Volatility (finance)

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Volatility most frequently refers to the standard deviation of the continuously compounded returns of a financial instrument with a specific time horizon. It is often used to quantify the risk of the instrument over that time period. Volatility is typically expressed in annualized terms, and it may either be an absolute number ($5) or a fraction of the mean (5%). Volatility can be traded directly in today's markets through options and variance swaps.

For a financial instrument whose price follows a Gaussian random walk, or Wiener process, the volatility increases as time increases. Conceptually, this is because there is an increasing probability that the instrument's price will be farther away from the initial price as time increases. However, rather than increase linearly, the volatility increases with the square-root of time as time increases, because some fluctuations are expected to cancel each other out, so the most likely deviation after twice the time will not be twice the distance from zero.

More broadly, volatility refers to the degree of (typically short-term) unpredictable change over time of a certain variable. It may be measured via the standard deviation of a sample, as mentioned above. However, price changes actually do not follow Gaussian distributions. Better distributions used to describe them actually have "fat tails" although their variance remains finite. Therefore, other metrics may be used to describe the degree of spread of the variable. As such, volatility reflects the degree of risk faced by someone with exposure to that variable.

Historical volatility (or ex-post volatility) is the volatility of a financial instrument based on historical returns. This phrase is used particularly when it is wished to distinguish between the actual volatility of an instrument in the past, and the current (ex-ante, or forward-looking) volatility implied by the market.

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[edit] Volatility for market players

When investing directly in a security, volatility is often viewed as a negative in that it represents uncertainty and risk. However, with other investing strategies, volatility is often desirable. For example, if an investor is short on the peaks, and long on the lows of a security, the profit will be greatest when volatility is highest.

In today's markets, it is also possible to trade volatility directly, through the use of derivative securities such as options and variance swaps. See Volatility arbitrage.

[edit] Volatility versus direction

Volatility does not measure the direction of a trend, only its magnitude. This is because when calculating standard deviation (or variance), all differences are squared, so that negative and positive differences are combined into one quantity. Two instruments with different volatilities may have the same expected return, but the instrument with higher volatility will have a larger swings in values at the end of a given period of time.

For example, a lower volatility stock may have an expected (average) return of 7%, with annual volatility of 5%. This would indicate returns from approximately -3% to 17% most of the time (19 times out of 20, or 95%). A higher volatility stock, with the same expected return of 7% but with annual volatility of 20%, would indicate returns from approximately -33% to 47% most of the time (19 times out of 20, or 95%)

[edit] Volatility over time

It's common knowledge that types of assets experience periods of high and low volatility. That is, during some periods prices go up and down quickly, while during other times they might not seem to move at all.

Periods when prices fall quickly (a crash) are often followed by prices going down even more, or going up by an unusual amount. Also, a time when prices rise quickly (a bubble) may often be followed by prices going up even more, or going down by an unusual amount.

The converse behavior, 'doldrums' can last for a long time as well.

Most typically, extreme movements do not appear 'out of nowhere'; they're presaged by larger movements than usual. This is termed autoregressive conditional heteroskedasticity. Of course, whether such large movements have the same direction, or the opposite, is more difficult to say. And an increase in volatility does not always presage a further increase—the volatility may simply go back down again.

[edit] Mathematical definition

The annualized volatility σ is the standard deviation σ of the instrument's logarithmic returns in a year.

The generalized volatility σT for time horizon T in years is expressed as:

\sigma_T = \sigma \sqrt{T}.\,

Therefore, if the daily logarithmic returns of a stock have a standard deviation of σSD and the time period of returns is P, the annualized volatility is

\sigma = {\sigma_{SD}\over\sqrt{P}} .\,

A common assumption is that P = 1 / 252 (there are 252 trading days in any given year). Then, if σSD = 0.01 the annualized volatility is

\sigma = {0.01 \over \sqrt{\frac{1}{252}}} = 0.1587.

The monthly volatility (i.e., T = 1 / 12 of a year) would be

\sigma_\text{month} = 0.1587 \sqrt{\frac{1}{12}} = 0.0458.

Note that the formula used to annualize returns is not deterministic, but is an extrapolation valid for a random walk process whose steps have finite variance. Generally, the relation between volatility in different time scales is more complicated, involving the Lévy stability exponent α:

\sigma_T = T^{1/\alpha} \sigma.\,

If α = 2 you get the Wiener process scaling relation, but some people believe α < 2 for financial activities such as stocks, indexes and so on. This was discovered by Benoît Mandelbrot, who looked at cotton prices and found that they followed a Lévy alpha-stable distribution with α = 1.7. (See New Scientist, 19 April, 1997].) Mandelbrot's conclusion is, however, not accepted by mainstream financial econometricians.

[edit] Quick-and-dirty (percentage) volatility measurement

Suppose you notice that a market price index which is approximately 10,000, moves about 100 points a day on average. This would constitute a 1% (up or down) daily movement. To annualize this, you can use the "rule of 16", that is, multiply by 16 to get 16% as the overall (annual) volatility. The rationale for this is that 16 is the square root of 256, which is approximately the number of actual trading days in a year. This uses the statistical result that the standard deviation of the sum of n independent variables (with equal standard deviations) is \sqrt{n} times the standard deviation of the individual variables. It also takes the average magnitude of the observations as an approximation to the standard deviation of the variables. Under the assumption that the variables are normally distributed with mean zero and standard deviation σ, the expected value of the magnitude of the observations is √(2/π)σ = 0.798σ, thus the observed average magnitude of the observations may be taken as a rough approximation to σ.

[edit] See also

[edit] References

Question book-new.svg
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  • Lin Chen (1996). Stochastic Mean and Stochastic Volatility -- A Three-Factor Model of the Term Structure of Interest Rates and Its Application to the Pricing of Interest Rate Derivatives. Blackwell Publishers. 

[edit] External links

v • d • e
Volatility
Modelling volatility
Trading volatility
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