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The Black-Scholes Model, Explained

By Brian Nibley · October 02, 2023 · 6 minute read

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The Black-Scholes Model, Explained

The Black-Scholes option pricing model is a mathematical formula used to calculate the theoretical price of an option. It’s a commonly-used formula for determining the price of contracts, and as such, can be useful for investors in the options market to know and have in their pocket for use.

But there are some important things to know about it, such as the fact that the model only applies to European options, and more.

What Is the Black-Scholes Model?

As mentioned, the Black-Scholes model is one of the most commonly used formulas for pricing options contracts. The model, also known as the Black-Scholes formula, allows investors to determine the value of options they’re considering trading.

The formula takes into account several important factors affecting options in an attempt to arrive at a fair market price for the derivative. The Black-Scholes options pricing model only applies to European options.


💡 Quick Tip: How do you decide if a certain trading platform or app is right for you? Ideally, the investment platform you choose offers the features that you need for your investment goals or strategy, e.g., an easy-to-use interface, data analysis, educational tools.

The History of the Black-Scholes Model

The Black-Scholes model gets its name from Myron Scholes and Fischer Black, who created the model in 1973. The model is sometimes called the Black-Scholes-Merton model, as Robert Merton also contributed to the model’s development. These three men were professors at the Massachusetts Institute of Technology (MIT) and University of Chicago.

The model functions as a differential equation that requires five inputs:

•   The option’s strike price

•   The amount of time until the option expires

•   The price of its underlying stock

•   Interest rates

•   Volatility

Modern computing power has made it easier to use this formula and made it more popular among those interested in stock options trading.

The model only works for European options, since American options allow contract holders to exercise at any time between the time of purchase and the expiration date. By contrast, European options come at cheaper prices and only allow the owner to exercise the option on the expiration date. So, while European options only offer a single opportunity to earn profits, American options offer multiple opportunities.

Recommended: American vs European Options: What’s the Difference?

What Does the Black-Scholes Model Tell?

The main goal of the Black-Scholes Formula is to determine the chances that an option will expire in the money. To this end, the model goes deeper than simply looking at the fact that a call option will increase when its underlying stock price rises and incorporates the impact of stock volatility.

The model looks at several variables, each of which impact the value of that option. Greater volatility, for example, could increase the odds the options will wind up being in the money before its expiration. The more time the investor has to exercise the option also increases the likelihood of it winding up in the money and lowers the present value of the exercise price. Interest rates also influence the price of the option, as higher rates make the option more expensive by decreasing the present value of the exercise price.

The Black-Scholes Formula

The Black-Scholes formula expresses the value of a call option by taking the current stock prices multiplied by a probability factor (D1) and subtracting the discounted exercise payment times a second probability factor (D2).

Explaining in exact detail what D1 and D2 represent can be difficult because the original research papers by Black and Scholes didn’t explain or interpret D1 and D2, and neither did the papers published by Merton. Entire research papers have been written on the subject of D1 and D2 alone.


💡 Quick Tip: If you’re an experienced investor and bullish about a stock, buying call options (rather than the stock itself) can allow you to take the same position, with less cash outlay. It is possible to lose money trading options, if the price moves against you.

Why Is the Black-Scholes Model Important?

The Black-Scholes option pricing model is so important that it once won the Nobel Prize in economics. Some even claim that this model is among the most important ideas in financial history.

Some traders consider the Black-Scholes Model one of the best methods for figuring out fair prices of European call options. Since its creation, many scholars have elaborated on and improved this formula. In this sense, Black and Scholes made a significant contribution to the academic world when it comes to math and finance.

Some claim that the Black-Scholes model has made a significant contribution to the efficiency of the options and stock markets. While designed for European options, the Black-Scholes Model can still help investors understand how an option’s price might react to its underlying stock price movements and improve their overall options trading strategies.

This allows investors to optimize their portfolios by hedging accordingly, making the overall markets more efficient. However, others assert that the model has increased volatility in the markets, as more investors constantly try to fine tune their trades according to the formula.

How Accurate Is the Black-Scholes Model?

Some studies have shown the Black-Scholes model to be highly predictive of options prices. This doesn’t mean the formula has no flaws, though.

The model tends to undervalue calls that are deeply in the money and overvalue calls that are deeply out of the money.

That means the model might assign an artificially low value to options that are much higher than the price of their underlying stock, while it may overvalue options that are far beneath the stock’s current value. Options that deal with stocks yielding a high dividend also tend to get mispriced by the model.

Assumptions of the Black-Scholes Model

There are also a few assumptions made by the model that can lead to less-than-perfect predictions. Some of these include:

•   The assumption that volatility and the risk- free rate within a stock remain constant

•   The assumption that stock prices are stable and large price swings don’t happen

•   The assumption that a stock doesn’t pay dividends until after an option expires

Recommended: How Do Dividends Work?

Such assumptions are necessary, even if they may negatively impact results. Relying on assumptions like these make the task possible, as only so many variables can reasonably be calculated.

Over the years, math scholars have elaborated on the work of Black and Scholes and made efforts to compensate for some of the gaps created by the original assumptions.

This leads to another flaw of the Black-Scholes model, unlike other inputs in the model, volatility must be an estimate rather than an objective fact. Interest rates and the amount of time left until the option expires are concrete numbers, while volatility has no direct numerical value.

The best a financial analyst can do is calculate an estimation of volatility by using something like the formula for variance. Variance is a measurement of the variability of an asset, or how much prices change from time to time. One common measurement of volatility is the standard deviation, which is equivalent to the square root of variance.

The Takeaway

The Black-Scholes option-pricing model is among the most influential mathematical formulas in modern financial history, and it may be the most accurate way to determine the value of a European call option. It’s a complicated formula that has some drawbacks that traders must understand, but it’s a useful tool for European options traders.

Given the Black-Scholes model’s complexity, it’s likely that many investors will never use it. That doesn’t mean it isn’t important to know or understand, of course, but many investors may not get much practical use out of it unless they delve deeper into the world of options trading.

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Photo credit: iStock/akinbostanci

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