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1.2 Asymptotic Notation (Big O)

1.5 Monte Carlo Simulation and Variance Reduction Techniques

**Literature Review****Methodology**

3.2 Theorems and Model Discussion

The research findings have demonstrated that finite difference methods offer a powerful tool for approximating option prices and hedging parameters, especially in cases where closed-form solutions are unavailable or impractical. The asymptotic analysis conducted in this study has provided valuable insights into the convergence properties and accuracy of finite difference approximations, shedding light on the optimal choice of discretization schemes and grid sizes for different option contracts. Furthermore, the combination of finite difference methods and Monte Carlo simulation with variance reduction techniques offers a comprehensive approach to hedge error analysis in option pricing. By leveraging the strengths of both methodologies, financial practitioners can obtain more robust and reliable estimates of option prices and hedge parameters, thereby enhancing their risk management capabilities and decision-making processes.

It is important to acknowledge the limitations of this study and areas for future research. While the asymptotic analysis provides valuable theoretical insights, further empirical validation is warranted to assess the robustness of the findings across different market conditions and asset classes. Additionally, exploring alternative variance reduction techniques and incorporating more sophisticated models of market dynamics could yield further improvements in hedge error analysis and option pricing accuracy. this study has advanced our understanding of the factors influencing hedge errors and provided practical tools for mitigating their impact on derivative pricing and risk management. As financial markets continue to evolve, the insights gained from this research will remain valuable for academics, practitioners, and policymakers striving to enhance the efficiency and stability of global financial systems.

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**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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