# volatility: Volatility In TTR: Technical Trading Rules

## Description

Selected volatility estimators/indicators; various authors.

## Usage

 1 volatility(OHLC, n = 10, calc = "close", N = 260, ...)

## 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. ... Arguments to be passed to/from other methods.

## Details

• Close-to-Close Volatility (calc="close")

sqrt(N) * runSD(ROC(Cl), 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.

sqrt(N/n * runSum(0.5 * log(Hi/Lo)^2 - (2*log(2)-1) * log(Cl/Op)^2, n))

• High-Low Volatility: Parkinson (calc="parkinson")
The Parkinson formula for estimating the historical volatility of an underlying based on high and low prices.

sqrt(N/(4*n*log(2)) * runSum(log(Hi/Lo)^2, n))

• 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.

sqrt(N/n * runSum(log(Hi/Cl) * log(Hi/Op) + log(Lo/Cl) * log(Lo/Op), n))

• 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.

sqrt(N/n * runSum(log(Op/lag(Cl,1))^2 + 0.5 * log(Hi/Lo)^2 - (2*log(2)-1) * log(Cl/Op)^2 , n))

• 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 α (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.

s <- sqrt(s2o + k*s2c + (1-k)*(s2rs^2))

s2o <- N * runVar(log(Op/lag(Cl,1)), n=n)

s2c <- N * runVar(log(Cl/Op), n=n)

s2rs <- volatility(OHLC, n, "rogers.satchell", N, ...)

k <- (alpha-1) / (alpha + (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.

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 links are currently dead, but can be accessed via internet archives.

Close-to-Close Volatility (calc="close"):
http://www.sitmo.com/eq/172

OHLC Volatility: Garman Klass (calc="garman.klass"):
http://www.sitmo.com/eq/402

High-Low Volatility: Parkinson (calc="parkinson"):
http://www.sitmo.com/eq/173

OHLC Volatility: Rogers Satchell (calc="rogers.satchell"):
http://www.sitmo.com/eq/414

OHLC Volatility: Garman Klass - Yang Zhang (calc="gk.yz"):
http://www.sitmo.com/eq/409

OHLC Volatility: Yang Zhang (calc="yang.zhang"):
http://www.sitmo.com/eq/417