Comprehensive Overview of Option Pricing and the Black-Scholes Model

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

2. Literature Review
3. Methodology

3.1 Model Assumption

3.2 Theorems and Model Discussion

4. Result Analysis
5. Conclusion and References

1.1 OPTION PRICING

Option price, also known as the premium, is the cost paid by the option buyer to the seller for the right to buy (call option) or sell (put option) an underlying asset at a predetermined price within a specified period.

Put and call options are financial instruments that grant the holder the right, but not the obligation, to buy (call option) or sell (put option) an underlying asset at a predetermined price (strike price) within a specified period (expiration date).

A call option gives the buyer the right to purchase the underlying asset at the strike price, making a profit if the asset's market price rises above the strike price before the option expires. This allows investors to benefit from potential price increases without owning the asset outright, providing leverage and risk management.

On the other hand, a put option grants the holder the right to sell the underlying asset at the strike price. Put options are valuable when the market price of the asset falls below the strike price, enabling the holder to sell at a higher price than the market value, thereby hedging against potential losses or profiting from a declining market. Options are widely used for speculation, hedging, and risk management in financial markets. Understanding the dynamics of options trading is crucial for investors to effectively manage risk and capitalize on market opportunities.

Time value, which is dependent on the anticipated volatility of the underlying asset, and intrinsic value, which quantifies the option's profitability, make up the price of an option.

The amount of time remaining before the option expires) The asset price, strike price, amount of time to expiration, volatility, and risk-free interest rate are some of the variables that determine an option's fair value. Finding the probability of an option being "in-the-money" or "out-of-the-money" at the time of execution is the main objective of option pricing [1]. For traders, investors, and financial institutions to make well-informed decisions about purchasing, disposing of, or hedging risks against certain underlying assets, option pricing is essential. It is possible to use the Partial Differential Equations (PDE) approach to option pricing issues. In other words, the price function can be calculated using a PDE's solution. One such approach is the Black-Scholes model framework, which describes the dynamics of option prices using a parabolic nonlinear PDE [2].

Although many changes have been suggested, the BS model has been the industry standard for estimating the fair value of options.

The Black-Scholes PDE for pricing a European call option is derived as:

where 𝐶 is the option price, 𝑡 is time, 𝜎 is the volatility of the underlying asset, 𝑆 is the spot price of the underlying asset, and 𝑟 is the risk-free interest rate.