Brownian Motion & The Stochastic Behaviour of Stocks

TL;DR

This paper explores the origins and mathematical modeling of Brownian motion, its application to stock price fluctuations, and evaluates the model's accuracy through backtesting with Apple stock data.

Contribution

It provides a comprehensive overview of Brownian motion's development and applies stochastic differential equations to model stock prices, including empirical validation.

Findings

The model accurately captures stock price fluctuations.

Backtesting shows a high correlation coefficient with actual stock data.

The paper demonstrates the relevance of Itô calculus in financial modeling.

Abstract

We begin by exploring the intuition of Brownian motion by explaining its birth through the observations of Robert Brown and later through Bachelier's work on its applications to the financial market and finally its rigorous and concretized form proposed by Norbert Wiener. The aforementioned motivates a stochastic differential equation to model the future price fluctuations of a stock traded wherein It\^o integration is prominent and consequently expanded upon. The final part of this paper focuses on the accuracy of the model by backtesting it with Apple stock and deriving a correlation coefficient.