Exploring the Role of Brownian Motion in Financial Modeling: A Stochastic Approach to the Black-Scholes Model for European Call Options

Mehul Zawar

2025

Abstract

Stochastic processes, particularly Brownian motion, have become foundational tools in financial modeling, enabling the development of more accurate and insightful representations of market behavior. This paper delves into the mathematical framework behind stochastic differential equations (SDEs) and their critical role in the Black-Scholes model, specifically focusing on its application to European call options. We explore the influence of key parameters, such as stock drift, volatility, and risk-free interest rate, on option pricing by incorporating Brownian motion (Wiener processes) into the model. Through this exploration, we provide a detailed analysis of how these stochastic components shape the dynamics of stock prices and the option's value over time. The stability of the Black-Scholes model is evaluated under various boundary conditions, revealing its robustness in financial modeling. However, limitations of the Black-Scholes approach, including assumptions regarding constant volatility and market efficiency, are discussed, and potential improvements are suggested. This paper underscores the significance of stochastic integration methods, including the Ito and Stratonovich calculus, in refining the modeling of financial systems, thereby offering a comprehensive understanding of the Black-Scholes framework's applicability and areas for enhancement.

Paper Citation

in Harvard Style

Zawar M. (2025). Exploring the Role of Brownian Motion in Financial Modeling: A Stochastic Approach to the Black-Scholes Model for European Call Options. In Proceedings of the 7th International Conference on Finance, Economics, Management and IT Business - Volume 1: FEMIB; ISBN 978-989-758-748-1, SciTePress, pages 90-104. DOI: 10.5220/0013446300003956

in Bibtex Style

@conference{femib25,
author={Mehul Zawar},
title={Exploring the Role of Brownian Motion in Financial Modeling: A Stochastic Approach to the Black-Scholes Model for European Call Options},
booktitle={Proceedings of the 7th International Conference on Finance, Economics, Management and IT Business - Volume 1: FEMIB},
year={2025},
pages={90-104},
publisher={SciTePress},
organization={INSTICC},
doi={10.5220/0013446300003956},
isbn={978-989-758-748-1},
}

in EndNote Style

TY - CONF

JO - Proceedings of the 7th International Conference on Finance, Economics, Management and IT Business - Volume 1: FEMIB
TI - Exploring the Role of Brownian Motion in Financial Modeling: A Stochastic Approach to the Black-Scholes Model for European Call Options
SN - 978-989-758-748-1
AU - Zawar M.
PY - 2025
SP - 90
EP - 104
DO - 10.5220/0013446300003956
PB - SciTePress