# volatility

##### Volatility

Selected volatility estimators/indicators; various authors.

- Keywords
- ts

##### Usage

`volatility(OHLC, n = 10, calc = "close", N = 260, mean0 = FALSE, ...)`

##### Arguments

- OHLC
Object that is coercible to xts or matrix and contains Open-High-Low-Close prices (or only Close prices, if

`calc="close"`

).- n
Number of periods for the volatility estimate.

- calc
The calculation (type) of estimator to use.

- N
Number of periods per year.

- mean0
Use a mean of 0 rather than the sample mean.

- …
Arguments to be passed to/from other methods.

##### Details

Close-to-Close Volatility (

`calc="close"`

) $$ \sigma_{cl} = \sqrt{\frac{Z}{n-2} \sum_{i=1}^{n-1}(r_i-\bar{r})^2} $$ $$where\;\; r_i = \log \left(\frac{C_i}{C_{i-1}}\right) $$ $$and\;\; \bar{r} = \frac{r_1+r_2+\ldots +r_{n-1}}{n-1} $$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. $$ \sigma = \sqrt{ \frac{Z}{n} \sum \left[ \textstyle\frac{1}{2}\displaystyle \left( \log \frac{H_i}{L_i} \right)^2 - (2\log 2-1) \left( \log \frac{C_i}{O_i} \right)^2 \right] } $$High-Low Volatility: Parkinson (

`calc="parkinson"`

) The Parkinson formula for estimating the historical volatility of an underlying based on high and low prices. $$ \sigma = \sqrt{ \frac{Z}{4 n \times \log 2} \sum_{i=1}^{n} \left(\log \frac{H_i}{L_i}\right)^2} $$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. $$ \sigma = \sqrt{ \textstyle\frac{Z}{n} \sum \left[ \log \textstyle\frac{H_i}{C_i} \times \log \textstyle\frac{H_i}{O_i} + \log \textstyle\frac{L_i}{C_i} \times \log \textstyle\frac{L_i}{O_i} \right] } $$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. $$ \sigma = \sqrt{ \textstyle\frac{Z}{n} \sum \left[ \left( \log \textstyle\frac{O_i}{C_{i-1}} \right)^2 + \textstyle\frac{1}{2}\displaystyle \left( \log \textstyle\frac{H_i}{L_i} \right)^2 - (2 \times \log 2-1) \left( \log \textstyle\frac{C_i}{O_i} \right)^2 \right] } $$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. $$ \sigma^2 = \sigma_o^2 + k\sigma_c^2 + (1-k)\sigma_{rs}^2 $$ $$ \sigma_o^2 =\textstyle \frac{Z}{n-1} \sum \left( \log \frac{O_i}{C_{i-1}}-\mu_o \right)^2 $$ $$ \mu_o=\textstyle \frac{1}{n} \sum \log \frac{O_i}{C_{i-1}} $$ $$ \sigma_c^2 =\textstyle \frac{Z}{n-1} \sum \left( \log \frac{C_i}{O_i}-\mu_c \right)^2 $$ $$ \mu_c=\textstyle \frac{1}{n} \sum \log \frac{C_i}{O_i} $$ $$ \sigma_{rs}^2 = \textstyle\frac{Z}{n} \sum \left( \log \textstyle\frac{H_i}{C_i} \times \log \textstyle\frac{H_i}{O_i} + \log \textstyle\frac{L_i}{C_i} \times \log \textstyle\frac{L_i}{O_i} \right) $$ $$ k=\frac{\alpha}{1+\frac{n+1}{n-1}} $$

##### Value

A object of the same class as `OHLC`

or a vector (if
`try.xts`

fails) containing the chosen volatility estimator values.

##### References

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 Also

See `TR`

and `chaikinVolatility`

for other
volatility measures.

##### Examples

`library(TTR)`

```
# NOT RUN {
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")
# }
```

*Documentation reproduced from package TTR, version 0.23-1, License: GPL-2*