table 4.2 Adding independent credibility values.

The total credibility Ctotai resulting from two independent credibility values and C2. The function Ctotai = C[T{C\) + 74 C2)] defines an addition operator for credibilities, based on Equations 4.1 and 4.2. The values in brackets, (0.5), are in fact indefinite limit values where C,ota] may converge to any value between 0 and I.

Ctotai C] =

0 0.25 0.5 0.75 1

C2 =

probability even if validity could be exactly defined. Credibility can be understood as a "possibility" in the sense of fuzzy logic as proposed by Zimmermann (1985), for example.

Credibility is not additive; the credibility of a scalar quote gained from two tests is not the sum of the credibilities gained from the individual tests. This follows from the definition of credibility between 0 and 1. The sum of two credibilities of, say, 0.75 would be outside the allowed domain.

For internal credibility computations based on different tests, an additive variable is needed to obtain the joint view of all tests. We define the additive trust capital T, which is unlimited in value. There is no theoretical limit for gathering evidence in favor of accepting or rejecting the validity hypothesis. Full validity corresponds to a trust capital of T = oo, full invalidity to T - -oo. We impose a fixed, monotonic relation between the credibility and the trust capital of a certain object

C(T) = -+ Z-, (4.1)

2 2 Vl + T

and the inverse relation

C - 1

T(C) = 2 (4.2)

VC (1 - C)

There are possible alternatives to this functional relationship. The chosen solution has some advantages in the formulation of the algorithm that will be shown later.

The additi vity of trust capitals and Equations 4.1 and 4.2 imply the definition of an addition operator for credibilities. Table 4.2 shows the total credibility resulting from two independent credibility values.

4.4.2 Filtering of Single Scalar Quotes: The Level Filter

There is only one analysis of a single quote called the level filter. Comparisons between quotes (done for a pair of quotes, treated in Section 4.4.3) are often more important in filtering than the analysis of a single quote.

The level filter computes a first credibility of the value of the filtered variable. This only applies to those volatile but mean-reverting time series where the levels as such have a certain credibility in the absolute sense-not only the level changes. Moreover, the timing of the mean reversion should be relatively fast. Interest rates or interest rates futures prices, for example, are mean-reverting only after time intervals of years; they appear to be freely floating within smaller intervals (see Ballocchi, 1996). For those rates and for other prices, level filtering is not suitable.

The obvious example for fast mean reversion and thus for using a level filter is the bid-ask spread, which can be rather volatile from quote to quote but tends to stay within a fixed range of values that varies only very slowly over time. For spreads, an adaptive level filter is at least as important as a pair filter that considers the spread change between two quotes.

The level filter first puts the filtered variable value x into the perspective of its own statistical mean and standard deviation. Following the notation of Section 3.3.8, the standardized variable x is defined by

MSD[ # , 2; x ] EMA[ # ; ( - x)2 \

where the mean value of x is also a moving average:

x = EMA[ A$r; x ] (4.4)

The time scale used for this computation is called Taking a business time scale & as introduced in Section 4.4.6 leads to better data cleaning than taking physical time. The variable dr denotes the configurable range of the kernel of the moving averages and should cover the time frame of the mean reversion of the filtered variable; a reasonable value for bid-ask spreads has to be chosen. The iterative computation of moving averages is explained in Section 3.3.5. Here and for all the moving averages of the filtering algorithm, a simple exponentially weighted moving average (EMA) is used for efficiency reasons.

A small \x\ value deserves high trust; an extreme \x\ value indicates an outlier with low credibility and negative trust capital. Before arriving at a formula for the trust capital as a function of x, the distribution of x has to be discussed. A symmetric form of the distribution is assumed at least in coarse approximution. This is ensured by the definition of the filtered variable jc, which is a mathematically transformed variable. The exact definition of x is deferred to Section 4.6.3 in the chosen structure of this chapter.

The amount of negative trust capital for outliers depends on the tails of the distribution at extreme (positive and negative) x 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 x~a where a is called the tail index. We know that density functions of level-filtered variables such as bid-ask spreads are fat-tailed (see Muller and Sgier, 1992). Determining the distribution and a in a moving sample would be a considerable task, certainly too heavy for filtering software. Therefore, we choose an approximate assumption on a that was found acceptable across many rates, filtered variable types and financial instruments: a = 4. This value is also used in the analogous pair filtering tool (e.g., for price changes, and discussed in Section 4.4.3).

For extreme events, the relation between credibility and trust capital, Equation 4.1, can be asymptotically expanded as follows

= for « - 1 (4.5)

Terms of order higher than (1/T)2 are neglected here. Defining a credibility proportional to x~a is thus identical to defining a trust capital proportional to x"!1. Assuming a - 4 , we obtain a trust capital proportional to x2. For outliers, this trust capital is negative, but for small x, the trust capital is positive up to a maximum value we define to be 1.

Now, we have the ingredients to come up with a formulation that gives the resulting trust capital of the ith quote according to the level filter:

Ti0 = l-§,2 (4.6)

where the index 0 of 7)o indicates that this is a result of the level filter only. The variable is x in a scaled and standardized form:

& = (4-7)

with a constant §o- Equation 4.6 together with Equation 4.7 is the simplest possible way to obtain the desired maximum and asymptotic behavior. For certain rapidly mean-reverting variables such as hourly or daily trading volumes, this may be enough.

However, the actual implementation for bid-ask spreads has some special properties. Filter tests have shown that these properties have to be taken into account in order to attain satisfactory spread filter results:

Quoted bid-ask spreads tend to cluster at "even" values (e.g., 10 basis points,) whereas the real spread may be an odd value oscillating in a range below the quoted value. A series of formal, constant spreads can therefore hide some substantial volatility that is not covered by the statistically determined denominator of Equation 4.3. We need an offset Ax2 to account for the typical hidden volatility in that denominator. A suitable choice is Ax2 = [constanti(x + constant2)]2.