Final_1.pdf

whereYtis the observation and htis the observation noise with zero mean
and variance of.
With this information we are now ready to apply the Kalman-filtering
ideas in a recipe-like fashion (as discussed in Chapter 4). We list the recipe
below once again for sake of continuity. The a priori estimate of the state at
timetgiven all observations up to time t– 1 is denoted as , and the
posteriori estimate of the state at time tgiven all the observations up to time
tis given as. The various steps are then as follows:

1. Evaluate and using the state equation.


2. Find the observation Ytand by observing the system.


3. Evaluate Kt, also known as the Kalman gain, which will be used to ob-
   tain the linear minimum error variance estimate.


4. Evaluate given by


5. Finally, evaluate

These steps are repeated for the next time step. The exact formulas to use for
the various steps are derived in simplified form in the appendix.

MODEL SELECTION

Note that based on the preceding scheme, we have multiple models for the
prediction equation. The differentiating factor amongst them is d, the lag
factor used in the estimation of the instantaneous error correction rates. For
each of the prediction equations, we can implement a unique Kalman filter-
implementation. We therefore need to choose the implementation that
is most appropriate; that is, we need to make an appropriate choice for the
parameterd.
The principle guiding our choice for the lag value dis the maintenance
of the delicate balance between prediction and observation. Relying overly
on the prediction would mean that we rely excessively on our model and do
not give enough weight to what is observed in practice. However, relying too
much on the observation would mean that we do not have any view what-
soever on where the spread must be and so must rely excessively on the noisy
observation. The right model choice achieves a happy medium between the
two extremes. Let us therefore look at how we can quantify this notion of a
happy medium.
The basic idea is to work with the two sources of error; namely, the
measurement/observation error and the state transition prediction error.
We denote the measurement cost as the sum of all measurement errors
and the prediction cost as the sum of all the prediction errors. More pre-
cisely, ifX 1 ,X 2 ,X 3 ,...,Xnare the estimated states of the Kalman filter,

var()Xˆtt|

ˆˆ

XKYXtt||−− 11 +−t()t tt

ˆ

Xtt|

var()Yt

Xˆtt|− 1 var()Xˆtt|− 1

ˆ

Xtt|

ˆ

Xtt|− 1

ση^2 t

194 RISK ARBITRAGE PAIRS