156 THE HANDBOOK OF PORTFOLIO MATHEMATICS
dollars and euros? Is a .30-point move at .4500 the same as a .40-point move
at .6000?
My personal opinion is that you are probably better off with the equal-
ized data. Often, the matter is moot, in that if a stock has moved from $20
per share to $100 per share and we want to determine the optimalf,we
want to use current data. The trades that occurred at $20 per share may not
be representative of the way the stock is presently trading, regardless of
whether they are equalized or not.
Generally, then, you are better off not using data where the underly-
ing was at a dramatically different price than it presently is, as the char-
acteristics of the way the item trades may have changed as well. In that
sense, the optimalfoff of the raw data and the optimalfoff of the equal-
ized data will be identical if all trades occurred at the same underlying
price.
So we can state that if it does matter a great deal whether you equalize
your data or not, then you’re probably using too much data anyway. You’ve
gone so far into the past that the trades generated back then probably
are not very representative of the next trade. In short, we can say that
it doesn’t much matter whether you use equalized data or not, and if it does,
there’s probably a problem. If there isn’t a problem, and there is a difference
between using the equalized data and the raw data, you should opt for the
equalized data. This does not mean that the optimalffigured off of the
equalized data would have been optimal in the past. It would not have been.
The optimalffigured off of the raw data would have been the optimal in the
past. However, in terms of determining the as-yet-unknown answer to the
question of what will be the optimalf(or closer to it tomorrow), the optimal
ffigured off of the equalized data makes better sense, as the equalized data
is a fairer representation of the distribution of possible outcomes on the
next trade.
Equations (4.10a) through (4.11) will give different answers depend-
ing upon whether the trade was initiated as a long or a short. For exam-
ple, if a stock is bought at 80 and sold at 100, the percentage gain is 25.
However, if a stock is sold short at 100 and covered at 80, the gain is
only 20%. In both cases, the stock was bought at 80 and sold at 100, but
the sequence—the chronology of these transactions—must be accounted
for. As the chronology of transactions affects the distribution of percent-
age gains and losses, we assume that the chronology of transactions in
the future will be more like the chronology in the past than not. Thus,
Equations (4.10a) through (4.11) will give different answers for longs and
shorts.
Of course, we could ignore the chronology of the trades (using 4.01c for
longs and using the exit price in the denominator of 4.01c for shorts), but
to do so would be to reduce the information content of the trade’s history.