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Figure 5.5: Fundamental Price and Quoted Price with Bid-Ask Bounces

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



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Figure 5.5: Fundamental Price and Quoted Price with Bid-Ask Bounces

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