to a deterioration of fit in favor of increased smoothness. Then again, past
another value of l, the fit error becomes insensitive to the lvalue. At this
point the function is more or less a straight line at the mean of all the points
(since this is the line that would lead to the lowest cost for the smoothing
function.
Our choice for lis right at the heel of the curve. If we move lto the left
of the point, then the fit error dominates the cost function. Moving to the
right of the point results in the smoothing term dominating the cost func-
tion. Choosing the value of lto be at the heel of the curve achieves a fine
balance between the two cost measures and is the value that we choose.
The function values for this value of lare evaluated and plotted in Fig-
ure 8.6. The profit profile is then computed from the regularized curve and
is as shown in Figure 8.7. The maximum is now easily read off from this
graph and determines where we place the threshold for our trading signal.
The maximum occurs at 0.75 times the standard deviation, which is also the
theoretical value, thus validating our approach.
134 STATISTICAL ARBITRAGE PAIRS
FIGURE 8.6 Regularized Plot of Counts.–0.2 0.3 0.8 1.3 1.80.2
0.3
0.4
0.5
0.1
0.0
Amount of Sigma Away from MeanCount Adjusted for Monotonicity
Regularized Curve