The Black-Scholes Model, Explained
Editor's Note: Options are not suitable for all investors. Options involve risks, including substantial risk of loss and the possibility an investor may lose the entire amount invested in a short period of time. Please see the Characteristics and Risks of Standardized Options.
Table of Contents
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.
But there are some important things to know about it, such as the fact that the model only applies to European-style options.
Key Points
• The Black-Scholes model is a mathematical formula used to calculate the theoretical price of an option.
• It is commonly used for pricing options contracts and helps investors determine the value of options they’re considering trading.
• The model takes into account factors like the option’s strike price, time until expiration, underlying stock price, interest rates, and volatility.
• The Black-Scholes model was created by Myron Scholes and Fischer Black in 1973 and is also known as the Black-Scholes-Merton model.
• While the model has some assumptions and limitations, it is considered an important tool for European options traders.
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 estimate 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 theoretical price for the derivative. The Black-Scholes options pricing model only applies to European options.
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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 researchers were affiliated with 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 is designed 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 may be priced differently and only allow the owner to exercise the option on the expiration date. So, while European options only offer a single opportunity to exercise, American option traders may choose any of the days up until and on expiration to exercise the option.
Recommended: American vs European Options: What’s the Difference?
What Does the Black-Scholes Model Tell?
The main goal of the Black-Scholes model is to estimate the theoretical price of a European-style contract, giving options traders a benchmark to compare against market prices. To this end, the model goes deeper than simply looking at the fact that the price of a call option may increase when its underlying stock price rises and incorporates the impact of stock volatility.
The model looks at several variables, each of which may impact the value of that option. Greater volatility, for example, could increase the option’s theoretical value since it may have a higher chance of seeing larger price moves. Similarly, more time to expiration may increase the model’s estimate of the option ending in the money, and may lower the present value of the exercise price. Interest rates also influence the price of the option, as higher rates can make the option more expensive by decreasing the present value of the exercise price.
The Black-Scholes Formula
The Black-Scholes formula estimates the theoretical value of a call option or put option using inputs such as current stock price, time to expiration, volatility, and interest rates. It 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 complex. They are part of the mathematical process used to estimate option prices in the market, and are often debated.
💡 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 impactful that it once won the Nobel Prize in economics. Some consider it a foundational idea in financial history.
Some traders use the Black-Scholes model to estimate theoretical values of European options. Since its creation, many scholars have elaborated on and improved this formula. The model is widely recognized as a landmark in mathematical finance.
Some analysts argue that the model has contributed to greater pricing efficiency of options and stock markets. While designed for European options, the Black-Scholes model can still offer insights into how theoretical option values respond to changes in core pricing factors, which may help inform investors’ overall options trading strategies.
Some traders use the model hedge against portfolio risk, which they believe may improve overall market efficiency. 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 effective at estimating theoretical options prices. This doesn’t mean the formula has no flaws, however.
The model tends to underestimate the value of deep in-the-money calls and overestimate calls that are deeply out of the money.
That means the model might assign an artificially low value to options that are significantly in the money, while it may overvalue options that are significantly out of the money. Options tied to stocks yielding a high dividend may also get mispriced by the model.
Recommended: How Do Dividends Work?
Assumptions of the Black-Scholes Model
There are also a few assumptions made by the model that can limit its real-world accuracy. Some of these include:
• The assumption that volatility and the risk-free rate remain constant over the option’s life
• The assumption that stock prices move continuously and without sudden jumps
• The assumption that a stock doesn’t pay dividends during the option’s life
Such assumptions are necessary to simplify the model, even though they may negatively impact results. Relying on assumptions makes the model mathematically tractable, as only so many variables can reasonably be calculated.
Over the years, quantitative researchers have expanded on the original models to address limitations introduced by its 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 fixed numerical value.
The best a financial analyst can do is estimate volatility using something like the formula for variance. Variance is a measurement of the variability of an asset, or how much its price changes from time to time. One common measurement of volatility is the standard deviation, which is calculated as the square root of variance.
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The Takeaway
The Black-Scholes option-pricing model is among the most influential mathematical formulas in modern financial history, and it may be one of the most accurate ways to determine the theoretical value of a European call option. It’s a complicated formula that has some drawbacks that traders should be aware of, but it’s a useful tool for European options traders.
Given the Black-Scholes model’s complexity, it’s likely that many investors may never apply it directly in their trading decisions. 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 options trading.
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FAQ
What is an example of the Black-Scholes method?
An example would be using the Black-Scholes formula to estimate the theoretical value of a European call option on a stock trading at $100, with a $105 strike price, 30 days to expiration, 20% volatility, and a 5% risk-free rate. The model would help determine the option’s theoretical worth under these conditions.
What is the 5 step method of Black-Scholes?
The five steps typically include: identifying the input values (stock price, strike price, time to expiration, volatility, and risk-free rate), calculating d1 and d2 (which are probability factors), finding the cumulative normal distribution values of d1 and d2, plugging the values into the Black-Scholes formula, and interpreting the result as the option’s theoretical price.
Is Black-Scholes still used?
Yes, the Black-Scholes model remains widely used as a foundational pricing tool for European options. Many traders and financial institutions still use it, though modifications or alternative models may be applied in complex or non-standard scenarios.
Why are Black-Scholes so important?
The Black-Scholes model helped transform how options are priced by offering a standardized, mathematically grounded method. Some argue that it has helped to improve market efficiency and risk management and pave the way for the modern derivatives market.
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Options involve risks, including substantial risk of loss and the possibility an investor may lose the entire amount invested in a short period of time. Before an investor begins trading options they should familiarize themselves with the Characteristics and Risks of Standardized Options . Tax considerations with options transactions are unique, investors should consult with their tax advisor to understand the impact to their taxes.
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