Final_1.pdf

The state diagram in Figure 11.1 comes with a few caveats. Very often,
the fact that the deal is going to be unsuccessful is known much before the
time it would take for successful completion, in which case, treating both the
payoffs as if they are known at the same time would not be appropriate. For
our purposes, however, we choose to reflect this difference in timing in the
value of STwhile keeping Ta constant. Later on in the multistep model for-
mulation, we will eliminate the need to estimate ST. Therefore, keeping Tthe
same for both the success and failure scenarios does not affect the outcome
of the model.
Recall that the initial cost of setting up the trade was zero. By the no ar-
bitrage condition the expected payoff is also zero. Writing out the equations,
we have

(11.1)

psuccess+pfailure= 1 (11.2)

wherepsuccessandpfailureare the risk-neutral probabilities of successful merger
and failed merger, respectively. Solving the two equations, we have

(11.3)

Note here that in the derivation of the one-step model we apply the
Arrow-Debreu ideas to the spread. For the curious mind, the same results
may be obtained by applying the Arrow-Debreu ideas to the individual
stocks in question, also. The derivation of the formulas for the model using
the individual stocks is presented in the appendix.
In Equation 11.3 for failure probability, all the values are known or ob-
servable except for ST, the spread in the future if the deal happens to break.
The value for this spread is anybody’s guess. Unless a reasonable value for
the deal break spread is known, the one-step model as it stands is of not
much use. We deal with this issue in the multistep model.

The Multistep Model

The multistep model relates the changes in the risk-neutral probability to the
dynamics of the spread movement. In this case, it is crucial to have an esti-
mate of deal break probability on the eve of the announcement. If we are
able to assess the probability of successful merger on the eve of the an-
nouncement, the value may be updated periodically based on the spread dy-
namics. This eliminates the need to guess at the deal break spread and thus
mitigates the problem of the one-step model. We now proceed to describe
the model.

πfailure=+eS erT()/() 0 −rTcash ST+cash

ππsuccess()()eSrT 00 ++cash failureeS SrT −=T 0

The Market Implied Merger Probability 177