What Do You Know About the Black-Scholes Option Pricing Model?

:::info Authors:

(1) Agni Rakshit, Department of Mathematics, National Institute of Technology, Durgapur, Durgapur, India ([email protected]);

(2) Gautam Bandyopadhyay, Department of Management Studies, National Institute of Technology, Durgapur, Durgapur, India ([email protected]);

(3) Tanujit Chakraborty, Department of Science and Engineering & Sorbonne Center for AI, Sorbonne University, Abu Dhabi, United Arab Emirates ([email protected]).

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Abstract and 1. Introduction

1.1 Option Pricing

1.2 Asymptotic Notation (Big O)

1.3 Finite Difference

1.4 The Black-Schole Model

1.5 Monte Carlo Simulation and Variance Reduction Techniques

1.6 Our Contribution

2. Literature Review
3. Methodology

3.1 Model Assumption

3.2 Theorems and Model Discussion

4. Result Analysis
5. Conclusion and References

\ Abstract— The Black-Scholes option pricing model remains a cornerstone in financial mathematics, yet its application is often challenged by the need for accurate hedging strategies, especially in dynamic market environments. This paper presents a rigorous analysis of hedge errors within the BlackScholes framework, focusing on the efficacy of finite difference techniques in calculating option sensitivities. Employing an asymptotic approach, we investigate the behavior of hedge errors under various market conditions, emphasizing the implications for risk management and portfolio optimization. Through theoretical analysis and numerical simulations, we demonstrate the effectiveness of our proposed method in reducing hedge errors and enhancing the robustness of option pricing models. Our findings provide valuable insights into improving the accuracy of hedging strategies and advancing the understanding of option pricing in financial markets.

The Black-Scholes option pricing model, a landmark in financial mathematics, revolutionized the way financial derivatives are valued and traded. However, despite its widespread adoption and utility, the model is not without limitations. One critical aspect that has garnered attention in recent years is the accuracy of hedging strategies based on the Black-Scholes framework. While the model provides a robust theoretical foundation for option pricing, the real-world application of its hedging strategies often falls short of expectations due to various factors such as transaction costs, market frictions, and model assumptions.

\ This research endeavors to address the issue of hedge errors within the Black-Scholes framework by employing an innovative dual approach. First, leveraging an asymptotic analysis of finite difference methods, we aim to elucidate the behavior of hedge errors in different market conditions and under varying model parameters. By exploring the asymptotic properties of finite difference approximations, we seek to provide a deeper understanding of the limitations inherent in the Black-Scholes model and its impact on hedging effectiveness.

\ Furthermore, recognizing the importance of practical implementation, this study integrates Monte Carlo simulation techniques for variance reduction in option pricing and hedging. Monte Carlo simulation offers a powerful computational tool for pricing complex derivatives and assessing their associated risks. By applying variance reduction techniques within the Monte Carlo framework, we aim to mitigate the impact of stochastic noise and improve the accuracy of hedging strategies, thereby enhancing risk management practices in financial markets.

\ The dual focus of this research on both theoretical analysis and practical implementation underscores its relevance to both academia and industry. By combining insights from asymptotic analysis and Monte Carlo simulation, we seek to bridge the gap between theoretical models and real-world applications, offering valuable insights for practitioners, financial institutions, and policymakers alike. Ultimately, this research contributes to the ongoing dialogue on the refinement and enhancement of quantitative models in finance, with implications for risk management, derivative pricing, and investment strategies in dynamic market environments.

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:::info This paper is available on arxiv under CC by 4.0 Deed (Attribution 4.0 International) license.

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