Selected volatility estimators/indicators; various authors.
volatility(OHLC, n = 10, calc = "close", N = 260, mean0 = FALSE, ...)A object of the same class as OHLC or a vector (if
try.xts fails) containing the chosen volatility estimator values.
Object that is coercible to xts or matrix and contains
Open-High-Low-Close prices (or only Close prices, if calc="close").
Number of periods for the volatility estimate.
The calculation (type) of estimator to use.
Number of periods per year.
Use a mean of 0 rather than the sample mean.
Arguments to be passed to/from other methods.
Joshua Ulrich
Close-to-Close Volatility (calc="close")
$$ \sigma_{cl} = \sqrt{\frac{N}{n-2} \sum_{i=1}^{n-1}(r_i-\bar{r})^2}
$$
$$where\;\; r_i = \log \left(\frac{C_i}{C_{i-1}}\right) $$
$$and\;\; \bar{r} = \frac{r_1+r_2+\ldots +r_{n-1}}{n-1} $$
OHLC Volatility: Garman and Klass (calc="garman.klass")
The
Garman and Klass estimator for estimating historical volatility assumes
Brownian motion with zero drift and no opening jumps (i.e. the opening =
close of the previous period). This estimator is 7.4 times more efficient
than the close-to-close estimator.
$$ \sigma = \sqrt{ \frac{N}{n} \sum
\left[ \textstyle\frac{1}{2}\displaystyle
\left( \log \frac{H_i}{L_i} \right)^2 - (2\log 2-1)
\left( \log \frac{C_i}{O_i} \right)^2 \right] }
$$
High-Low Volatility: Parkinson (calc="parkinson")
The Parkinson formula for estimating the historical volatility of
an underlying based on high and low prices.
$$ \sigma = \sqrt{ \frac{N}{4 n \times \log 2} \sum_{i=1}^{n}
\left(\log \frac{H_i}{L_i}\right)^2}
$$
OHLC Volatility: Rogers and Satchell (calc="rogers.satchell")
The Roger and Satchell historical volatility estimator allows for non-zero
drift, but assumed no opening jump.
$$ \sigma = \sqrt{ \textstyle\frac{N}{n} \sum \left[
\log \textstyle\frac{H_i}{C_i} \times \log \textstyle\frac{H_i}{O_i} +
\log \textstyle\frac{L_i}{C_i} \times \log \textstyle\frac{L_i}{O_i} \right] }
$$
OHLC Volatility: Garman and Klass - Yang and Zhang
(calc="gk.yz")
This estimator is a modified version of the Garman
and Klass estimator that allows for opening gaps.
$$ \sigma = \sqrt{ \textstyle\frac{N}{n} \sum \left[
\left( \log \textstyle\frac{O_i}{C_{i-1}} \right)^2 +
\textstyle\frac{1}{2}\displaystyle
\left( \log \textstyle\frac{H_i}{L_i} \right)^2 - (2 \times \log 2-1)
\left( \log \textstyle\frac{C_i}{O_i} \right)^2 \right] }
$$
OHLC Volatility: Yang and Zhang (calc="yang.zhang")
The Yang
and Zhang historical volatility estimator has minimum estimation error, and
is independent of drift and opening gaps. It can be interpreted as a
weighted average of the Rogers and Satchell estimator, the close-open
volatility, and the open-close volatility.
Users may override the default values of \(\alpha\) (1.34 by default) or
\(k\) used in the calculation by specifying alpha or k in
..., respectively. Specifying k will cause alpha to be
ignored, if both are provided.
$$ \sigma^2 = \sigma_o^2 + k\sigma_c^2 + (1-k)\sigma_{rs}^2
$$
$$ \sigma_o^2 =\textstyle \frac{N}{n-1} \sum
\left( \log \frac{O_i}{C_{i-1}}-\mu_o \right)^2
$$
$$ \mu_o=\textstyle \frac{1}{n} \sum \log \frac{O_i}{C_{i-1}} $$
$$ \sigma_c^2 =\textstyle \frac{N}{n-1} \sum
\left( \log \frac{C_i}{O_i}-\mu_c \right)^2
$$
$$ \mu_c=\textstyle \frac{1}{n} \sum \log \frac{C_i}{O_i} $$
$$ \sigma_{rs}^2 = \textstyle\frac{N}{n} \sum \left(
\log \textstyle\frac{H_i}{C_i} \times \log \textstyle\frac{H_i}{O_i} +
\log \textstyle\frac{L_i}{C_i} \times \log \textstyle\frac{L_i}{O_i}
\right)
$$
$$ k=\frac{\alpha-1}{alpha+\frac{n+1}{n-1}}
$$
The following sites were used to code/document these
indicators. All were created by Thijs van den Berg under the GNU Free
Documentation License and were retrieved on 2008-04-20. The original
links are dead, but can be accessed via internet archives.
Close-to-Close Volatility (calc="close"):
https://web.archive.org/web/20100421083157/http://www.sitmo.com/eq/172
OHLC Volatility: Garman Klass (calc="garman.klass"):
https://web.archive.org/web/20100326172550/http://www.sitmo.com/eq/402
High-Low Volatility: Parkinson (calc="parkinson"):
https://web.archive.org/web/20100328195855/http://www.sitmo.com/eq/173
OHLC Volatility: Rogers Satchell (calc="rogers.satchell"):
https://web.archive.org/web/20091002233833/http://www.sitmo.com/eq/414
OHLC Volatility: Garman Klass - Yang Zhang (calc="gk.yz"):
https://web.archive.org/web/20100326215050/http://www.sitmo.com/eq/409
OHLC Volatility: Yang Zhang (calc="yang.zhang"):
https://web.archive.org/web/20100326215050/http://www.sitmo.com/eq/409
See TR and chaikinVolatility for other
volatility measures.
data(ttrc)
ohlc <- ttrc[,c("Open","High","Low","Close")]
vClose <- volatility(ohlc, calc="close")
vClose0 <- volatility(ohlc, calc="close", mean0=TRUE)
vGK <- volatility(ohlc, calc="garman")
vParkinson <- volatility(ohlc, calc="parkinson")
vRS <- volatility(ohlc, calc="rogers")
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