Consider the following two graphs.
They depict time series of prices for two hypothetical
assets. We may think of the price series on the left as more
risky. We say it is the more "volatile" of the two. We formalize this by
defining volatility as follows. Let
where log denotes a natural logarithm. However, simple returns are sometimes used. This is especially true in the context of portfolio theory. If we assume that returns are conditionally homoskedastic, definition [1] is precise. However, if they are conditionally heteroskedastic, we need to clarify the definition. Does volatility at time t–1 represent the unconditional standard deviation of the time t log return? Or does it represent the standard deviation of the time t log return conditional on information available at time t–1? The answer is the latter. To emphasize this, we might express definition [1] as
where the preceding superscript t–1 indicates that the standard deviation is conditional on information available at time t–1. Another issue in defining volatility is that of the unit of time on which it is based. The standard deviation of a stock's price return over a day might be .01. Over a year, it might be .16. Accordingly, for any process, we might speak of its daily volatility, weekly volatility, annual volatility, etc. All would be distinct notions. This leads to the question of whether, given a volatility based upon one time unit, there is a way to convert it to an equivalent volatility based upon another time unit. As a general rule, the answer is no. To understand why, consider Exhibit 2.
The first time series is a realization of a mean reverting process. There is about as much uncertainty in the price a day in the future as there is a month in the future. If the one-day volatility is .02, then the monthly volatility might also be .02. The second time series is a realization of a random walk. The price is more uncertain a month in the future than it is a day in the future. If the daily volatility is .02, then the monthly volatility would be .05. The third time series is a realization of a process that tends to follow long-term trends. Because of those trends, there is far more uncertainty in prices a month in the future than a day in the future. If the daily volatility is .02, the monthly volatility might be .08. As this example illustrates, volatilities for different units of time are fundamentally different notions. There is no direct relationship between, say a weekly volatility and an annual volatility. However, there is an exception to this observation. The exception is called the square root of time rule. If fluctuations in a stochastic process from one period to the next are independent (i.e., there are no serial correlations or other dependencies) volatility increases with the square root of the unit of time. Any price that follows a random walk, Brownian motion or geometric Brownian motion satisfies this independence condition. The square root of time rule is exact if volatilities are based upon log returns. It is approximately correct if volatilities are based upon simple returns. Let's consider an example. Suppose a price has .04 monthly volatility. If follows a random walk, so, by the square root of time rule, its annual volatility is
Consider another example. A price has .24 annual volatility. What is its daily volatility, assuming it follows a geometric Brownian motion? We can apply the square root of time rule, but this raises an question. In converting from a year to a day, should we count actual days (including weekends and holidays) or should we count only trading days? The latter approach seems more plausible because prices cannot change on days that no trading takes place. In fact, empirical research supports this approach. In most markets, price fluctuations from one trading day to the next appear typically not to be much larger if there is an intervening weekend or holiday as when there is not. Returning to our example, in the United States, there are about 252 trading days in a year. Accordingly, our price will have daily volatility of
Volatilities play an important role in financial engineering and especially in the valuation of various forms of options. In their landmark (1973) paper, Black and Scholes derived a formula for pricing a vanilla put or call option on a non-dividend paying stock. That formula requires as inputs
All of these quantities are (typically) observable in the marketplace—except the volatility. Accordingly, financial engineering has spawned a tremendous need to estimate volatilities for underliers. Those underliers were first prices—say prices of stocks or commodities—but they have come to include quantities such as exchange rates, interest rates or even weather conditions. A standard way to estimate a volatility for a given underlier is to use the price of an option on that underlier. Suppose a call option on the underlier is actively trade, so the option's price is readily obtainable. Then, by applying a suitable option pricing formula—in a sense backwards—we calculate the annual volatility that would have to be input into the option pricing formula to obtain that price for the option. In this manner, we obtain the volatility implied by the option price—what is called the implied volatility for the underlier. Such an implied volatility can then be used to price other options on that same underlier—perhaps options that are not actively traded or for which prices are otherwise not readily available. In practice, this is what financial engineers do to price a variety of derivative instruments—obtain implied volatilities from quoted prices for certain derivatives on an underlier so that they can price other derivatives on that same underlier. However, the process tends to be more involved than this. In practice, different implied volatilities may be obtained for a given underlier depending upon the strike or expiration of the option from which they are obtained. See the article volatility skew. Another approach to estimating volatilities is to apply techniques of time series analysis to historical data for the variable whose volatility is to be estimated. Volatilities calculated in this manner are called historical volatilities. Historical volatilities are routinely used in applications other than financial engineering—such as value-at-risk or portfolio theory—where volatilities are required for quantities on which options are not traded. They might also be used by financial engineers for underliers for which implied volatilities are unavailable—perhaps because options are not actively traded on those underliers. Financial engineers also might use historical volatilities as a "reality check" to supplement implied volatilities.
Historical volatilities are usually calculated from daily data. This means that they are daily volatilities. Because volatilities are usually quoted on an annual basis (especially for option pricing) such daily historical volatilities are routinely converted to an annual basis by applying the square root of time rule. This is done even if conditions for applying that rule are not satisfied. The resulting volatilities are referred to as annualized volatilities—as opposed to annual volatilities—to alert people to the fact that this is just a quoting convention. A question that frequently arises is whether implied or historical volatilities offer a better indication of market risk. The answer is that each has its strengths as well as limitations. Implied volatilities are often referred to as a "market consensus" of volatility—an indication of risk that combines the insights of many market participants. For the most part, this is a reasonable interpretation. However, implied volatilities are essentially prices. They can be biased by such things as bid-ask spreads as well as supply and demand for options. For example, at the height of his speculative activity in 1995, Nick Leeson was selling so many Nikkei options that he drove that implied volatility far below its historical levels. Historical volatility, on the other hand, reflects actual market fluctuations. However, the data upon which an historical volatility is based may be stale—perhaps encompassing a period not reflective of current market conditions. For this reason, implied volatilities tend to be more responsive to current market conditions.
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be a 
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represent prices, accumulated values, exchange rates, interest rates, etc.
The volatility of the process at time t–1 is defined as the
the 
