# Volatility and Implied Volatility

Volatility, as applied to options, is a statistical measurement of the rate of price changes in the underlying asset: the greater the changes in a given time period, the higher the volatility. The volatility of an asset will influence the prices of options based on that asset, with higher volatility leading to higher option premiums. Option premiums depend, in part, on volatility because an option based on a volatile asset is more likely to go into the money before expiration. On the other hand, a low volatile asset will tend to remain within tight limits in its price variation, which means that an option based on that asset will only have a significant probability of going into the money if the underlying price is already close to the strike price. Thus, volatility is a measure of the uncertainty in the expected future price of an asset.

An option premium consists of time value, and it may also consist of intrinsic value if it is in the money. Volatility only affects the time value of the option premium. How much volatility will affect option prices will depend on how much time there is left until expiration: the shorter the time, the less influence volatility will have on the option premium, since there is less time for the price of the underlying to change significantly before expiration.

Higher volatility increases the delta for out-of-the-money options while decreasing delta for in-the-money options; lower volatility has the opposite effect. This relationship holds because volatility has an effect on the probability that the option will finish in the money by expiration: higher volatility will increase the probability that an out-of-the-money option will go into the money by expiration, whereas an in-the-money option could easily go out-of-the-money by expiration. In either case, higher volatility increases the time value of the option so that intrinsic value, if any, is a smaller component of the option premium.

## Implied Volatility is Not Volatility — It Measures the Demand Over Supply for a Particular Option

Because volatility obviously has an influence on option prices, the Black-Scholes model of option pricing includes volatility as a component plus the following factors:

- strike price in relation to the underlying asset price;
- the amount of time remaining until expiration;
- interest rates, where higher interest rates increase the call premium but lower the put premium;
- dividends, where a higher dividend paid by the underlying asset lowers a call premium but increases the put premium.

The Black-Scholes formula calculates only a theoretical price for a call premium; the theoretical price for a put premium can be calculated through the put-call parity relationship. However, the actual value — the market price — of an option premium will be determined by the instantaneous supply and demand for the option.

When the market is active, the following factors are known:

- the actual option premium
- strike price
- time until expiration
- interest rates
- any dividend

Therefore, volatility can be calculated with the Black-Scholes equation or from another option-pricing model by plugging in the known factors into the equation and solving for the volatility that would be required to yield the market price of the call premium. This is what is known as **implied volatility**. Implied volatility does not have to be calculated by the trader, since most option trading platforms provide it for each option listed.

Implied volatility makes no predictions about future price swings of the underlying stock, since the relationship is tenuous at best. Implied volatility can change very quickly, even without any change in the volatility of the underlying asset. Although implied volatility is measured the same as volatility, as a standard deviation percentage, it does not actually reflect the volatility either of the underlying asset or even of the option itself. It is simply the demand over supply for that particular option, and nothing more.

Generally, in a rising market, calls will generally have a higher implied volatility while puts will have a lower implied volatility; in a declining market, puts will have a higher implied volatility over calls. This reflects the increased demand for calls in a rising market and a rising demand for puts in a declining market.

A rise in the implied volatility of a call will decrease the delta for an in-the-money option, because it has a greater chance of going out-of-the-money, whereas for an out-of-the-money option, a higher implied volatility will increase the delta, since it will have a greater probability of finishing in the money.

Implied volatility is not present volatility nor future volatility. It is simply the volatility calculated from the market price of the option premium. There is an indirect connection between historical volatility and implied volatility, in that historical volatility will have a large effect on the market price of the option premium, but the connection is only indirect; implied volatility is directly affected by the market price of the option premium, which, in turn, is influenced by historical volatility. Implied volatility is the volatility that is implied by the current market price of the option premium. That implied volatility does not represent the actual volatility of the underlying asset can be seen more clearly by considering the following scenario: a trader wants to either buy or sell a large number of options on a particular underlying asset. A trader may want to sell because he needs the money; perhaps, it is a pension fund that needs to make payments on its pension obligations. Now, a large order will have a direct influence on the pricing of the option, but it would have no effect on the price of the underlying. It is clear to see that the price change in the option premium is not effected by any changes in the volatility of the underlying asset, because the buy or sell orders are for the option itself, not for the underlying asset. As a further illustration, the implied volatility for puts and calls and for option contracts with different strike prices or expiration dates that are all based on the same underlying asset will have different implied volatilities, because the different options will each have a different supply-demand equilibrium. This is what causes the volatility skew and volatility smile. Thus, implied volatility is not a direct measure of the volatility of the underlying asset.

Implied volatility varies with the change in the supply-demand equilibrium, which is why it measures the supply and demand for a particular option rather than the volatility of the underlying asset. For instance, if a stock is expected to increase in price, then the demand for calls will be greater than the demand for puts, so the calls will have a higher implied volatility, even though both the calls and the puts are based on the same underlying asset. Likewise, puts on indexes, such as the S&P 500, may have a higher implied volatility, since there is a greater demand by fund managers who wish to protect their position in the underlying stocks. At the same time, the same fund managers generally sell calls on the indexes to finance the purchase of puts on the same index; such a spread is referred to as a collar. This lowers the implied volatility on the calls while increasing the implied volatility for the puts.

Because implied volatility is a measure of the instantaneous demand-supply equilibrium, it can indicate that an option is either over- or under-priced relative to the other factors that determine the option premium, but only if implied volatility is not higher because of major news or because of an impending event, such as FDA approval for a drug or the results of an important court case. Likewise, implied volatility may be low because the option is unlikely to go into the money by expiration. If implied volatility is high because of an impending event, then it will decline after the event, since the uncertainty of the event is removed; this rapid deflation of implied volatility is sometimes referred to as a **volatility crush**.

