Volatility

From Exampleproblems

For volatility in chemistry, see Volatility(Chemistry)

Volatility is the standard deviation of the change in value 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 (100$ ± 5$) or a fraction of the initial value (100$ ± 5%).

For a financial instrument whose return follows a Gaussian random walk, or Wiener process, the volatility increases by the square-root of time 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. Mathematically, this is a direct result of applying Ito's lemma to the random process.

Historical 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 volatility implied by the market.

1 Volatility for market players
2 Volatility versus direction
3 Volatility over time
4 Defined
5 See also
6 External links

Volatility for market players

Volatility is often viewed as a negative in that it represents uncertainty and risk. However, volatility can be good in that if one shorts on the peaks, and buys on the lows one can make money, with greater money coming with greater volatility. The possibility for money to be made via volatile markets is how short term market players like day traders make money, and is in contrast to the long term investment view of buy and hold.

It is also possible to trade volatility directly, through the use of derivative securities such as options. See Volatility arbitrage.

Volatility versus direction

Volatility does not imply direction. (This is due to the fact that all changes are squared.) An instrument that is more volatile is likely to increase or decrease in value more than one that is less volatile.

For example, a checking account has low volatility. It won't lose 50% in a year. But then, it won't gain 50% in a year, either.

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 can seem to move almost not at all for a long time.

An extreme is a stock market crash or bubble. A time when prices fall quickly is often followed by prices going down even more, or going up an unusual amount. As well, a time when prices rise quickly is often may be followed by prices going up even more, or going down an unusual amount.

The converse is, '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. Of course, whether such large movements have the same direction, or opposite, is much more difficult to say.

This is termed autoregressive conditional heteroskedasticity.

Defined

The annualized volatility $σ$ is proportional to standard deviation $σ S D$ of the instrument's returns by the square-root of time period of the returns :

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

where $P$ is time period in years of returns. The generalized volatility $σ T$ for time horizon $T$ is expressed as:

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

For example, if the daily returns of a stock have a standard deviation of 0.01 and there are 252 trading days in a year, then the time period of returns is 1/252 and annualized volatility is

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

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

$\sigma_{month} = 0.1587 \sqrt{1/12} = 0.0458$ .

Note that the formula used to annualize returns is not deterministic, but it is an extrapolation valid for a Wiener process. Generally, the relation between volatility in different time scales is more complicate, involving Lévy stability exponent.