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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 Black-Scholes model is a cornerstone of modern financial theory, providing a mathematical framework for pricing European-style options. Developed by Fischer Black, Myron Scholes, and Robert Merton in the early 1970s, the model revolutionized the field of quantitative finance.

Mathematically, the Black-Scholes model calculates the price of a European call option, which gives the holder the right to buy an underlying asset at a specified price (the strike price) on or before a specified date (the expiration date). The model assumes that the price of the underlying asset follows geometric Brownian motion, characterized by a constant volatility.

The Black-Scholes formula for the price of a European call option is given by:

where

๐ถ = ๐ถ๐๐๐ ๐๐๐ก๐๐๐ ๐๐๐๐๐

๐ = ๐ถ๐ข๐๐๐๐๐ก ๐๐๐๐๐ ๐๐ ๐กโ๐ ๐ข๐๐๐๐๐๐ฆ๐๐๐ ๐๐ ๐ ๐๐ก

๐พ = ๐๐ก๐๐๐๐ ๐๐๐๐๐

๐ = ๐ ๐๐ ๐ ๐๐๐๐ ๐๐๐ก๐๐๐๐ ๐ก ๐๐๐ก๐

๐ก = ๐๐๐๐ ๐๐ ๐๐ฅ๐๐๐๐๐ก๐๐๐

๐ = ๐ถ๐ข๐๐ข๐๐๐ก๐๐ฃ๐ ๐๐๐ ๐ก๐๐๐๐ข๐ก๐๐๐ ๐๐ข๐๐๐ก๐๐๐ ๐๐ ๐กโ๐ ๐ ๐ก๐๐๐๐๐๐ ๐๐๐๐๐๐ ๐๐๐ ๐ก๐๐๐๐ข๐ก๐๐๐

The formula derived from the Black-Scholes model computes the theoretical price of a call or put option based on the aforementioned factors. It considers the probability distribution of potential future asset prices and discounts expected payoffs back to the present value using the risk-free interest rate [5].

**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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