# Black Scholes Model

## What is the Black Scholes Model?

The Black Scholes Model is a mathematical model that is used to estimate the price of stock options. It provides investors with the ability to hedge their investment risk by eliminating the option's option to sell the underlying asset at a set price. The Black-Scholes equation, which is based on a PDE, and its associated formula are used to provide a theoretical estimate of the correct price for European stock options. This model is important because it shows that an option has a unique and unchangeable price regardless of the risk associated with its underlying security and expected return.

## See Also

- Option Pricing: The Black-Scholes Model is primarily known for its role in option pricing. This topic encompasses the broader subject of determining the theoretical value of an option, which the Black-Scholes Model specifically addresses for European options.
- European-Style Option: The Black-Scholes Model is explicitly designed for European-style options, which can only be exercised at expiration. An understanding of what a European option is helps contextualize the model's use cases.
- Binomial Option Pricing Model: This is another method used for option pricing, which utilizes a binomial tree to represent possible future stock prices. It serves as an alternative and sometimes complementary approach to the Black-Scholes Model.
- Volatility Smile: The volatility smile is a phenomenon where implied volatilities of options differ for equal moneyness. It's an observed pattern that sometimes emerges from market prices and challenges some of the assumptions in the Black-Scholes Model.
- Implied Volatility: Implied volatility is a metric derived from the market price of an option and can be thought of as the market's expectation of future volatility. The Black-Scholes Model uses volatility as a key input, and implied volatility is often back-calculated using the model.
- Greeks (Delta, Gamma, Theta, Vega, Rho): These are measures of an option's price sensitivity to various factors, like changes in the underlying asset's price or volatility. The Black-Scholes Model provides the mathematical foundation for calculating these "Greeks."
- Risk-Free Rate: The risk-free rate is another essential input in the Black-Scholes Model. It represents the return on a theoretically risk-free investment and is usually based on government bond yields.
- Merton's Extension to the Black-Scholes Model: Robert Merton, who shared the Nobel Prize with Black and Scholes, expanded the original model to account for dividend payments. It's a direct extension of the foundational Black-Scholes work.
- Financial Derivatives: Options are a type of financial derivative, which are contracts that derive their value from an underlying asset. A broader understanding of derivatives gives context to the importance of models like Black-Scholes in finance.