a threshold value is then estimated by simulating trades on the generated
data. The profits thus calculated could then be used to determine the opti-
mal value for the threshold.
Commentary
Reflecting on the deliberations so far, it appears that the closer we wish to
model the spread to reality, the more complicated the models get. The com-
putational methods used in these situations in turn get increasingly involved.
It seems that it would be quite a formidable challenge to simplify the ap-
proach, does it not? Then again, it may be that we are not looking at the
problem with the right perspective. Let us therefore restate what we wish to
accomplish.
The purpose of the whole exercise is to come up with a reasonable and
reliable approach to decide the threshold values. So far, the assumption has
been that in order to design the threshold values it is necessary to know the
dynamics of spread behavior intimately and have parametric models (mod-
els where knowledge of a few parameters gives us a complete description of
the dynamics) describing their behavior. That, however, need not be the
case. We would remain happy campers if we could come up with a reason-
able band design without having to worry about modeling the dynamics of
the spread using parametric models. So, for our purposes we resort to non-
parametric methods where we estimate the profit profile function directly
from the sample realization of the spread; that is, the spread series observed
in the recent past. Obviously, it does not require much persuasion to sub-
scribe to the argument that we get more of the proverbial bang for the buck
on adopting the nonparametric approach.
As mentioned earlier, the idea in the nonparametric approach is to esti-
mate the profit function directly from the sample realization of the spread.
This would preclude us from having to use relatively involved computa-
tional schemes to estimate the parameters of the model had we gone the
parametric route. But we have only one sample realization, and so relying
completely on this realization would bias our results too much to this real-
ization. We address this issue of bias and describe a reasonable method for
threshold design in the following section, nonparametric approach.
Nonparametric Approach
One of the key issues that relate to estimating the profit function directly is
the size of the spread sample. The larger the size of the sample the more con-
fident we can be that the sample truly represents all aspects of the behavior
of the spread. The statement above has in it a built-in assumption of ergod-
126 STATISTICAL ARBITRAGE PAIRS