icity.Ergodicityhas a rigorous mathematical definition and proof, but for
our purposes, it may be simply stated as follows: A large sample size is ef-
fectively equivalent to having multiple realizations of the series of smaller
sample sizes. Therefore, with large sample sizes for the spread, we can be
reasonably confident that the effects of bias to the sample at hand have been
mitigated.
However, a large sample size may be a luxury that we are not afforded.
Recall that the spread dynamics are directly linked to the fundamentals of
the firm and its valuation. These fundamentals are dynamic and continue to
evolve with time. Therefore, the observations of the spread far back in the
past may or may not be of significance, depending on whether the funda-
mentals of the firms involved remained more or less the same. It is therefore
likely that we may find ourselves in a situation where we need to estimate
the profit function from a relatively small sample data set.
We will soon describe the steps to overcome this issue and propose an
approach to evaluate a close approximation to the profit function. To con-
vince the skeptical reader that this approach actually produces reasonable
results, we will apply the approach to the white noise case. We already
know the true functional form of the profit in the white noise case and
therefore the true optimal value for the threshold. The threshold value esti-
mated using this approach can now be compared with the true value, thus
providing us with a validation of the nonparametric approach suggested.
Trading Design 127
THE PROOF OF THE PUDDING
In the Mark Twain classic A Connecticut Yankee in King Arthur’s
Court, Merlin, the wizard in Arthur’s court claims to have the ability
to foresee things and know the unknown. The Yankee, pragmatic as he
is, challenges Merlin to guess what he (the Yankee) is holding in his
hand. As the Yankee knows what he is holding in his hand, this would
serve as a ready test case to verify Merlin’s claim.
In fact, the idea of the proof of the pudding is in the eating is
standard fare in the area of signal processing to demonstrate the effi-
cacy of an estimation algorithm. A sample data set for which the pa-
rameters are known is submitted to the estimation algorithm. The
performance of the estimation algorithm is measured by how closely
the algorithm guesses the known true value of the desired parameters.
In this case, we apply the approach to the white noise case. The value
obtained from the estimation can be compared with the true value to
provide us some evidence of the efficacy of the approach.