The increment by 1 stands for the new quote, which is not yet included in the window. The computation of Qj is explained at the end of Section 4.5.3. The resulting value of Q is inserted in Equation 4.19.

The trust capital of the new, i,h quote is computed additively as follows

Ti = cievei7;o+ ]T CjT.j (4.32)

j=i-n

T! is not termed 7} because it is not yet the final trust capital in some cases. Equation 4.32 is a weighted sum with weights Cj = C(Tj) from Equation 4.1, which are the current credibilities of the n other quotes of the window.

The number n of quotes used for comparison to the ith quote has an influence on the trust capital and thus the credibility. The higher the value of n, the higher the trust capital according to Equation 4.32 (provided that we are in a series of good data). This effect reflects the fact that the more comparisons to other quotes, the more certain our judgment on credibility. However, the effect of increasing n by adding more and more remote quotes is marginal. The remoteness of quotes implies a high term proportional to ( ?/ )3 in the denominator of Equation 4.13, so the resulting T-,j values are close to zero. The choice of n is further discussed in Section 4.5.4.

Equation 4.32 is a conservative concept insofar as it judges the credibility of a new quote in the light of the previously obtained credibilities Cj of the earlier quotes. In the case of an unusually large real move or price jump, new quotes on a new level might be rejected for a prolonged time. To prevent this, there is special treatment of "after-jump" situations, which may lead to a correction of the resulting trust capital , and a quicker acceptance of a new level after a jump.

The first step of the after-jump algorithm is to identify the location of apossible real jump within the scalar filtering window. This is done during the computation of Equation 4.32. At every j, we test whether the incomplete sum of that equation

j=i-n

is less than the critical value

Tcrit = M Qevel Tj0 - 1 (4.34)

(where fi is defined below). At the same time, we test > 0 (this indicates having reached a new, stable level after the jump rather than an outlier). At the first j where both conditions are satisfied, we conclude that a value jump must have taken place somewhere before quote j - 1. Although this jump certainly happened before quote j, we define 7jump = j because this is the index of the first quote where we have a reason to believe that the jump was real. In order to

validate this possible real value jump, we initialize an alternative trust capital T"

* = Tori,-0.5+ M(7;.;al.-rcri,) (4.35)

We dilute the normal trust capital Tjj by a small dilution factor / . When the filter is initialized (before seeing some 10 acceptable quotes), we choose a slightly larger ix value in order to prevent the filter from being trapped by an initial outlier. The offset term -0.5 in Equation 4.35 prevents the alternative hypothesis from being too easily accepted. For all values of j > jjUmp, we set

Tj = p Tj (4.36)

and insert these diluted trust capitals Tj of old quotes in Equation 4.1. The resulting credibilities Cj are used to complete the computation of the alternative trust capital T":

j-./jump

analogous to Equation 4.32.

Now, we decide whether to take the normal, conservative trust capital T! or the alternative T". The resulting, final trust capital is

( Tj if TJ" > , and TJ" > 0 h ~ \T! otherwise (*>

The alternative solution prevails if its trust capital exceeds 0 and the trust capital of the conservative solution. The trust capital 7/ of the new quote is the end result of a pure filter test. Tn the case of a normal update, the window has to be updated.

4.5.3 Updating the Scalar Window

A new quote affects the trust capitals of the old quotes of the window. The most dramatic change happens in the case of accepting the alternative hypothesis according to Equation 4.38. In this case, a real value jump is acknowledged, which leads to a major reassessment of the old quotes. First, the pairwise trust capital of quote comparisons across the jump is diluted

T - j ij for j < ./jump (A „,

convy - j j.. otherwjse

In the normal case with no jump, Tconjj = . Afterward, the quotes after the newly detected jump get a new opportunity

T. j I 7i if > ./jump and 7) < 0

y.new - j j. otherwise (4-4U)

In the case of a jump, this new value 7ynew replaces Tj.

In every case, whether there is a jump or not, the trust capitals of all quotes are finally updated additively following Equation 4.32

7),new = Tj + Cj Tcon,tj , for j = i - n ... i - 1 (4.41)

where , = (7}) follows from Equation 4.1 by substituting 7} from Equation 4.38. The result 7yinew of Equation 4.41 is replacing the old value Tj. I( should also be clarified that the diluted values Tj from Equation 4.36 are never directly used to modify the trust capitals Tj.

In historical filtering, Equations 4.39 through 4.41 may lead to the rehabilitation of an initially rejected old quote. Even in real-time filtering, the corrected trust capital of an old quote indirectly contributes to the filtering of new quotes through Equation 4.32 and through the use of only sufficiently credible old quotes in the statistics of adaptive filtering.

The valid-quote age Qj of all the old quotes is also updated

Gy.new = Qj + Cj , for ; = i - ... i - 1 (4.42)

where C, = C(Tj). The more credible the new quote, the higher the increment of the valid-quote age Qj.

After all these updates, the new quote with index i and with its newly computed trust capital Tj is inserted in the window as its newest member, with the valid-quote age Qj initialized to zero.

4.5.4 Dismissing Quotes from the Scalar Window

The window does not grow infinitely. At the end of a normal update as described in Section 4.5.3, a rule for dismissing scalar quotes is applied. There are three criteria for obtaining a properly sized window: (1) a sufficient time interval, (2) a sufficient number of quotes, and (3) a sufficient overall credibility of all scalar quotes. These criteria are listed here in the sequence of increasing importance.

In our general quote dismissal rule, we use the product of the criteria. At the end of an update with a new quote, the following condition for dismissing the oldest quote (with index i - n) is evaluated:

(&i -#; „+, ) 2 ECwj > W (4-43)

The sum of credibilities, the overall credibility, is the most important criterion and is therefore raised to the sixth power. This exponent is a parameter as many others; the value 6 has been found optimal in tests of samples of different data frequencies and qualities. The configuration parameter W defines the sufficient size of the window and has the dimension of a time. The parameter W is somehow related to the parameter v of Equation 4.13, which determines a filtering range. Choosing a very large W when v is limited does not add value because the distant quotes have a negligible weight in this case.