volatility {TTR}

Volatility

Description

Selected volatility estimators/indicators; various authors.

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{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}}

Value

A object of the same class as OHLC or a vector (if try.xts fails) containing the chosen volatility estimator values.

Author(s)

Joshua Ulrich

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

 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")