The Black-Scholes model, also known as the Black-Scholes-Merton (BSM) model, is a significant concept in modern financial theory. It is a mathematical equation that estimates the theoretical value of derivatives, such as options contracts, based on various factors, including time and risk. Developed in 1973, it remains a widely used method for pricing options contracts.
History of the Black-Scholes Model
The BSM model was the brainchild of Fischer Black, Robert Merton, and Myron Scholes, who introduced it in 1973. It was the first mathematical approach for determining an option contract’s theoretical value, using variables such as current stock prices, expected dividends, the option’s strike price, expected interest rates, time to expiration, and expected volatility. The model was introduced in a 1973 paper by Black and Scholes and later expanded upon by Merton. In 1997, Scholes and Merton were awarded the Nobel Memorial Prize in Economic Sciences for their work on the model.
How Black Scholes Model Works
The BSM model presumes that financial instruments, such as stocks or futures contracts, will exhibit a lognormal distribution of prices, following a random walk with constant drift and volatility. The model requires five variables: volatility, the price of the underlying asset, the strike price of the option, the time until the option’s expiration, and the risk-free interest rate. With these variables, the model calculates the price of a European-style call option.
Black-Scholes Assumptions
The Black Scholes model makes several assumptions:
- No dividends are paid out during the life of the option.
- Markets are random, meaning market movements cannot be predicted.
- There are no transaction costs in buying the option.
- The risk-free rate and volatility of the underlying asset are known and constant.
- The returns of the underlying asset are normally distributed.
- The option is European and can only be exercised at expiration.
The Black Scholes Model Formula
The Black Scholes formula is computed by multiplying the stock price by the cumulative standard normal probability distribution function. Then, the strike price’s net present value (NPV) multiplied by the cumulative standard normal distribution is subtracted from the resulting value of the previous calculation.
Here’s the mathematical representation of it:
Where,
C = call option price
N = CDF of the normal distribution
St = spot price of an asset
K = strike price
r = risk-free interest rate
t = time to maturity
σ = volatility of the asset
Benefits of the Black-Scholes Model
- It provides a theoretical framework for pricing options, allowing investors and traders to determine the fair price of an option using a structured, defined methodology.
- It enables risk management by allowing investors to understand their risk exposure to different assets.
- It can be used for portfolio optimisation by providing a measure of the expected returns and risks associated with different options.
- It enhances market efficiency and transparency as traders and investors are better able to price and trade options.
- It streamlines pricing, allowing for greater consistency and comparability across different markets and jurisdictions.
Here are some specific examples of how the Black Scholes model can benefit investors:
- To determine the fair price of an option. The Black-Scholes model can be used to calculate the theoretical price of an option, which can then be compared to the market price to see if the option is undervalued or overvalued.
- To hedge against risk. By buying an option, an investor can protect themselves against the risk of the underlying asset price falls. For example, if an investor owns shares in a company, they could buy a put option on those shares to protect themselves against the risk of the share price falling below a certain level.
- To create trading strategies. By combining options with other financial instruments, investors can create strategies to generate profits in different market conditions.
Limitations of the Black-Scholes Model
Despite its benefits, the Black-Scholes model has some limitations:
- It is only used to price European options and does not account for the fact that American options could be exercised before the expiration date.
- It assumes dividends and risk-free rates are constant, which may not always be the case.
- It assumes volatility remains constant over the option’s life, which is often not the case as volatility fluctuates with the level of supply and demand.
- It makes several other assumptions, such as no transaction costs or taxes, constant risk-free interest rates for all maturities, and no riskless arbitrage opportunities. These assumptions can lead to prices that deviate from actual results.
- It is a “black box” model. This means that it is not always clear how the model arrives at its results. This can make it difficult to use the model to understand the underlying factors that are affecting option prices.
FAQs
What is the Black-Scholes Model?
The Black-Scholes Model, also known as the Black-Scholes-Merton Model, is a mathematical model used to calculate the theoretical price of options. It was developed by economists Fischer Black and Myron Scholes, with contributions from Robert Merton.
How does the Black-Scholes Model work?
The Black-Scholes Model works by using five key variables: the current price of the asset, the strike price of the option, the time until the option expires, the risk-free interest rate, and the volatility of the asset. The model assumes that markets are efficient, returns are normally distributed, and there are no transaction costs.
What are the assumptions of the Black-Scholes Model?
The Black-Scholes Model makes several key assumptions. These include:
- • The risk-free rate and volatility of the underlying are known and constant.
- • The returns on the underlying asset are normally distributed.
- • Markets are efficient, meaning they reflect all available information.
- • There are no transaction costs or taxes.
- • The underlying asset does not pay a dividend.
What are the limitations of the Black-Scholes Model?
The Black-Scholes Model, while widely used, has several limitations. It assumes that volatility is constant and known, which is often not the case in real-world markets. It also assumes that returns are normally distributed, which may not always hold true. Additionally, it does not account for dividends paid by the underlying asset.
How is the Black-Scholes Model used in finance?
The Black-Scholes Model is primarily used to price European options and derivatives. It’s also used in the creation of trading strategies involving these financial instruments. Despite its limitations, it remains a fundamental tool in financial markets due to its simplicity and the insights it provides into the factors influencing option pricing.
