Modelagem estocástica de ativos e derivativos financeiros: do movimento browniano à equação de Black–Scholes

Abstract

This work aims to discuss the main theoretical developments in the field of financial asset behavior modeling. It begins by presenting Bachelier’s model for Brownian motion, then proceeds to take the continuous limit and derive the Fokker–Planck equation. A derivation of the Central Limit Theorem is also included. Subsequently, stochastic calculus is introduced through its distinctions from deterministic calculus, with particular emphasis on Itô’s Lemma. The Black–Scholes equation is presented using different theoretical and economic arguments, and its solution is provided. Some limitations of the models are discussed, and subsequent developments are indicated. The study concludes that these models hold importance not only from a historical perspective but also from a methodological one, and they remain central to the field, with increasingly refined modifications aimed at understanding financial market phenomena.