Precificação de opções de ações: modelagem do processo estocástico que gera os preços de opções européias

Abstract

Option pricing plays a fundamental role in financial markets, being essential for strategic decision-making by investors (both institutional and individual) and risk managers. In this context, the volatility of the underlying asset is one of the main parameters of the Black-Scholes model, which is widely used for pricing European options. However, volatility is not directly observable, requiring appropriate estimation techniques. This study focuses on demonstrating the formulation of this model—from the economic assumptions about the functioning of financial markets to the mathematical derivations (using Fourier Transform, Diffusion Equation, and Chapman-Kolmogorov Equation)—as well as on the application of methods to calculate an asset’s volatility. Models such as Geometric Brownian Motion and the Ornstein-Uhlenbeck process are explored to estimate the stochastic process that drives asset prices. To achieve this, an algorithm based on maximum likelihood estimation using the transition density of the process is employed.