How Is Implied Volatility Used in the Black-Scholes Formula?

Reviewed by Thomas J. CatalanoFact checked by Ryan Eichler

Implied volatility is derived from the Black-Scholes formula and it can provide significant benefits to investors. It’s an estimate of the future variability for an asset underlying an options contract. The Black-Scholes model is used to price options and it assumes that the price of the underlying asset follows a geometric Brownian motion with constant drift and volatility.

The inputs for the Black-Scholes equation are volatility, the price of the underlying asset, the strike price of the option, the time until the expiration of the option, and the risk-free interest rate. It’s theoretically possible for options sellers to set rational prices for the options they’re selling using these variables.

Key Takeaways

• Plugging all the other variables into the Black-Scholes equation, including the option price, yields the implied volatility estimate.
• It’s referred to as implied volatility because it’s the expected volatility implied by the options market.
• Implied volatility has some drawbacks related to the volatility smile and illiquidity.
• Implied volatility can be more accurate than historical volatility when dealing with upcoming events such as quarterly earnings reports and dividend declarations.

Calculating Implied Volatility

Black-Scholes can be used to determine any single variable when all other variables are known.

The options market is reasonably well developed so we already know the market prices for many options. Plugging the option’s price into the Black-Scholes equation along with the price of the underlying asset, the strike price of the option, the time until expiration of the option, and the risk-free interest rate allows you to solve for volatility. The solution is the expected volatility implied by the option price.

Important

An estimate is only as good as the inputs that are used to obtain it. The best implied volatility estimates are derived from at-the-money options on heavily traded securities.

Assumptions

The Black-Scholes model makes several assumptions that may not always be correct. The model assumes that volatility is constant but in reality, it’s often moving. The Black-Scholes model is limited to European options which may only be exercised on the last day but American options can be exercised at any time before expiration.

Black-Scholes and the Volatility Skew

The Black-Scholes equation assumes a lognormal distribution of price changes for the underlying asset. This distribution is also known as a Gaussian distribution. Asset prices often have significant skewness and kurtosis so high-risk downward moves happen more often in the market than a Gaussian distribution predicts.

The assumption of lognormal underlying asset prices should show that implied volatilities are similar for each strike price according to the Black-Scholes model. Implied volatilities for at-the-money options have been lower than those further out of the money or far in the money since the 1987 market crash. The market prices have a higher likelihood of a sharp downward move.

This has led to the volatility skew. A smile or skewed shape can be seen when the implied volatilities for options with the same expiration date are mapped out on a graph. This phenomenon is also known as a volatility smile. An uncorrected Black-Scholes model isn’t always sufficient for accurately calculating implied volatility (IV) due to volatility smiles.

Historical vs. Implied Volatility

The shortcomings of the Black-Scholes method have led some investors to place more importance on historical volatility rather than implied volatility. Historical volatility is the realized volatility of the underlying asset over a previous period. It’s determined by measuring the standard deviation of the underlying asset from the mean during that time.

Standard deviation is a statistical measure of the variability of price changes from the mean price change. This estimate differs from the Black-Scholes method’s implied volatility because it’s based on the actual volatility of the underlying asset. But using historical volatility also has some drawbacks. Volatility shifts as markets go through different regimes so historical volatility may not be an accurate measure of future volatility.

Implied Volatility and Upcoming Events

The most significant benefit of implied volatility for investors is that it may be a more accurate estimate of future volatility in some cases. Implied volatility takes into account all the information used by market participants to determine prices in the options market rather than just past prices.

Quarterly earnings reports may be the best example of this. Stock prices sometimes jump up dramatically on positive earnings news. Investors know this so they’re willing to pay more for options as quarterly earnings announcements approach. Implied volatility also goes up near these dates as a result. Dividend declarations, quarterly earnings, and other upcoming events can’t directly influence any volatility estimate that’s based entirely on past prices.

Liquidity Issues

Implied volatility can be extremely inaccurate when options markets aren’t sufficiently liquid. Lack of liquidity tends to make market prices less stable and less rational. In extreme cases, mistakes made by a single amateur trader can lead to wildly irrational options prices in an illiquid market. Estimates will also be inaccurate if those prices are used to estimate implied volatility. That can be a serious problem because many parts of the options market suffer from a lack of liquidity.

What’s the Difference Between European and American Options?

It’s all a matter of timing. A buyer of either option can buy or sell the asset at a set price by a given deadline but they’re not required to do so. They can buy or sell the American option at any time up until that deadline. The holder of a European option must wait until the deadline.

What Is Historical Volatility?

Historical volatility is past tense. It measures returns over a period that’s already occurred. Averages are used in the calculation: the difference between the average price during that time and the average deviation.

What Is an Illiquid Market?

An illiquid market is one in which sellers are forced to hold onto assets rather than sell them or reduce their price because the market is ailing. This might be due to overall market panic. No one is willing to buy. It can occur because of poor market conditions or simply a lack of interested buyers.

The Bottom Line

The Black-Scholes formula calculates an estimate of implied volatility in the options market. It has its drawbacks including potential illiquidity but it tends to be more accurate than historical volatility when dealing with upcoming events. Other inputs include the price of the asset, the strike price of the option, the remaining time until expiration, and the risk-free interest rate.

As with any investment tool, you’ll want to be sure that you fully understand its mechanisms and implications before you invest based on its viability. This is particularly the case if you’re new to trading.

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