Figure 5.6: Range-Based Variance Proxy (top) and Squared Returns (bottom)
Tải bản đầy đủ - 0trang
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