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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

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• 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

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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

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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

Figure 5.4: Forecast of Daily, Weekly and Monthly S&P 500 Volatility