spread can then be evaluated on a tick-by-tick basis and the sum of squared
returns on the spread could be used as a measure of observation variance. In
the absence of tick data we use the following approximation to estimate the
realized volatility measure. Initially, the four spreads corresponding to the
open high low and close prices for the day of the two stocks are computed.
Note that the times in which the two stocks registered their highs could be
different; nevertheless, we treat them as if the highs were registered simulta-
neously. The sum of squared returns on the spread is calculated on the two
possible spread paths; namely, open-high-low-close and open-low-high-
close. The smaller of the two values is then chosen to be the variance of the
observation. The question the reader might ask at this point is, “Is this re-
ally the variance of the error in the observation?” It is probably not. How-
ever, the variances thus calculated are not used to price any instrument. It
should suffice that the measured variances are proportional to the actual
error variances and that the relative order of the errors associated with the
observations is preserved. This ensures that the weighting of the observa-
tions in the evaluation of the state estimate in done in a consistent manner
resulting in state estimates that seem to be satisfactory.
To get an intuitive feel for the volatility measure, consider the following.
As the spread gets closer to zero, the bid-ask spread of the individual stocks
measured as a percentage of the spread between the two stocks becomes
higher. It then gets increasingly harder to tell whether the change in spread
is real or due to microstructure or bid-ask effects. Thus, as the spread re-
duces, we should expect an increase in its variance.
APPLYING THE KALMAN FILTER
Summarizing the discussions so far, in order to apply the Kalman filtering
approach to smooth the spread, we went through the exercise of modeling
the prediction equation and the observation equation. The Kalman state in
our situation corresponds to the logarithm of the true spread, and the ob-
servation corresponds to the logarithm spread as observed in the market.
The state equation as applicable in our situation is given as
Xt=Xt–1+∆t–1+et (12.7)whereXtis the state at time t,∆t–1is the time derivative of Xt–1, and etis the
state noise. The time derivative ∆t–1may be estimated by measuring the av-
erage change in the spread over different lag values. The observation equa-
tion is given by
Yt=Xt+ht (12.8)Spread Inversion 193