implication is that under static interest rate conditions and a constant rate
of information arrival at the market, the rate of decay of the spread would
be a constant. However, the various milestones leading to the merger like
the filing of documents, the proxy vote, and so forth, all happen at discrete
times. So, how realistic is the assumption of a constant rate of information
flow? It turns out that it is not terribly unrealistic. When the merger an-
nouncement is made and events are set into motion, it creates an expecta-
tion in the marketplace as to the timing of the various milestones leading to
the merger. If a day passes without any news contrary to expectation, then
it strengthens the expectations. In that sense, the mere passage of time may
be construed as providing information leading to the strengthening of ex-
pectations—one of the very rare cases in the market where no news is con-
sidered good news.
A constant rate of information flow, according to the model, carries
with it the implication that the logarithm of the spread approaches zero in a
linear fashion. In fact, this would be a very reasonable assumption, if the
deal were to proceed without any hiccups. However, given the vicissitudes
of the deal process, such an assumption might prove to be untenable. Nev-
ertheless, it may still be reasonable to assume that the logarithm of the
spread is piecewise linear; that is, the rate of spread reduction is about the
same as observed in the recent past. Therefore, we model the spread at the
next time instant to be the current spread decremented by the instantaneous
rate of reduction of the spread. The state equation is given as
Xt=Xt–1+∆t–1+et (12.2)where∆t–1is the instantaneous rate of change of the spread. Note that Equa-
tion 12.2 can also be viewed as the first two terms of a Taylor expansion of
the spread about Xt–1.
This, therefore, brings us to the requirement of specifying a scheme to
evaluate the instantaneous rate of change of the spread, ∆t–1. A naive method
would be to use the correction rate of the past time step given by
(12.3)A little more sophisticated method would be to take the minimum variance
weighted average of the past two correction rates given by
(12.4)
wherertis the weight. Alternately, one could obtain an estimate of the cor-
rection rate by taking the mean correction rate over the past dtime steps;
that is,
∆tttt tt−−−−− 11122 =−rX()ˆˆ||X +−() 1 r Xttt tt()ˆ ˆ−−−− 2233 ||−X∆ttttt−−−−− 11122 =−XXˆˆ||Spread Inversion 191