Implied volatility is the parameter component of an option pricing model, such as the Black-Scholes model, which gives the market price of an option. Implied volatility shows how the marketplace views where volatility should be in the future.

Since implied volatility is forward-looking, it helps us gauge the sentiment about the volatility of a stock or the market. However, implied volatility does not forecast the direction in which an option is headed. In this article, we'll review an example of how implied volatility is calculated using the Black-Scholes model and we'll discuss two different approaches to calculate implied volatility.

### Key Takeaways

- Implied volatility is one of several components of the Black-Scholes formula, a mathematical model that estimates the pricing variation over time of financial instruments, such as options contracts.
- The five other inputs of the Black-Scholes model are the market price of the option, the underlying stock price, the strike price, the time to expiration, and the risk-free interest rate.
- The iterative search is one method using the Black-Scholes formula to calculate implied volatility.
- A trader can compare historical volatility with implied volatility to potentially determine if there is an underlying event that might impact a stock’s price.

## The Black-Scholes Formula

The Black-Scholes model, also called the Black-Scholes-Merton model, was developed by three economists—Fischer Black, Myron Scholes, and Robert Merton in 1973. It is a mathematical model that projects the pricing variation over time of financial instruments, such as stocks, futures, or options contracts. From this model, the three economists derived the Black-Scholes formula.

Since its introduction, the Black-Scholes formula has gained in popularity and was responsible for the rapid growth in options trading. Investors widely use the formula in global financial markets to calculate the theoretical price of European options (a type of financial security). These options can only be exercised at expiration.

The Black-Scholes model does not take into account dividends paid during the life of the option.

## Implied Volatility Inputs

Implied volatility is not directly observable, so it needs to be solved using the five other inputs of the Black-Scholes model, which are:

- The market price of the option.
- The underlying stock price.
- The strike price.
- The time to expiration.
- The risk-free interest rate.

Implied volatility is calculated by taking the market price of the option, entering it into the Black-Scholes formula, and back-solving for the value of the volatility. But there are various approaches to calculating implied volatility. One simple approach is to use an iterative search, or trial and error, to find the value of implied volatility.

## The Iterative Search

Suppose that the value of an at-the-money call option for Walgreens Boots Alliance, Inc. (WBA) is $3.23 when the stock price is $83.11, the strike price is $80, the risk-free rate is 0.25%, and the time to expiration is one day. Implied volatility can be calculated using the Black-Scholes model, given the parameters above, by entering different values of implied volatility into the option pricing model.

For example, start by trying an implied volatility of 0.3. This gives the value of the call option of $3.14, which is too low. Since call options are an increasing function, the volatility needs to be higher. Next, try 0.6 for the volatility; that gives a value of $3.37 for the call option, which is too high. Trying 0.45 for implied volatility yields $3.20 for the price of the option, and so the implied volatility is between 0.45 and 0.6.

The iterative search procedure can be done multiple times to calculate the implied volatility. In this example, the implied volatility is 0.541, or 54.1%.

## Historical Volatility

Historical volatility, unlike implied volatility, refers to realized volatility over a given period and looks back at past movements in price. One way to use implied volatility is to compare it with historical volatility.

From the example above, if the volatility in WBA is 23.6%, we look back over the past 30 days and observe that the historical volatility is calculated to be 23.5%, which is a moderate level of volatility. If a trader compares this to the current implied volatility, the trader should become aware that there may or may not be an event that could affect the stock's price.

## The Bottom Line

The Black-Scholes formula has been proven to result in prices very close to the observed market prices. And, as we've seen, the formula provides an important basis for calculating other inputs, such as implied volatility. While this makes the formula quite valuable to traders, it does require complex mathematics. Fortunately, traders and investors who use it do not need to do these calculations. They can simply plug the required inputs into a financial calculator.