Figure 5.5: Fundamental Price and Quoted Price with Bid-Ask Bounces
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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 â 2012 by Peter Christoffersen
The Sparse RV Estimator
40
Here we try to construct an s-minute grid (where s ≥ 1)
instead of a 1-minute grid so that our new RV estimator
would be
• It is sometimes denoted as the Sparse RV estimator as
opposed to the previous All RV estimator
• The question is how to choose the parameter s?
• The larger the s the less likely we are to get a biased
estimate of volatility,
• But the larger the s the fewer observations we are using and
so the more noisy our estimate will be
Elements of Financial Risk Management Second Edition © 2012 by Peter Christoffersen
The Sparse RV Estimator
41
• The choice of s clearly depends on the specific asset
• For very liquid assets we should use an s close to 1 and for
illiquid assets s should be much larger
• If liquidity effects manifest themselves as a bias in
estimated RVs when using a high sampling frequency then
that bias should disappear when the sampling frequency is
lowered (when s is increased)
Elements of Financial Risk Management Second Edition â 2012 by Peter Christoffersen
The Sparse RV Estimator
Volatility signature plots provide a convenient
graphical tool for choosing s:
• First compute RVst+1 for values of s going from 1
to 120 minutes.
• Second, scatter plot the average RV across days on
the vertical axis against s on the horizontal axis.
• Third, look for the smallest s such that the average
RV does not change much for values of s larger
than this number
Elements of Financial Risk Management Second Edition © 2012 by Peter Christoffersen
42
The Sparse RV Estimator
• In markets with wide bid–ask spreads the average
RV in the volatility signature plot will be
downward sloping for small s
• But for larger s the average RV will stabilize at the
true long run volatility level
• We want to choose the smallest s for which the
average RV is stable. This will avoid bias and
minimize variance.
Elements of Financial Risk Management Second Edition â 2012 by Peter Christoffersen
43
The Sparse RV Estimator
In markets where trading is thin, new information
is only slowly incorporated into the price
• Intraday returns will have positive autocorrelation
resulting in an upward sloping volatility signature
plot
• To compute RV, choose the smallest s for which
the average RV has stabilized
Elements of Financial Risk Management Second Edition © 2012 by Peter Christoffersen
44
The Average RV Estimator
• Let us use the volatility signature plot to chose s=15 in
the Sparse RV so that we are using a 15-minute grid for
prices and squared returns to compute RV
• The first Sparse RV will use a 15-minute grid starting
with the 15-minute return at midnight, call it RVs,1t+1
• The second will also use a 15-minute grid but this one
will be starting one minute past midnight, call it RVs,2t+1
and so on until the 15th Sparse RV, which uses a 15minute grid starting at 14 minutes past midnight, call it
RVs,15t+1
Elements of Financial Risk Management Second Edition © 2012 by Peter Christoffersen
45
The Average RV Estimator
46
• We thus use the fine 1-minute grid to compute 15 Sparse
RVs at the 15-minute frequency
• We used the 1-minute grid but we have used it to
compute 15 different RV estimates, each based on 15minute returns, and none of which are materially affected
by illiquidity bias.
• By simply averaging the 15 sparse RVs we get the
Average RV estimator
Elements of Financial Risk Management Second Edition © 2012 by Peter Christoffersen
RV Estimators with Autocovariance
Adjustments
• To avoid RV bias we can try to model and then correct
for the autocorrelations in intraday returns that are
driving the volatility bias
• Assume that the fundamental log price is observed with
an additive i.i.d. error term, u, caused by illiquidity so
that
Elements of Financial Risk Management Second Edition © 2012 by Peter Christoffersen
47
RV Estimators with Autocovariance
Adjustments
48
• In this case the observed log return will equal the true
fundamental returns plus an MA(1) error:
• Due to the MA(1) measurement error our simple squared
return All RV estimate will be biased. The All RV in this
case is defined by
Elements of Financial Risk Management Second Edition â 2012 by Peter Christoffersen
RV Estimators with Autocovariance
Adjustments
49
As the measurement error u has positive variance the RVmt+1
estimator will be biased upward
• If the measurement error is of the MA(1) form then only the
first-order autocorrelations are nonzero
• Therefore we can easily correct the RV estimator as
follows:
• where we have added the cross products from the
adjacent intraday returns.
Elements of Financial Risk Management Second Edition © 2012 by Peter Christoffersen