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 1.5 MONTE CARLO SIMULATION AND VARIANCE REDUCTION TECHNIQUES Monte Carlo simulation is a powerful computational technique used in various fields, including finance, engineering, physics, and statistics, to approximate complex systems and processes through repeated random sampling. It relies on the principles of randomness and statistical inference to estimate unknown quantities or simulate the behavior of systems that may be too intricate to model analytically. At its core, Monte Carlo simulation involves generating many random samples from a specified probability distribution, using these samples to simulate the system under consideration, and then analyzing the results to draw conclusions or make predictions [6]. One of the key challenges in Monte Carlo simulation is achieving accurate and efficient estimates while keeping computational costs manageable [7]. Variance reduction techniques are strategies employed to improve the efficiency and precision of Monte Carlo simulations by reducing the variability of the estimates obtained. Importance Sampling is a variance reduction technique that aims to improve the efficiency of Monte Carlo simulations by focusing the random samples on regions of the probability space where the integrand has the most significant contributions. Instead of sampling from the original distribution, importance sampling involves sampling from a modified distribution that places more emphasis on the relevant regions, thereby reducing the variance of the estimator [8][9]. Vector Random Variable technique is another variance reduction method commonly used in Monte Carlo simulations. It involves transforming correlated random variables into independent ones by utilizing techniques such as Cholesky decomposition or eigenvalue decomposition. By transforming the variables into an uncorrelated set, the variance of the estimator can be reduced, leading to more accurate results with fewer samples [10]. Antithetic Variates is a simple yet effective variance reduction technique that exploits negative correlation between pairs of random variables. It involves generating paired samples such that one sample is the negative of the other [11]. By averaging the results obtained from each pair, the variance of the estimator is reduced, resulting in more precise estimates with fewer random samples. Control Variates is a variance reduction technique that leverages known relationships between the variable of interest and another related variable, known as the control variate. By incorporating the control variate into the simulation, the variance of the estimator can be reduced, leading to more efficient estimates [12]. Control variates are chosen such that they are correlated with the variable of interest and have known expectations, facilitating the estimation process. Overall, variance reduction techniques play a crucial role in improving the accuracy and efficiency of Monte Carlo simulations. By implementing these techniques, practitioners can obtain more reliable estimates with fewer computational resources, making Monte Carlo simulation a valuable tool for decision-making and problem-solving in diverse fields. 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 1.5 MONTE CARLO SIMULATION AND VARIANCE REDUCTION TECHNIQUES Monte Carlo simulation is a powerful computational technique used in various fields, including finance, engineering, physics, and statistics, to approximate complex systems and processes through repeated random sampling. It relies on the principles of randomness and statistical inference to estimate unknown quantities or simulate the behavior of systems that may be too intricate to model analytically. At its core, Monte Carlo simulation involves generating many random samples from a specified probability distribution, using these samples to simulate the system under consideration, and then analyzing the results to draw conclusions or make predictions [6]. One of the key challenges in Monte Carlo simulation is achieving accurate and efficient estimates while keeping computational costs manageable [7]. Variance reduction techniques are strategies employed to improve the efficiency and precision of Monte Carlo simulations by reducing the variability of the estimates obtained. Importance Sampling is a variance reduction technique that aims to improve the efficiency of Monte Carlo simulations by focusing the random samples on regions of the probability space where the integrand has the most significant contributions. Instead of sampling from the original distribution, importance sampling involves sampling from a modified distribution that places more emphasis on the relevant regions, thereby reducing the variance of the estimator [8][9]. Vector Random Variable technique is another variance reduction method commonly used in Monte Carlo simulations. It involves transforming correlated random variables into independent ones by utilizing techniques such as Cholesky decomposition or eigenvalue decomposition. By transforming the variables into an uncorrelated set, the variance of the estimator can be reduced, leading to more accurate results with fewer samples [10]. Antithetic Variates is a simple yet effective variance reduction technique that exploits negative correlation between pairs of random variables. It involves generating paired samples such that one sample is the negative of the other [11]. By averaging the results obtained from each pair, the variance of the estimator is reduced, resulting in more precise estimates with fewer random samples. Control Variates is a variance reduction technique that leverages known relationships between the variable of interest and another related variable, known as the control variate. By incorporating the control variate into the simulation, the variance of the estimator can be reduced, leading to more efficient estimates [12]. Control variates are chosen such that they are correlated with the variable of interest and have known expectations, facilitating the estimation process. Overall, variance reduction techniques play a crucial role in improving the accuracy and efficiency of Monte Carlo simulations. By implementing these techniques, practitioners can obtain more reliable estimates with fewer computational resources, making Monte Carlo simulation a valuable tool for decision-making and problem-solving in diverse fields. 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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