tional VAR methodologies will apply. However, when the deal is on, the risk
is mostly due to the spread volatility, and the value at risk is best obtained
by measuring the spread volatility. Thus, based on the view we choose to
adopt, we now have two VAR values. We now reconcile between the two
views and arrive at a final VAR number by weighing each scenario. Fortu-
nately for us, the consensus probabilities of deal success and failure provide
us with a ready mechanism by which to weigh the two views.
Putting things more precisely, each of the two views may be associated
with a return distribution describing the probability of return of the trade in
the next time period. The overall probability distribution is then a weighted
sum of the two distributions, the weights corresponding to the probability
of deal success and deal failure, respectively. Distributions that are con-
structed as a weighted sum of other distributions are called mixed density
distributions.
To better understand the model, let us consider the process of drawing
a sample from a mixture distribution. This is actually a two-step process. In
the first step, we randomly choose the distribution from the available set of
distributions. The choice is guided by the probability weights assigned to
each distribution in the set. In the next step, we draw a random sample from
the chosen distribution.
Applying the mixture model to our situation, the probability of an out-
comexdollars or lower is given by the distribution function shown in Equa-
tion 11.9
(11.9)whereΦspreadis the distribution of spread returns, and ΦstockPortfoliois the dis-
tribution of the returns assuming that there is no deal. The failure and suc-
cess probabilities serve as weights. Thus, given a value of xdollars, we can
find the probability that the returns will be less than or equal to xusing the
above formula.
Note that VAR value is that value of xthat results in the probability
value of 2.5 percent in Equation 11.9. Additionally, the distribution function
like every other distribution function is a nondecreasing function. As a con-
sequence, the VAR value may be deduced by applying a standard binary
search method. Thus, knowledge of the consensus probability estimates of
deal success and failure may be used to calculate the VAR in risk arbitrage
trades.
Event Risk Management
Oftentimes in risk arbitrage, the outcome of merger success or deal break
may hinge on a single event like the shareholders’ vote or a court decision.
Fx()=+ππsuccessΦΦspread()x failure stockPortfolio()xThe Market Implied Merger Probability 185