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

Implied volatility is derived from the Black-Scholes formula, and using it can provide significant benefits to investors. Implied volatility is an estimate of the future variability for the asset underlying the options contract. The Black-Scholes model is used to price options. The model assumes 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 expiration of the option, and the risk-free interest rate. With these variables, it is theoretically possible for options sellers to set rational prices for the options that they are selling.

Calculating Implied Volatility

As with any equation, Black-Scholes can be used to determine any single variable when all the other variables are known. The options market is reasonably well developed at this point, 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 allow one to solve for volatility. This solution is the expected volatility implied by the option price. Therefore, it is called implied volatility.

Assumptions

The Black-Scholes model makes several assumptions that may not always be correct. The model assumes that volatility is constant. In reality, it is often moving. The Black-Scholes model is limited to European options, which may only be exercised on the last day. However, 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. Often, asset prices have significant skewness and kurtosis. That means high-risk downward moves happen more often in the market than a Gaussian distribution predicts.

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

That has led to the presence of the volatility skew. When the implied volatilities for options with the same expiration date are mapped out on a graph, a smile or skew shape can be seen. This phenomenon is also known as a volatility smile. Due to volatility smiles, an uncorrected Black-Scholes model is not always sufficient for accurately calculating implied volatility (IV).

Historical vs. Implied Volatility

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

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, as it is based on the actual volatility of the underlying asset. However, using historical volatility also has some drawbacks. Volatility shifts as markets go through different regimes. Thus, 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 of the information used by market participants to determine prices in the options market, instead of just past prices.

The best example of this may be quarterly earnings reports. Stock prices sometimes jump up dramatically on positive earnings news. Investors know this, so they are willing to pay more for options as quarterly earnings announcements approach. As a result, implied volatility also goes up near those dates. Dividend declarations, quarterly earnings, and other upcoming events cannot directly influence any volatility estimate based entirely on past prices.

Liquidity Issues

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