Realised volatility estimators

Measuring volatility you cannot see

Volatility is never observed directly, only estimated. The simplest estimator is close-to-close: take the standard deviation of daily log returns and annualise it by multiplying by $\sqrt{252}$ . It uses one number per day and ignores everything that happened between the open and the close. With intraday prices you can do much better. The realised variance of a day is the sum of its squared intraday returns, $RV_t=\sum_{i=1}^{m}r_{t,i}^{2}$ , and as the sampling gets finer it approaches the true variance of the day. In practice the finest sampling backfires, because bid-ask bounce and rounding add noise that inflates $RV$ ; returns every 5 minutes, or an average over several offset grids, are a common compromise.

RV_t=\sum_{i=1}^{m}r_{t,i}^{2},\qquad \hat\sigma^{2}_{P}=\frac{\big(\ln(H/L)\big)^{2}}{4\ln 2},\qquad \sigma_{A}=\sigma_{D}\sqrt{252},\qquad RV_{t+1}=\beta_0+\beta_dRV_t+\beta_wRV_{t}^{(5)}+\beta_mRV_{t}^{(22)}+\varepsilon

(1)

Without intraday data, the daily high $H$ and low $L$ still carry information about the day's path. Parkinson's estimator $\hat\sigma^{2}=(\ln(H/L))^{2}/(4\ln2)$ is about five times more efficient than close-to-close, and Garman-Klass adds the open and the close. They assume continuous trading, so they are biased down when prices jump or when the overnight gap is large. The reason to want a better measure is forecasting: the squared daily return is an unbiased but extremely noisy proxy for the day's variance, one draw from a wide distribution, while $RV$ averages many draws. The HAR-RV model of Corsi regresses tomorrow's $RV$ on today's value, the average of the last 5 days and the average of the last 22 days. Three terms are enough to reproduce the slow, long-memory decay of volatility, and the model is hard to beat as a benchmark.

Two small estimatesFour intraday returns of

0.10\%

-0.20\%

0.15\%

and

0.05\%

give

RV=0.01+0.04+0.0225+0.0025=0.075

in percent squared, a volatility of

0.27\%

for the stretch they cover. A day with high

102

and low

99

gives

\ln(102/99)=0.0299

, so Parkinson's variance is

0.0299^{2}/(4\ln2)=0.000321

and the daily volatility is

1.79\%

, which annualises to

1.79\%\cdot\sqrt{252}\approx28.5\%

. A HAR forecast with

\beta=(0.1,0.3,0.4,0.2)

and

RV

values

2

(today),

1.5

(5-day mean) and

1

(22-day mean) is

0.1+0.6+0.6+0.2=1.5

Close-to-close: standard deviation of daily returns times $\sqrt{252}$ ; it ignores the path within the day.
Realised variance sums squared intraday returns; very fine sampling adds microstructure noise.
Range estimators (Parkinson, Garman-Klass) use high, low, open and close; HAR-RV forecasts from daily, weekly and monthly averages.

Lab · what's graded

Realised variance15 pts
Annualising15 pts
Parkinson's estimator15 pts
A HAR forecast15 pts
Why not sample every second10 pts
Match the vocabulary15 pts
A better proxy than the squared return15 pts

Measuring volatility you cannot see

RV_t=\sum_{i=1}^{m}r_{t,i}^{2},\qquad \hat\sigma^{2}_{P}=\frac{\big(\ln(H/L)\big)^{2}}{4\ln 2},\qquad \sigma_{A}=\sigma_{D}\sqrt{252},\qquad RV_{t+1}=\beta_0+\beta_dRV_t+\beta_wRV_{t}^{(5)}+\beta_mRV_{t}^{(22)}+\varepsilon

(1)

Two small estimatesFour intraday returns of

0.10\%

-0.20\%

0.15\%

and

0.05\%

give

RV=0.01+0.04+0.0225+0.0025=0.075

in percent squared, a volatility of

0.27\%

for the stretch they cover. A day with high

102

and low

99

gives

\ln(102/99)=0.0299

, so Parkinson's variance is

0.0299^{2}/(4\ln2)=0.000321

and the daily volatility is

1.79\%

, which annualises to

1.79\%\cdot\sqrt{252}\approx28.5\%

. A HAR forecast with

\beta=(0.1,0.3,0.4,0.2)

and

RV

values

2

(today),

1.5

(5-day mean) and

1

(22-day mean) is

0.1+0.6+0.6+0.2=1.5

Close-to-close: standard deviation of daily returns times $\sqrt{252}$ ; it ignores the path within the day.
Realised variance sums squared intraday returns; very fine sampling adds microstructure noise.
Range estimators (Parkinson, Garman-Klass) use high, low, open and close; HAR-RV forecasts from daily, weekly and monthly averages.

Lab · what's graded

Realised variance15 pts
Annualising15 pts
Parkinson's estimator15 pts
A HAR forecast15 pts
Why not sample every second10 pts
Match the vocabulary15 pts
A better proxy than the squared return15 pts

Measuring volatility you cannot see

Lab · what's graded

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Realised volatility estimators

Measuring volatility you cannot see

Lab · what's graded

Log in to run your code.