Final_1.pdf

known as regularization. The following section contains an in-depth de-
scription of the regularization process. Upon completion of the regular-
ization process, the resulting count function may be used to calculate the
profit profile and the maximum value read off as the optimal threshold
value to use.

Regularization

Let us restate the problem for sake of clarity. We are attempting to estimate
the frequency function for level crossing given a sample realization. Such
problems involving the estimation of functions given data fall under the
general subject area of inverse problem theory. Regularization is one of the
most basic ideas of this theory. The fundamental idea in regularization is
that of two cost measures. The first cost measure quantifies the degree of
agreement of the computed function to the given data. The second cost
measure quantifies the deviation from a known property of the function like

130 STATISTICAL ARBITRAGE PAIRS

FIGURE 8.3 Profitability of Thresholds: No Inventory Constraints.

Amount of Sigma Away from Mean

0.00 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0

Profit Profile

0.00

0.05

0.10

0.15

0.20