Saying this in words, the upper bound for the average is the average of
the upper bounds. It is easy to see that the same argument is also true for
lower bounds. Exploiting this idea, we compute the bounds on the maxi-
mum and minimum for each day in the pricing period and use the average
of the bounds as the bound for the average. This is illustrated in the follow-
ing example.
Example
Consider the bidder stock selling at $80, 15 days into the pricing period con-
sisting of a total of 20 days. The realized average price in the 15 days for the
bidder stock is $79. The volatility of the bidder stock on an annualized basis
is 32 vol. The value of a target share is fixed at $60. We plan on shorting
100,000 bidder shares. So, what are the bounds on the number of target
shares that we would end up holding in the end?
Step 1: List the given information.
The daily volatility = 2.01%
Number of days to end of pricing period, N= 5 days
Current average price avgc = $ 79
Current closing price pclose = $80
Total days in pricing period, T = 20 days
Fixed value of target stock ptgt = 60.0
Number of bidder shares nB = 100,000 sharesStep 2: We now compute the individual bounds in Table 10.2.
TABLE 10.2 Bounds on Size of Target Position.
Log of Log of Price Price
Scaling Incremental Upper Lower Upper Lower
Days Value Bound Bound Bound Bound Bound
nelog(UpBound) elog(lowBound)
1 1.000 0.0402 4.4222 4.3418 83.2815 76.8478
2 1.414 0.0569 4.4389 4.3252 84.6798 75.5788
3 1.732 0.0696 4.4517 4.3124 85.7687 74.6192
4 2.000 0.0804 4.4624 4.3016 86.6976 73.8198
5 2.236 0.0899 4.4719 4.2921 87.5243 74.7976
+ 2 σ n − 2 σ nn 2 σ n log()pclose log()pcloseσ= 32 252Trade Execution 165