Realized Volatility

Realized volatility is a measure of how much the price of an asset actually moved over a completed period, calculated from the returns observed in that period. It is backward-looking: it reports what happened, not what is expected. The figure is normally expressed as an annualized percentage so that periods of different lengths can be compared.

The term is used in contrast with implied volatility, which is the market's expectation of future movement extracted from option prices. It is closely related to historical volatility, and the two terms are often used interchangeably. Where a distinction is drawn, realized volatility refers to the estimate built from squared returns, frequently intraday returns, without subtracting the average return.

The standard calculation from $n$ returns is:

• $r_i$ is the logarithmic return in period $i$, that is, the natural log of the price divided by the previous price.
• $n$ is the number of returns in the measurement window.
• $A$ is the number of such periods in a year, commonly 252 for daily returns.

The sum of squared returns on its own is called realized variance. The mean return is treated as zero because over short intervals it is tiny compared with the size of typical moves.

Suppose a share has five daily log returns of 1.0, -1.5, 0.5, 2.0 and -1.0 percent. As decimals, their squares are 0.0001, 0.000225, 0.000025, 0.0004 and 0.0001, which sum to 0.00085. Dividing by 5 gives an average daily variance of 0.00017. Multiplying by 252 gives an annualized variance of 0.04284, and the square root is about 0.207. The realized volatility over the week was therefore about 20.7 percent a year.

A daily closing price hides everything that happened during the day. Research by Torben Andersen, Tim Bollerslev, Francis Diebold and others from the late 1990s showed that summing squared intraday returns gives a far more precise estimate of each day's volatility, in effect turning volatility from something hidden into something that can be observed.

In theory, the finer the sampling the better the estimate. In practice, prices observed at very short intervals bounce between the bid and the ask and are affected by discrete tick sizes, and this microstructure noise inflates the estimate. A common compromise is to sample every five minutes. More refined estimators correct for the noise, and range-based measures such as the Parkinson volatility estimator use the high and low of each period instead of closing prices. Work at this level relies on tick-by-tick data.

Comparing the two is the basis of most volatility trading. An option's price reflects implied volatility, while the profit from hedging that option depends on the volatility that is subsequently realized. A trader who buys an option and practises delta hedging gains if realized volatility over the option's life exceeds the implied level paid, and loses if it falls short. Volatility arbitrage and gamma scalping are built on this relationship.

On average, implied volatility on equity indices has been higher than the volatility later realized. The gap is known as the volatility risk premium and is interpreted as the price investors pay for protection against sharp falls. Some contracts pay out directly on the realized figure. A variance swap, for example, settles on the difference between realized variance over its life and a strike fixed at the start.

Realized volatility is an input to many routine calculations. Risk models use recent realized values to estimate value at risk and to set margin. Volatility targeting strategies scale positions down when realized volatility rises and up when it falls, with the aim of holding portfolio risk steady. Performance measures such as the Sharpe ratio divide return by realized volatility.

Realized series are also the raw material for volatility forecasting. Volatility clusters, with turbulent days tending to follow turbulent days, so recent realized values carry information about the near future. Models in the GARCH family exploit this, as does the heterogeneous autoregressive model proposed by Fulvio Corsi, which forecasts from daily, weekly and monthly averages of realized volatility.

The result depends heavily on choices made by the person calculating it. A 10-day window reacts quickly but is noisy, while a 90-day window is smooth but slow to register change. Sampling frequency, the treatment of overnight gaps and the annualization factor all alter the figure, so two published numbers for the same asset may differ.

Realized volatility also treats upward and downward moves alike, although investors generally care more about losses. It summarizes the size of moves but not their shape, and says nothing about skewness or the frequency of extreme days. Finally, it describes the past. A calm period can end abruptly, and low realized volatility is not evidence that risk is low.