Table of Links 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 Literature Review
Methodology 3.1 Model Assumption 3.2 Theorems and Model Discussion Result Analysis
Conclusion and References 1.3 FINITE DIFFERENCE Finite difference methods are numerical techniques used to approximate derivatives and solve differential equations by discretizing the domain into a grid of points. This approach is based on Taylor series expansions, where derivatives are expressed as combinations of function values at nearby points. Consider a function 𝑓(𝑥) and its derivative 𝑓′(𝑥) then finite difference approximation of the derivative at a point 𝑥𝑖 is given by: In this plot, the tangent line represents the true derivative 𝑓 ′ (𝑥0) , Meanwhile, the secant line represents the finite difference approximation. As the step size ℎ decreases, the secant line becomes closer to the tangent line, demonstrating the convergence of the finite difference approximation to the true derivative as ℎ approaches zero. The application of finite difference methods in option pricing allows for more accurate simulations of market dynamics and estimation of option prices. These methods enable practitioners to account for various factors affecting option values, such as changes in asset prices, volatility, and interest rates. Finite difference methods offer flexibility and scalability, making them suitable for pricing various types of options and constructing hedging strategies. Despite their computational complexity, advancements in numerical algorithms and computing power have made finite difference methods increasingly accessible and efficient for option pricing applications [4]. Authors:
(1) Agni Rakshit, Department of Mathematics, National Institute of Technology, Durgapur, Durgapur, India (spiritualagnimath.statml@gmail.com);
(2) Gautam Bandyopadhyay, Department of Management Studies, National Institute of Technology, Durgapur, Durgapur, India (gbandyopadhyay.dms@nitdgp.ac.in);
(3) Tanujit Chakraborty, Department of Science and Engineering & Sorbonne Center for AI, Sorbonne University, Abu Dhabi, United Arab Emirates (tanujit.chakraborty@sorbonne.ae). This paper is available on arxiv under CC by 4.0 Deed (Attribution 4.0 International) license. Table of Links Abstract and 1. Introduction Abstract and 1. Introduction Abstract and 1. Introduction 1.1 Option Pricing 1.1 Option Pricing 1.2 Asymptotic Notation (Big O) 1.2 Asymptotic Notation (Big O) 1.3 Finite Difference 1.3 Finite Difference 1.4 The Black-Schole Model 1.4 The Black-Schole Model 1.5 Monte Carlo Simulation and Variance Reduction Techniques 1.5 Monte Carlo Simulation and Variance Reduction Techniques 1.6 Our Contribution 1.6 Our Contribution Literature Review Methodology Literature Review Literature Review Literature Review Methodology Methodology 3.1 Model Assumption 3.1 Model Assumption 3.2 Theorems and Model Discussion 3.2 Theorems and Model Discussion Result Analysis Conclusion and References Result Analysis Result Analysis Result Analysis Conclusion and References Conclusion and References Conclusion and References 1.3 FINITE DIFFERENCE Finite difference methods are numerical techniques used to approximate derivatives and solve differential equations by discretizing the domain into a grid of points. This approach is based on Taylor series expansions, where derivatives are expressed as combinations of function values at nearby points. Consider a function 𝑓(𝑥) and its derivative 𝑓′(𝑥) then finite difference approximation of the derivative at a point 𝑥𝑖 is given by: In this plot, the tangent line represents the true derivative 𝑓 ′ (𝑥0) , Meanwhile, the secant line represents the finite difference approximation. As the step size ℎ decreases, the secant line becomes closer to the tangent line, demonstrating the convergence of the finite difference approximation to the true derivative as ℎ approaches zero. The application of finite difference methods in option pricing allows for more accurate simulations of market dynamics and estimation of option prices. These methods enable practitioners to account for various factors affecting option values, such as changes in asset prices, volatility, and interest rates. Finite difference methods offer flexibility and scalability, making them suitable for pricing various types of options and constructing hedging strategies. Despite their computational complexity, advancements in numerical algorithms and computing power have made finite difference methods increasingly accessible and efficient for option pricing applications [4]. Authors: (1) Agni Rakshit, Department of Mathematics, National Institute of Technology, Durgapur, Durgapur, India (spiritualagnimath.statml@gmail.com); (2) Gautam Bandyopadhyay, Department of Management Studies, National Institute of Technology, Durgapur, Durgapur, India (gbandyopadhyay.dms@nitdgp.ac.in); (3) Tanujit Chakraborty, Department of Science and Engineering & Sorbonne Center for AI, Sorbonne University, Abu Dhabi, United Arab Emirates (tanujit.chakraborty@sorbonne.ae). Authors: Authors: (1) Agni Rakshit, Department of Mathematics, National Institute of Technology, Durgapur, Durgapur, India (spiritualagnimath.statml@gmail.com); (2) Gautam Bandyopadhyay, Department of Management Studies, National Institute of Technology, Durgapur, Durgapur, India (gbandyopadhyay.dms@nitdgp.ac.in); (3) Tanujit Chakraborty, Department of Science and Engineering & Sorbonne Center for AI, Sorbonne University, Abu Dhabi, United Arab Emirates (tanujit.chakraborty@sorbonne.ae). This paper is available on arxiv under CC by 4.0 Deed (Attribution 4.0 International) license. This paper is available on arxiv under CC by 4.0 Deed (Attribution 4.0 International) license. available on arxiv available on arxiv

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Finite Difference Methods: A Numerical Approach to Option Pricing and Derivatives

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