# The historical and implied volatility of options

In my previous article, I discussed the **option Greeks delta, gamma, vega and theta**. Understanding of these option Greeks gives traders insight into the pricing of options, which thus is essential for an options trader. The option Greeks show what impact passage of time or an increase in volatility, for example, have on the option premium. In order to go a step deeper into the option theory, I’m going to discuss the concept of **volatility**. volatility basically comes in two flavours: the **historical volatility** and **implied volatility**. Both concepts will I thoroughly discuss.

## Historical volatility

The term volatility refers to the fluctuation of the price of an instrument. Historical volatility, abbreviated HR, refers to the realised fluctuation. It is simply the price from the past. Historical volatility, therefore, says something about the past and not directly about the future. Yet, it is important to see how the price has developed in the past period. In many cases this provides a clear guideline for expected future price developments.

The chart below shows the AEX index with its historical volatility underneath it. What you may notice first, is that the historical volatility is displayed as a percentage. This is the historical volatility of the last 30 days on an annual basis. The movement in the last 30 trading days is thus about 16%, when converted on a yearly basis. In early February, when the results were greater, volatility peaked above 20%. In contrast, it was actually very quiet on the financial markets in July 2014, when volatility reached a low of 8%.

The fact that volatility in July amounted to 8%, has no further direct predictive value on the market. It’s just a representation of how volatile the AEX was at that time. For example, this can be explained by the summer months, in which the results are lower, historically speaking.

However, it is logical that when the fluctuation in a certain period has been limited, it is used as a basis for the expected future volatility. So when historical volatility is low, it is generally the case for the expected volatility. Simply put: between historical volatility and expected volatility exists a positive correlation. If volatility on the stock markets rises, then expected volatility also increases. In other words, the historical volatility and implied volatility are important to each other.

## Implied volatility

Where historical volatility refers to the past, implied volatility is the expectation about the future. In the **Black-Scholes option model** implied volatility is the missing variable. The other variables, including the price of the underlying asset, the maturity of the option, the strike price and the interest rate, are given. Based on implied volatility, the price of an option is determined. This is the expected volatility of the underlying asset until the expiration date. By the way, there is one more variable that is not fixed, namely the expected dividends. Broadly speaking, this is more or less fixed, though an increase or decrease of the dividend can have a major influence on the option premiums.

The implied volatility is thus crucial for pricing options. This value is determined by the market. By this I mean that it is a question of supply and demand, among other things. If there is much demand for put options, for example, the implied volatility of these options increases. The price of the options will quickly rise as demand gets higher. Ultimately, it is the market makers who constantly issue options prices. They are the *liquidity providers* who offer bid and ask prices and thereby determine the implied volatility of the options.

## Percentage

As pointed out earlier, volatility, both historical and implied, is expressed as a percentage. This percentage refers to the annualized volatility. To convert the volatility from annual to daily, you divide it by 16. Now the question is of course: why 16?

In a year are slightly more than 250 trading days. For convenience, let’s assume 256 trading days, the volatility on a daily basis can easily be calculated using the square root of the number of trading days. The root of 256 is 16. This way you can reconvert from annual to daily basis.

## Interpretation of the implied volatility

The implied volatility of an option reflects the expected volatility of the underlying asset until the expiration date of the option. With an implied volatility of 16%, the volatility on a daily basis equals 1%. This is based on one standard deviation. This means that in 68.2% (twice 34.1%, because the fluctuation can be both positive and negative) of the trading days a movement of 1% or less is expected. For the remaining 31.8% of the trading days higher volatility is expected. This is easy to explain on the basis of the normal distribution.

Based on the normal distribution it is expected that in 27.2% (twice 13.6%) of the trading days, volatility will be between one and two standard deviations. In this example, that means a movement between 1% and 2%, and -1% and -2%. In 4.2% (twice 2.1%) of trading days a fluctuation between 2% and 3%, and -2% and-3% is expected. Finally, in 0.2% of the trading days volatility is expected to be greater than 3% or -3%.

As an example I take a stock that is at € 10, for which an at-the-money call option has a volatility of 16%. The picture above shows that the option is priced with an expected certainty of over 99% that the share price at the next trading day will be between € 9.70 and € 10.30.

## Volatility skew

Options with the same maturity and different strike prices are not priced using the same implied volatility. This means that the options are valued at a different amount and volatility. Thus, an at-the-money put option is generally priced at a lower implied volatility than an out-of-the-money put option. This phenomenon, created after the crash in 1987, is described as the **volatility skew**.

For an options trader, it is important to understand the consequences of the volatility skew when making trading decisions. When out-of-the-money put options are priced at a higher implied volatility, these options are relatively expensive compared to at-the-money put options, for example.

Out-of-the-money put options with a relatively high implied volatility can be compared to a fire insurance on a house. The probability that a fire breaks out, is very small. But if it happens, the damage is huge. This can be extended to the world of trading. When the stock market goes down fast, like during a crash, then these out-of-the-money put options provide protection against losses in your portfolio.

## Term structure

Where the **volatility skew** indicates the difference in implied volatility for options with the same expiration date, the **term structure** indicates the difference in implied volatility for options with the same strike price and different expiration dates.

Options with the same strike price and different expiration dates have a different implied volatility and are therefore priced differently. This effect is mainly explained by the estimations of the market about the underlying value based on future events. It is common for the implied volatility to increase as the quarterly figures near. After the market has received the figures and the news is reflected in the price, the implied volatility decreases again.

For short-term options implied volatility is more volatitle than for long-term options. The term structure of volatility indicates the relationship of implied volatility and maturity. In this way, option traders interpret whether certain option months are cheap or expensive. It is, for example, not uncommon for the summer months to be priced at lower implied volatility than, say, September and October. During in the summer months it is generally calm on the financial markets and the expected price fluctuations are therefore smaller.

The table below shows the implied volatility of at-the-money AEX options with expiration dates of 10 – to 100 days. Due to the recent increase in uncertainty on the financial markets implied volatility has increased, which is clearly visible in the short-term AEX options. Over a longer period, it is expected that volatility decreases slightly and the market will enter calmer waters.

Maturity | Implied volatility |
---|---|

10 days | 19.55% |

40 days | 19.73% |

70 days | 18.10% |

100 days | 17.58% |

In recent years, we have mainly seen the reverse image. The options with the shortest maturity were priced at the lowest volatility, while the long-term options have higher volatility. The stock market slowly moved up with some pullbacks from time to time. During these type of shocks implied volatility is rising the fastest for short term options. The shorter the maturity of an option, the greater the influence of the current market movement on the implied volatility of the option, because the closer is the expiration date. The implied volatility shows the expected fluctuation until expiration and when duration is short, unexpected fluctuations have the biggest impact on implied volatility.

## Conclusion

The matter on both historical and implied volatility is not easy, but it is extremely useful for an options trader.

In the next few articles I will be discussing various option strategies, which include the c**overed call / covered put, vertical spread, straddle, strangle, iron condor and butterfly**.