This dependency on the measured and predicted states requires both the
measurement model and the prediction model to be fairly precise. Having
one of them to be very precise and the other to be erroneous leads us to
overly rely on one or the other and is likely to produce a mediocre result.
Thus, in some sense there is a trade-off to be made between the observation
and prediction costs, and the reduction of one of the costs at the expense of
the other is highly undesirable.
With this motivation, we define the cost function associated with a
Kalman filter to be
cost function = measurement cost + prediction cost (12.10)This cost function serves to keep the system honest. If in an attempt to re-
duce the cost function, we try to reduce the prediction cost, it would be all
right as long as it does not increase the measurement cost and vice versa. The
best choice for our prediction equation is therefore the one that results in
the minimum value for this cost function.
To demonstrate the approach, let us apply it to a real-life situation. Fig-
ure 12.1 is a plot of the spread and the corresponding Kalman smoother for
various lags. The bidder in this case is McKesson Inc., and the target is HBO
196 RISK ARBITRAGE PAIRS
FIGURE 12.1 Kalman Filter Implementations (MCK-HBOC).log(Spread) log(Spread)10 30 50–3–2–101210 30 50–3–2–1012log(Spread) log(Spread)10 30 50–3–2–101210 30 50–3–2–1012