How is implied volatility used in the Black-Scholes formula?

Implied volatility is derived from the Black-Scholes formula and is an important element for how the value of options are determined. Implied volatility is a measure of the estimation of the future variability for the asset underlying the option contract. The Black-Scholes model is used to price options. The model assumes the price of the underlying assets follows a geometric Brownian motion with constant drift and volatility. Implied volatility is the only input of the model not directly observable. The Black-Scholes equation must be solved to determine the implied volatility. The other inputs for the Black-Scholes equation are 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.

The Black-Scholes model makes a number of assumptions that may not always be correct. The model assumes volatility is constant, when in reality it is often moving. The model further assumes efficient markets are based on a random walk of asset prices. The Black-Scholes model is limited to European options that can only be exercised on the last day as opposed to American options that can be exercised at any time before expiration.

The Black-Scholes equation assumes a lognormal distribution of price changes for the underlying asset. This is also known as a Gaussian distribution. Often, asset prices have significant skewness and kurtosis. This means high-risk downward moves often 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. However, since the market crash of 1987, 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 phenomena is the market is pricing in a greater likelihood of a high volatility move to the downside in the markets.

This 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. Thus, the Black-Scholes model is not efficient for calculating 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. The standard deviation is a statistical measure of the variability of price changes from the mean price change. This differs from the implied volatility determined by the Black-Scholes method, 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.