Example
Days in pricing period n = 20
Fixed value of target stock Pr = $60
Closing price of bidder stock
onith day, pt = $80.5
Total shares of bidder
to short nB = 100,000
Shares of bidder to short for
the next day = = 100,000/20 = 5000
Shares of target to buy for
the next day = = 5000 ×(80.5/60.0) = 6708Bounds on the Position Size
It was noted in the last section that when trading during the pricing period,
an arbitrageur has to be willing to replace the uncertainty in exchange ratio
with an uncertainty in position. However, there is still an interest in evalu-
ating some bounds on the position size of the target shares. In this section,
we derive some bounds on the ratio and show how it can be converted to
bounds on the position size.
To see how we can estimate bounds on the ratio, consider the fact that
the average of the closing prices of the stock is always between the maxi-
mum and the minimum prices. Thus, a very simple bound could be based on
that. If we assume a log-normal process for the price movement of the bid-
der stock, that is, the logarithm of the prices executes a Brownian motion,
the probability distribution for the maximum of a Brownian motion in time
tis given as
(10.9)
wheresin the equation is the volatility of the Brownian motion and Φis the
cumulative density function of the normal distribution. The formula repre-
sents the probability that the Brownian motion will have a maximum value
less than or equal to xin the time duration t. By symmetry we can expect the
distribution of the minimum to be a flipped version of distribution of the
maximum for values less than 0. A plot of the distribution functions for the
maximum and minimum is shown in Figure 10.3.
Reading the graph, one can say that the maximum value of the Brownian
motion is less than the standardized value of 2.0 with 95-percent accuracy,
F
x
tmax= ,x
210 φ −≥
σn
np
pB
i
T×
n
nBTrade Execution 163