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Figure 5.6: Range-Based Variance Proxy (top) and Squared Returns (bottom)

Figure 5.6: Range-Based Variance Proxy (top) and Squared Returns (bottom)

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67



Range-based Proxies for Volatility

• Range-based volatility proxy does not make use of the

daily open and close prices

• Assuming that the asset log returns are normally

distributed with zero mean and variance, 2; a more

accurate range-based proxy can be derived as



Elements of Financial Risk Management Second Edition © 2012 by Peter Christoffersen



68



Figure 5.7:Autocorrelation of Range-Based Variance Proxy

and Autocorrelation of Squared Returns



Elements of Financial Risk Management Second Edition © 2012 by Peter Christoffersen



Range-based Proxies for Volatility

• In the more general case where the mean return is not

assumed to be zero the following range-based volatility

proxy is available



• All of these proxies are derived assuming that true

variance is constant

• For example, 30 days of high, low, open, and close

information can be used to estimate the (constant)

volatility for that period

Elements of Financial Risk Management Second Edition © 2012 by Peter Christoffersen



69



Forecasting Volatility Using the Range



70



• Now we are going to use RPt in the HAR model

• Several studies show that the log range is close to normally

distributed as:



• The strong persistence of the range and log normal

property suggest a log HAR model of the form



Elements of Financial Risk Management Second Edition © 2012 by Peter Christoffersen



Forecasting Volatility Using the Range

• where we have that



• The range-based proxy can also be used as a regressor

in GARCH-X models, for example



Elements of Financial Risk Management Second Edition © 2012 by Peter Christoffersen



71



Forecasting Volatility Using the Range

• A purely range-based model can be defined as



• Finally, a Realized-GARCH style model (RangeGARCH) can be defined via



Elements of Financial Risk Management Second Edition â 2012 by Peter Christoffersen



72



Forecasting Volatility Using the Range



73



The Range-GARCH model can be estimated using

bivariate maximum likelihood techniques using historical

data on return, Rt, and on range proxy, RPt

• ES and VaR can be constructed in the RP-based models by

assuming that zt+1 is i.i.d. normal where zt+1 = Rt+1/ t+1 in

the GARCH-style models or

in the

HAR model.



Elements of Financial Risk Management Second Edition © 2012 by Peter Christoffersen



74



Range Based vs Realized Variance

• For very liquid securities the RV modeling approach is

useful as the intraday returns gives a very reliable estimate

of today’s variance, which in turn helps forecast

tomorrow’s variance

• The GARCH estimate of today’s variance is heavily model

dependent, whereas the realized variance for today is

calculated from today’s squared intraday returns



Elements of Financial Risk Management Second Edition © 2012 by Peter Christoffersen



Range Based vs Realized Variance



75



• Shortcomings of realized variance approach includes

• It requires high-quality intraday returns to be feasible

• It is easy to calculate daily realized volatilities from 5minute returns, but it is difficult to construct at 10-year data

set of 5-minute returns

• Realized variance measures based on intraday returns can

be noisy

• This is especially true for securities with wide bid–ask

spreads and infrequent trading.

• However, range-based variance measure is relatively

immune to the market microstructure noise

Elements of Financial Risk Management Second Edition © 2012 by Peter Christoffersen



Range Based vs Realized Variance



76



• The true maximum can easily be calculated as the

observed maximum less one half of the bid–ask spread

• The true minimum as the observed minimum plus one

half of the bid–ask price

• The range-based variance measure thus has clear

advantages in less liquid markets

• In the absence of trading imperfections, range-based

variance proxies can be shown to be only about as useful

as 4-hour intraday returns



Elements of Financial Risk Management Second Edition © 2012 by Peter Christoffersen



GARCH Variance Forecast Evaluation

Revisited



77



• The realized variance measure can be used for evaluating

the forecasts from variance models.

• If only squared returns are available then we can run the

regression

• where 2t+1/t is the forecast from the GARCH model

• With RV-based estimates we can run the regression

• where we have used the Average RV estimator as an

example

Elements of Financial Risk Management Second Edition © 2012 by Peter Christoffersen



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Figure 5.6: Range-Based Variance Proxy (top) and Squared Returns (bottom)

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