Final_1.pdf

(12.5)

Thus, the process of evaluating the instantaneous rate of reduction of the
spread could vary depending on d, the number of time steps in the past that
we use in our estimation. Consequently, we could have different state equa-
tions corresponding to different values of d, the lag parameter, and for each
such state equation, a version of the Kalman filter could be implemented.
But which one of the state equations is most suited for our purpose? Let
us defer this question for now and address it a little later in the chapter.
Another point that is noteworthy in the modeling of the prediction
equation is the fact that the actual equation used is different for each time
step. The exact values for the coefficients in the prediction equation are es-
timated at each time step based on the instantaneous rate of spread reduc-
tion. This is a little different from typical Kalman filter implementations that
have a fixed prediction equation with fixed coefficients. We are now ready
to move on to the observation equation.

The Observation Equation

The observation at a given time instant is the logarithm of the spread. Asso-
ciated with an observation is a measure of the error. The magnitude of this
error is quantified in terms of the error variance. The observation equation
is written as

Yt=Xt+ht (12.6)

whereYtis the logarithm of the observed spread at time tandhtis the ob-
servation error with zero mean and variance. We now therefore need to
estimate the variance of our observation.
To estimate the variance of the observation, we draw on the notion of
realized volatility. Realized volatility is essentially an empirical volatility
measure that sums the squared tick-by-tick returns over a given period. The
construction of the realized volatility measure is based on some theoretical
results of integrated volatility defined in the context of continuous stochas-
tic processes. A detailed discussion on realized volatility measures and its
scaling properties can be found in the reference material.
Bear in mind that if we use the realized volatility as a measure of the
variance in the observation, then we would need access to tick data. The

ση^2 t

∆

∆

t

t t t t tdtd t dt d

t

tt tdtd

XX X X

d

XX

d

−

− − − − − − −− −−

−

− − −− −−

=

()− +... +()−

=

()−

1

11 2 2 1 1

1

11 1 1

ˆˆ,,ˆˆ

ˆˆ

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192 RISK ARBITRAGE PAIRS