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 2. LITERATURE REVIEW Classical methods, such as delta hedging, rely on continuous trading assumptions and deterministic models, often failing to capture the complexities of real market dynamics. Researchers have explored alternative techniques to improve hedge effectiveness under realistic conditions. Finite difference methods have gained traction for their flexibility and ability to accommodate discrete trading. These techniques discretize the underlying asset's price and time, allowing for more accurate simulations of market behavior. Previous studies have applied finite difference schemes to option pricing, demonstrating their efficacy in capturing market dynamics and reducing hedge errors. One notable contribution in this domain is the work of Brennan and Schwartz (1976), who introduced the concept of delta-gamma hedging to account for nonlinearities in option pricing. Their approach extended traditional delta hedging by incorporating second-order derivatives, offering a more robust framework for risk management. Subsequent research has built upon this foundation, exploring higherorder derivatives and advanced numerical techniques to further enhance hedge accuracy. Longstaff and Schwartz (1988) [13] explored numerical methods for option pricing, focusing on Monte Carlo simulation techniques. Their research demonstrated the efficacy of simulation-based approaches in capturing complex market dynamics and estimating hedge errors. Broadie and Glasserman [14] (1993) investigated Monte Carlo methods for option pricing, emphasizing variance reduction techniques to improve computational efficiency. Their work contributed to the development of more accurate and scalable numerical algorithms for estimating hedge errors. Andersen and Broadie (2001) [14] proposed the primal-dual simulation algorithm for pricing American options, integrating duality theory and simulation techniques. Their research offered insights into efficient methods for hedging American-style derivatives and managing associated hedge errors. Gatheral's (2006) [15] book provided a comprehensive overview of volatility surfaces and their implications for option pricing and hedging. The work synthesized theoretical concepts with practical insights, offering guidance on managing hedge errors in real-world trading environments. Avellaneda and Stoikov (2010) [16] examined high-frequency trading strategies in limit order book markets, addressing the challenges of latency and market impact. Their research shed light on the dynamics of hedge errors in fast-paced trading environments and the importance of adaptive hedging strategies. Joshi's (2015) textbook provided a comprehensive overview of mathematical finance, covering topics such as stochastic calculus, derivative pricing, and risk management [16]. The work served as a foundational resource for understanding hedge error approximation techniques within the broader context of quantitative finance. These works represent a chronological progression of research efforts aimed at improving hedge error approximation in the Black-Scholes option pricing model, culminating in the proposed asymptotic approach using finite difference methods outlined in this paper. 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 2. LITERATURE REVIEW Classical methods, such as delta hedging, rely on continuous trading assumptions and deterministic models, often failing to capture the complexities of real market dynamics. Researchers have explored alternative techniques to improve hedge effectiveness under realistic conditions. Finite difference methods have gained traction for their flexibility and ability to accommodate discrete trading. These techniques discretize the underlying asset's price and time, allowing for more accurate simulations of market behavior. Previous studies have applied finite difference schemes to option pricing, demonstrating their efficacy in capturing market dynamics and reducing hedge errors. One notable contribution in this domain is the work of Brennan and Schwartz (1976), who introduced the concept of delta-gamma hedging to account for nonlinearities in option pricing. Their approach extended traditional delta hedging by incorporating second-order derivatives, offering a more robust framework for risk management. Subsequent research has built upon this foundation, exploring higherorder derivatives and advanced numerical techniques to further enhance hedge accuracy. Longstaff and Schwartz (1988) [13] explored numerical methods for option pricing, focusing on Monte Carlo simulation techniques. Their research demonstrated the efficacy of simulation-based approaches in capturing complex market dynamics and estimating hedge errors. Broadie and Glasserman [14] (1993) investigated Monte Carlo methods for option pricing, emphasizing variance reduction techniques to improve computational efficiency. Their work contributed to the development of more accurate and scalable numerical algorithms for estimating hedge errors. Andersen and Broadie (2001) [14] proposed the primal-dual simulation algorithm for pricing American options, integrating duality theory and simulation techniques. Their research offered insights into efficient methods for hedging American-style derivatives and managing associated hedge errors. Gatheral's (2006) [15] book provided a comprehensive overview of volatility surfaces and their implications for option pricing and hedging. The work synthesized theoretical concepts with practical insights, offering guidance on managing hedge errors in real-world trading environments. Avellaneda and Stoikov (2010) [16] examined high-frequency trading strategies in limit order book markets, addressing the challenges of latency and market impact. Their research shed light on the dynamics of hedge errors in fast-paced trading environments and the importance of adaptive hedging strategies. Joshi's (2015) textbook provided a comprehensive overview of mathematical finance, covering topics such as stochastic calculus, derivative pricing, and risk management [16]. The work served as a foundational resource for understanding hedge error approximation techniques within the broader context of quantitative finance. These works represent a chronological progression of research efforts aimed at improving hedge error approximation in the Black-Scholes option pricing model, culminating in the proposed asymptotic approach using finite difference methods outlined in this paper. 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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