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38 High values of bidask spreads are worse in usability than low spreads, by nature. Thus the quote deviations from the mean as defined by Equation 4.3 are judged with bias. Deviations to the high side (x( > 0) are penalized by a factor phigh, whereas no such penalty is applied against low spreads. For some (minor) financial instruments, many quotes are posted with zero spreads (i.e., bid quote = ask quote). This is discussed in Section 4.6.1. In some cases, zero spreads have to be accepted, but we set a penalty against them as in the case of positive Xj. We obtain the following refined definition of  if X( < 0 and no zerospread case ?° xi . (48) Phigh  if */ > 0 or in a zerospread case where x, comes from a modified version of Equation 4.3, r  x (4.9) yEMA[ A&r; (xx)2] + Ax~l~ The constant §o determines the size of an x that is just large enough to neither increase nor decrease the credibility. Equation 4.8 is general enough for all meanreverting filterable variables. If we introduced meanreverting variables other than the bidask spread, a good value for Axmin would probably be much smaller or even 0, phigh around one and £o larger (to tolerate volatility increases in absence of a basic volatility level Ax2. ). 4.4.3 Pair Filtering: The Credibility of Returns The pairwise comparison of scalar quotes is a central basic filtering operation. The algorithm makes pairwise comparisons also for quotes that are not neighbors in the series, as explained in Section 4.5. Pair filtering contains several ingredients, the most important one being the change filter. Its task is to judge the credibility of a variable change ( return if the variable is a price). The time difference between the two quotes plays a role, so the time scale on which it is measured has to be specified. The criterion is adaptive to the statistically expected volatility estimate and therefore uses some results from a moving statistical analysis. The change of the filtered variable x from the jh to the Ith quote is Ax,j = xi  Xj (4.10) The variable x may be the result of a transformation in the sense of Section 4.6.3. The time difference of the quotes is j, measured on a time scale to be discussed in Section 4.4.6.
The expected variance V(Ai}) of x around zero is determined by the online statistics as described in Section 4.4.4. The relative change is defined by Axr, Hij = , 1 (4.11) with a positive constant §o, which has a value of around 5.5 and is further discussed later. Low § values deserve high trust, extreme £ values indicate low credibility and negative trust capital; at least one of the two compared quotes must be an outlier. The remainder of the algorithm is similar to that of the level filter as described in Section 4.4.2, using the relative change instead of the scaled standardized variable . The amount of negative trust capital for outliers depends on the density function of changes Ax, especially the tail of the distribution at extreme Ax or § values. A reasonable assumption is that the credibility of outliers is approximately the probability of exceeding the outlier value, given the distribution function. This probability is proportional to %~a, where a is the tail index of a fattailed distribution. We know that distributions of highfrequency price changes are indeed fattailed (see Dacorogna et al, 2001a). Determining the distribution and a in a moving sample would be a considerable task beyond the scope of fi ltering software. Therefore, we make a rough assumption on a that is good enough across many rates, filtered variable types and financial instruments. For many price changes, a good value is around a % 3.5, according to Dacorogna et al. (2001a) and Miiller et al. (1998). As in Section 4.4.2, we generally use a  4 as a realistic, general approximation. As in Section 4.4.2 and together with Equation 4.5, we argue that the trust capital should asymptotically be proportional to £2 and arrive at a formula that gives the trust capital as a function of §: = = (4.12) which is analogous to Equation 4.6. This trust capital, depending only on §, is called U to distinguish it from the final trust capital T that is based on more criteria. At £ = 1, Equation 4.12 yields a zero trust capital, neither increasing nor decreasing the credibility. Intuitively, a variable change of a few standard deviations might correspond to this undecided situation; smaller variable changes lead to positive trust capital, larger ones to negative trust capital. In fact, the parameter £o of Equation 4.11 should be configured to a high value, leading to a rather tolerant behavior even if the volatility V is slightly underestimated. The trust capital 11 tj from Equation 4.12 is a sufficient concept under the best circumstances, independent quotes separated by a small time interval. In the general case, a modified formula is needed to solve the following three special pair filtering problems. 1. Filtering should stay a local concept on the time axis. However, a quote has few close neighbors and many more distant neighbors. When the additive
trust capital of a quote is determined by pairwise comparisons to other quotes as explained in Section 4.5.2, the results from distant quotes must not dominate those from the close neighbors; the interaction range should be limited. This is achieved by defining the trust capital proportional to ( #)~3 (assuming a constant §) for asymptotically large quote intervals 2. For large ?, even moderately aberrant quotes would be too easily accepted by Equation 4.12. Therefore, the aforementioned decline of trust capital with growing Aft is particularly important in the case of positive trust capital. Negative trust capital, on the other hand, should stay strongly negative even if & is rather large. The new filter needs a selective decline of trust capital with increasing At?: fast for small § (positive trust capital), slow for large § (negative trust capital). This treatment is essential for data holes or gaps, where there are no (or few) close neighbor quotes. 3. Dependent quotes: if two quotes originate from the same source, their comparison can hardly increase the credibility (but it can reinforce negative trust in the case of a large §). Tn Section 4.4.5, we introduce an independence variable /, , between 0 (totally dependent) and 1 (totally independent). The two last points imply a certain asymmetry in the trust capital; gathering evidence in favor of accepting a quote is more delicate than evidence in favor of rejecting it. All of these concerns can be taken into account in an extended version of Equation 4.12. This is the final formula for the trust capital from a change filter: Tij = {%, » ,1 ) = /*T (4.13) where hi if < 1 1 if % > 1 (4 4) The independence /, , is always between 0 and 1 and is computed by Equation 4.23. The variable d is a quote density explained in Section 4.4.4. The configurable constant v determines a sort of filtering interaction range in units of the typical quote interval 1 /d). Table 4.3 shows the behavior of the trust capital according to Equation 4.13. The trust capital converges to zero with an increasing quote interval Ad much more rapidly for small variable changes £ than for large ones. For small Adij and /, , = 1, Equation 4.13 converges to Equation 4.12. The approach of Equation 4.13 has been tested for almost all available types of financial data, not only FX. We find that it works for all data types with the same values of the parameters.
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