A comprehensive introduction to the Kalman filter is provided in Chap-
ter 4. However, for sake of continuity we will summarize the method here
in a few sentences. The idea of Kalman filtering revolves around the notion
of state of the system. The process involves a sequence of state predictions
followed by observations. The predictions are then reconciled with the ob-
servations to obtain the best estimate of the state. Applying the idea to our
situation, let the state correspond to the logarithm of the spread. This
process then translates as follows. First, we make a prediction of the next
value for the log-spread and follow this with an observation after the elapsed
time. The predicted log-spread and the observed log-spread are then recon-
ciled to form the best estimate of the spread at that time instance. The
process is then repeated for the next time step.
Based on the discussion here, it is now obvious that Kalman filter de-
sign for our problem involves two main steps. First, we need to design the
prediction method that we plan to use, to provide us with an a priori
estimate of the state. Next, we need to model the observation process. The
outcome from the observation modeling process is a means of observing
the state along with a method to estimate the error variance in the obser-
vation. Once we have the prediction and observation, the Kalman filtering
process provides us with the means to reconcile between them to come up
with an estimate of the state; that is, the logarithm of the spread. This
value can in turn be used to estimate the spread implied probability of deal
break. Let us now discuss the modeling process for the prediction and ob-
servation equations.
The Prediction Equation
The prediction equation is essentially an equation specifying the state tran-
sition for the Kalman filter. Since in our case the state corresponds to the
logarithm of the spread, the state equation in our case involves coming up
with a prediction scheme for the value of the spread at the next time step.
We of course have at our disposal all the past values of the spread right up
to the current time step. Let us therefore discuss how we can achieve that.
In order to motivate our choice for the state equation, we recall the dif-
ference equation from the previous chapter.
(12.1)
The left-hand side of the equation is equivalent to the rate of information
flow. The right-hand side is the return in excess of the risk-free rate. The
log()ππfailuret^21 −log()failuret =− −rt() 21 t +log()Stt 21 −log()S190 RISK ARBITRAGE PAIRS