Figure 5.4: Forecast of Daily, Weekly and Monthly S&P 500 Volatility
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Heterogeneous Autoregressions (HAR)
• The HAR model can capture the leverage effect by simply
including the return on the right-hand side
• In the daily log HAR we can write
• This can also easily be estimated using OLS.
• Notice that because the model is written in logs the
variance forecast will not go negative
•
will always be a positive
number
Elements of Financial Risk Management Second Edition © 2012 by Peter Christoffersen
31
Heterogeneous Autoregressions (HAR)
• The stylized facts of RV suggested that we can assume
• Under this assumption, we can compute Value-at-Risk by
• where RVmt+1/t is provided by either the ARMA or HAR
forecasting models
• Expected Shortfall is computed by
Elements of Financial Risk Management Second Edition © 2012 by Peter Christoffersen
Combining GARCH and RV
32
• Here we try to incorporate the rich information in RV into
a GARCH modeling framework
• Consider the basic GARCH model:
• Given the information on daily RV we could augment the
GARCH model with RV as follows:
• In this GARCH-X model, RV is the explanatory variable
can be estimated using the univariate MLE approach
Elements of Financial Risk Management Second Edition â 2012 by Peter Christoffersen
Combining GARCH and RV
A shortcoming of the GARCH-X approach is that
a model for RV is not specified.
• This means that we cannot use the model to
forecast volatility beyond one day ahead.
• The more general so-called Realized GARCH
model is defined by
• where t is the innovation to RV
Elements of Financial Risk Management Second Edition © 2012 by Peter Christoffersen
33
Combining GARCH and RV
34
• This model can be estimated by MLE when assuming that Rt
and t have a joint normal distribution.
• The Realized GARCH model can be augmented to include a
leverage effect as well.
• In the Realized GARCH model the VaR and ES would
simply be
• and
Elements of Financial Risk Management Second Edition © 2012 by Peter Christoffersen
The All RV Estimator
35
• As discussed before, in the ideal case with ultra-high
liquidity we have m = 24 * 60 observations available
within a day
• We can calculate an estimate of the daily variance from the
intraday squared returns simply as
• This estimator is sometimes known as the All RV
estimator because it uses all the prices on the
1minute grid
Elements of Financial Risk Management Second Edition â 2012 by Peter Christoffersen
The All RV Estimator
36
Figure 5.5 uses simulated data to illustrate one of the
problems caused by illiquidity when estimating asset
price volatility.
• We assume the fundamental asset price, Sfund, follows the
simple random walk process with constant variance
• Where e = 0.001 in Figure 5.5
• The observed price fluctuates randomly around the bid
and ask quotes that are posted by the market maker
Elements of Financial Risk Management Second Edition â 2012 by Peter Christoffersen
The All RV Estimator
37
We observe
• where Bt+j/m is the bid price, which is the fundamental
price rounded down to the nearest $1/10
• At+j/m is the ask price, which is the fundamental price
rounded up to the nearest $1/10.
• It+j/m is an i.i.d. random variable, which takes the values 1
and 0 each with probability 1/2.
• It+j/m is thus an indicator variable of whether the observed
price is a bid or an ask price
Elements of Financial Risk Management Second Edition © 2012 by Peter Christoffersen
Figure 5.5: Fundamental Price and Quoted Price with
Bid-Ask Bounces
Elements of Financial Risk Management Second Edition â 2012 by Peter Christoffersen
38
The All RV Estimator
39
Figure 5.5 shows that the observed intraday price can be
very noisy compared with the smooth fundamental but
unobserved price.
• The bidask spread adds a layer of noise on top of the
fundamental price.
• If we compute RVmt+1 from the high-frequency Sobst+j/m then
we will get an estimate of 2 that is higher than the true
value because of the inclusion of the bid-ask volatility in
the estimate
Elements of Financial Risk Management Second Edition © 2012 by Peter Christoffersen