According to the Black-Scholes model, the value of an option is determined on the basis of five factors:

1) the value of the underlying asset (S)

2) strike price (X)

3) volatility (σ)

4) the time before the expiration (t)

5) risk-free interest rate (r)

All factors are described in the section “Intrinsic and time value of an option”. However, only the volatility value of σ is uncertain and is of greatest interest to option traders. The price of the underlying asset S and the interest rate r are known to the market. The expiration date of option t and the strike X price are written in advance in the contract. The only stumbling block is volatility.

How can the volatility of the price of the underlying asset be calculated or precisely predicted before the option expires? The logical point of departure is to analyze the price movements of the asset over a certain period in the past. Other things being equal, the more volatile the price of an asset, the greater the risk assumed by the writer of the option (or the greater the opportunity presented to the buyer) and the greater the time value of the option.

**Historical realized volatility**

Historical volatility is calculated as the standard deviation of daily returns on the underlying asset for a certain period of time. In practice, traders and analysts use the volatility value in annual terms. The calculation is based on the percentage change in the price of the underlying asset, not the absolute price change. This allows us to compare the volatility of assets that are traded at different price levels. The standard deviation is the most common unit of measurement in statistics relative to the average value.

The key issue in measuring historical volatility is to determine the historical range on the basis of which share dynamics should be analyzed. Should we consider the behavior of the stock over the last few months or several years? Is this historical period typical for an asset? Was high uncertainty prevalent or was the market in an upward phase? Should recent events be given greater weight when historical volatility is calculated?

A more fundamental problem for the option trader is predicting future volatility of the asset before expiration, since the expected payout (and therefore the premium required by the option writer) is based not on historical volatility, but on the future price volatility of the underlying asset. It is therefore necessary for the trader to predict the volatility based on historical data as well as taking into account potentially important events that may occur before the expiration date. Sometimes, trader may decide that the price of the underlying asset will be subject to significant fluctuations, showing a higher level of volatility than historical data indicates. In other cases, the trader will adhere to the view that price fluctuations will decrease after a period of high volatility.

**Implied volatility**

Instead of using historical volatility to determine the fair value of an option, traders take the option's market prices and find the volatility value that corresponds to that option price. This implied volatility is then compared to the historical volatility and future predictions.

If the trader believes that the actual price movements of the asset will be higher than the volatility level corresponding to the market price of the option, it makes sense to buy the option because its price is relatively cheap. If the trader believes that the market overestimates the future volatility of the asset price, then the option should be sold at the market price as the premium received will exceed the expected payout of the option.

For example, if a trader sells an option with relatively high implied volatility, and then the market forecast of future volatility decreases (with the other four parameters unchanged), then the trader can buy the same option back at a cheaper price.