Could This Be The Most Famous Equation In Finance?

The Black-Scholes options pricing formula,

The Black-Scholes equation is probably the most famous equation in finance. Financial theory largely relies upon this equation. Using this equation, we estimate the theoretical value of options and other investment instruments, taking into account the effects of time and other risk factors. It was developed in 1973 and is still considered one of the best methods of pricing options contracts for software algorithms.

Important Takeaways 💡

• The Black-Scholes model, also known as the Black-Scholes-Merton (BSM) model, is widely used to calculate the price of options contracts.
• Five input variables are needed to calculate the Black-Scholes model: the strike price of an option, the current price of an underlying stock, the time until expiration, and the volatility level.
• Even though Black-Scholes models are generally accurate, certain assumptions can cause prices to differ from real-life results.
• In the standard BSM model, only European options are priced, since American options might be exercised before the expiration date.

Although there are many methods of valuing options, none are as famous as the famous model that Fischer Black and Myron. The Black-Scholes model gives rise to a formula that can be used to calculate the theoretical value of an option. Calls and puts are calculated by Black-Scholes by taking the following variables into account:

• Strike price
• Underlying asset price
• Time until expiration
• Asset price volatility
• Risk free rate

A key idea behind the model is to hedge the options in an investment portfolio by buying and selling the underlying asset (such as stocks) in the right way and as a result, eliminate risk. Later, the method was referred to as “continuously revised delta hedging”, which was adapted by many of the world’s most prominent investment banks. The purpose of this blog is to explain the mathematical foundation, assumptions, and implications of the Black-Scholes equation.

Here’s How The Black-Scholes Model Works 📋

According to Black-Scholes, instruments such as stock shares or futures contracts will have a lognormal distribution of prices following a random walk with constant volatility and drift. The equation uses this assumption along with other important variables to calculate the price of a call option.

In order for Black-Scholes to work, five variables must be provided. The inputs include volatility, the price of the underlying asset, the strike price of the option, the time until the option expires, and the risk-free interest rate. Using these variables, options sellers can theoretically establish rational prices for the options that they are selling.

Furthermore, the model predicts that heavily traded assets will exhibit a downward geometric Brownian motion with constant volatility. As it pertains to stock options, the model incorporates constant price fluctuations of the stock, price time value, strike price, and time until expiration.

Black-Scholes Assumptions

Based on the Black-Scholes equation, the following assumptions are made:

• Dividends are not paid during the life of the option.
• The market moves at random (in other words, they are unpredictable).
• Buying the option has no transaction costs.
• An asset’s risk-free rate and volatility are known and constant.
• Underlying assets have log-normal distributions of returns.
• European options can only be exercised at expiration.

Although the original Black-Scholes model didn’t account for dividends paid during the option’s life, modern versions are frequently modified to include dividends by calculating the ex-dividend date value of the underlying stock. The model is also modified by many option-selling market makers to account for the effects of options that may be exercised before expiration.

The Black-Scholes Model Formula Explained

Admittedly, by calculating the formula yourself, it can be difficult and intimidating due to the complicated mathematics involved.

Fortunately for investors, it’s not necessary for you to understand the math to use Black-Scholes models in your own strategies. 👈

Today’s options traders have access to a software like Option Experts, and many trading platforms that offer tools for options analysis, including indicators and technology that perform the calculations and display the options pricing values.

To calculate the Black-Scholes call option formula, multiply the stock price by the cumulative standard normal probability distribution function. After that, the net present value (NPV) of the strike price is divided by the cumulative standard normal distribution, and the difference is subtracted from the value of the previous calculation.

As a mathematical notation that might put you to sleep 😴

Determining Volatility Skew

Due to the fact that asset prices cannot be negative (they are bound by zero), Black-Scholes assumes stock prices follow a lognormal distribution.

It is quite common to observe significant right skewness and kurtosis (fat tails) in asset prices. As a result, downward moves in the market are more likely to occur than a normal distribution would predict.

Using the Black-Scholes model, implied volatilities should be similar for each strike price when underlying asset prices are assumed to be lognormal. However, since the 1987 market crash, at-the-money implied volatility has been lower than implied volatility for options further out of the money or far in the money. The reason behind this is that the market is pricing in a greater likelihood of a high volatility move to the downside.

This has resulted in the presence of the volatility skew. It is possible to see a smile or skew shape if implied volatilities are mapped out for options with the same expiration date. This means that Black-Scholes is inefficient at calculating implied volatility.

So, What Are the Inputs for Black-Scholes Model?

A Black-Scholes equation consists of certain inputs, including volatility, the price of the underlying asset, the strike price, the expiration date, and the risk-free rate. Options sellers could theoretically set rational prices for options based on these variables.

Are There Assumptions in Black-Scholes?

Certain assumptions are made in the Black-Scholes model. One of the major disadvantages is that the option is European and can only be exercised at expiration. Other assumptions include that no dividends are paid during the life of the option; market movements cannot be predicted; there are no transaction costs in buying the option; the risk-free rate and volatility of the underlying are known and constant; and returns from the underlying asset are log-normally distributed.

Do Black-Scholes Models Have Limitations?

As a rule of thumb, the Black-Scholes model only prices European options and does not consider the possibility of American options being exercised before expiration. Furthermore, the model assumes dividends, volatility, and risk-free rates are constant over the option’s life.

The absence of tax, commissions, trading costs, or taxes can also lead to valuations that differ from actual results.

American options 🇺🇸

American options (so-called ‘continuous timeline instruments’) allow for exercise at any time, making them very difficult to deal with in comparison with European options (“point in time instruments”). As a result, the optimal exercise policy must be factored in when solving the Black-Scholes partial differential equation. Black-Scholes’ equation provides no closed form solution for American options. The following are exceptions:

• The American call option price is the same for European call options on assets that do not pay dividends (or other payouts). This is because it is optimal to not exercise the option in this case.
• There are some American call options which pay one known dividend in their lifetime that should be exercised early. The optimal time to exercise the option would be just before the stock goes ex-dividend, according to a solution given in closed-form by the so-called Roll-Geske-Whaley method (Roll, 1977; Geske, 1979; 1981; Whaley, 1981):

To determine if exercising the option early is the best idea, first check if the following inequality is satisfied.

Disclaimer

The team does not consist of mathematical economists, but rather software engineers. That being said, none of our articles serve as financial advice. Those interested in reading more about options trading should look at The Big Short by Michael Lewis.