However, implied volatility that is merely due to the normal statistical fluctuation of supply and demand for a particular option may be used to increase profits or decrease losses, especially for an option spread. If an option has high implied volatility, then it may contract later on, reducing the time value of the option premium in relation to the other price determinants; likewise, low implied volatility may have resulted from a temporary decline in demand or a temporary increase in supply that may revert to the average later. So high implied volatility will tend to decline, while low implied volatility will tend to increase over the lifetime of the option. Thus, implied volatility may be an important consideration when setting up option spreads, where maximum profits and losses are determined by how much was paid for long options and how much was received for short options. When selecting long options for a spread, some consideration should be given to selecting strike prices that have lower implied volatilities, while strike prices for short options should have higher implied volatilities. This lowers the debit when paying for long spreads while increasing the credit received for selling short spreads.

Although the implied volatility varies widely among different assets, including different stocks, different indexes, different futures contracts, and so on, the volatility of an index will usually be less than the volatility of individual assets, since an index is a measure of the price changes of all of the individual components of the index, where assets with greater volatility will be offset by other assets with lower volatility.

Implied volatility, like volatility, is calculated as an **annual standard deviation**, expressed as a percentage, that can be used to compare implied volatility of different options that are not only based on the same asset, but also on different assets, including stocks, indexes, or futures. Moreover, the other factors of the option-pricing model, such as interest and dividends, are also usually expressed as an annual percentage. Most trading platforms calculate the implied volatility for the different options.

The **standard deviation** is a statistical measure of the variability — and, therefore, of volatility — of an underlying asset and can be useful in predicting the probability that the asset will be within a particular price range. In a **normal distribution**, which characterizes the price variation of most assets, 68.3% of price changes of the underlying asset over a 1-year period will be within 1 standard deviation of the mean, 95.4% will be within 2 standard deviations, and 99.7% will be within 3 standard deviations. Volatility determines how wide the standard deviation is. If there is little variability, then the normal distribution will be much narrower, whereas for a highly variable asset, the normal distribution would be much flatter, where 1 standard deviation would encompass a wider variability in pricing over a unit of time. So if a stock has a mean price of $100, and has a volatility percentage of 15%, then during the course of the year, the price of the stock will stay within ±$15 for 68.3% of the time.

## Vega and Other Measures of Volatility

Vega measures a change in the theoretical option price caused by a 1-point change in implied volatility. For instance, an option with a vega of .01 will increase by $10 per contract (which consists of 100 shares) for each point increase in volatility and will lose $10 per contract for each 1% decline in volatility. For the short position, vega would have the exact opposite effect, where a 1-point increase in volatility would decrease the value of the short option by $10.

Most options have both intrinsic value and time value. Intrinsic value is a measure of how much the option is in the money; the time value is equal to the option premium minus the intrinsic value. Thus, time value depends on the probability that the option will go into the money or stay into the money by expiration. Volatility only affects the time value of an option. Therefore, vega, as a measure of volatility, is greatest when the time value of the option is greatest and least when it is least. Because time value is greatest when the option is at the money, that is also when volatility will have the greatest effect on the option price. And just as time value diminishes as an option moves further out of the money or into the money, so goes vega.

There are general measures of volatility that represent volatility of entire markets. The Chicago Board Option Exchange (CBOE) Volatility Index (VIX) measures the implied volatility on the options based on the S&P 500 index. This is not the same as implied volatility of the underlying stocks that compose the S&P 500 index, but as a measure of the implied volatility of the options on those stocks. VIX is also known as a **fear index**, because it presumably measures the amount of fear in the market; in actuality, it probably causes fear rather than reflecting fear, because higher fluctuations in the supply and demand for the options creates more uncertainty. Other measures of general volatility include the NASDAQ 100 Volatility Index (VXN), which measures the volatility of the NASDAQ 100, which includes many high-tech companies. The Russell 2000 Volatility Index (RVX) measures the volatility of the index composed of the 2000 stocks in the Russell 2000 Index.

## Volatility Skew

**Volatility skew** is a result of different implied volatilities for different strike prices and for whether the option is a call or put. Volatility skew further illustrates that implied volatility depends only on the option premium, not on the volatility of the underlying asset, since that does not change with either different strike prices or option type.

How the volatility skew changes with different strike prices depends on the type of skew, which is influenced by the supply and demand for the different options. A **forward skew** is exhibited by higher implied volatilities for higher strike prices. A **reverse skew** is one with lower implied volatilities for higher strike prices. A **smiling skew** is exhibited by an implied volatility distribution that increases for strike prices that are either lower or higher than the price of the underlying. A **flat skew** means that there is no skew: implied volatility is the same for all strike prices. The options of most underlying assets exhibit a reverse skew, reflecting the fact that slightly out-of-the-money options have a greater demand than those that are in the money. Furthermore, out-of-the-money options have a higher time value, so volatility will have a greater effect for options that only have time value. Thus, a call and a put at the same strike price will have different implied volatilities, since the strike price will likely differ from the price of the underlying, demonstrating yet again that implied volatility is not the result of the volatility of the underlying asset.

Options with the same strike prices but with different expiration months also exhibit a skew, with the near months generally showing a higher implied volatility than the far months, reflecting a greater demand for near-term options over those with later expirations.