distribution, and stochastic differential equations to establish a rigorous mathematical framework for asset price movements.

A key focus is placed on the arbitrage argument, which ensures that financial markets remain free from riskless profit opportunities. The no-arbitrage condition is then used to derive the Black-Scholes partial differential equation, which governs the pricing of options. Through a series of transformations, the equation is reduced to the heat equation, allowing for an analytical solution to be obtained. Finally, the study applies this methodology to derive explicit pricing formulas for European call and put options, highlighting the impact of volatility, risk-free interest rates, and time to maturity on option values. By formalizing the theoretical framework with stochastic calculus and arbitrage pricing theory, this research provides a robust foundation for the application of call option models in financial engineering.