Final_1.pdf

the standardized value being. Thus, we can say with 95-percent con-

fidence that the average value of the Brownian motion will be between

.
We now combine the preceding information with the fact that the mean
of a sample is definitely lower than the maximum of the sample. The maxi-
mum can now be used as an upper bound for the mean. In a similar manner,
the minimum serves as a lower bound for the ratio. The bounds on the ratio
can quickly be translated into bounds for the target share quantity by mul-
tiplying the ratio bounds with the Bidder shares quantity. Thus, an estimate
of the maximum and minimum price during the pricing period may be used
to calculate the bounds on the exchange ratio.
We, however, believe that we could obtain a tighter bound than the one
just shown. To see that, let us consider the average of n numbers x 1 ,x 2 ,...,xn.
Now let the upper bounds for each of the numbers be b 1 ,b 2 ,...,bn. Then

(10.10)

xx x

n

bb b

n

12 ++...+nn≤ 12 ++...+

xtx=− 2 σ and

x

σ t

164 RISK ARBITRAGE PAIRS

FIGURE 10.3 Distribution of the Max and Min of Brownian Motion.

–3 –2 –1 0 1 2 3

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

Cummulative Probability

Standardized Variable

Distribution of Minimum Distribution of Maximum