The Black-Scholes Model: A Fundamental Framework for Option Pricing

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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

1.4 THE BLACK-SCHOLE MODEL

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


𝐢 = πΆπ‘Žπ‘™π‘™ π‘œπ‘π‘‘π‘–π‘œπ‘› π‘π‘Ÿπ‘–π‘π‘’


𝑆 = πΆπ‘’π‘Ÿπ‘Ÿπ‘’π‘›π‘‘ π‘π‘Ÿπ‘–π‘π‘’ π‘œπ‘“ π‘‘β„Žπ‘’ π‘’π‘›π‘‘π‘’π‘Ÿπ‘™π‘¦π‘–π‘›π‘” π‘Žπ‘ π‘ π‘’π‘‘


𝐾 = π‘†π‘‘π‘Ÿπ‘–π‘˜π‘’ π‘π‘Ÿπ‘–π‘π‘’


π‘Ÿ = π‘…π‘–π‘ π‘˜ π‘“π‘Ÿπ‘’π‘’ π‘–π‘›π‘‘π‘’π‘Ÿπ‘’π‘ π‘‘ π‘Ÿπ‘Žπ‘‘π‘’


𝑑 = π‘‡π‘–π‘šπ‘’ π‘œπ‘“ 𝑒π‘₯π‘π‘–π‘Ÿπ‘Žπ‘‘π‘–π‘œπ‘›


𝑁 = πΆπ‘’π‘šπ‘’π‘™π‘Žπ‘‘π‘–π‘£π‘’ π‘‘π‘–π‘ π‘‘π‘Ÿπ‘–π‘π‘’π‘‘π‘–π‘œπ‘› π‘“π‘’π‘›π‘π‘‘π‘–π‘œπ‘› π‘œπ‘“ π‘‘β„Žπ‘’ π‘ π‘‘π‘Žπ‘›π‘‘π‘Žπ‘Ÿπ‘‘ π‘›π‘œπ‘Ÿπ‘šπ‘Žπ‘™ π‘‘π‘–π‘ π‘‘π‘Ÿπ‘–π‘π‘’π‘‘π‘–π‘œπ‘›


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].

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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).

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This paper is available on arxiv under CC by 4.0 Deed (Attribution 4.0 International) license.

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