3.2.5 Bid-Ask Spread

In bid-ask price pairs, the ask price is higher than the bid price.6 The bid-ask spread is their difference. A suitable variable for research studies is the relative

spread s(tj):

s(tj) = log/?ask(rv-) - logA)id(O) (3.12)

where j is still the index of the original inhomogeneous time series. This definition has similar advantages as the definition of the logarithmic price x, Equation 3.4. The nominal spread (pask - ¹d) s n unts °f tne underlying price, whereas the relative spread is dimensionless; relative spreads from different markets can directly be compared to each other. In the FX market, another advantage is obvious. If the relative spread of USD-JPY, for example, is s, the relative spread of JPY-USD is also s, because the roles of bid and ask are interchanged. Results of spread studies are invariant under inversion of the rate. Other spread definitions do not have this perfect symmetry. The relative spread is sometimes just called the "spread" if its relative nature is obvious from the context.

For spot interest rates, we can adapt the relative spread definition in the same sense as Equation 3.5:

s(tj) = log[l +/ask(0)]-log[l +¹«j)] (3.13)

The relative spread is a positive bounded quantity that has a strongly skewed distribution. This can be a problem for certain types of analysis. A further transformation leads to the "log spread," logs(f;). For bid-ask prices, the log spread is

\ogs(tj) = logtlogpaskOV) - logPbid(O)] (3-14)

Muller and Sgier (1992) have shown that its distribution is much less skewed and closer to symmetric than the distribution of s.

The bid-ask spread reflects the transaction and inventory costs and the risk of the institution that quotes the price. On the side of the traders who buy or sell at a quoted price, the spread is the only source of costs as intraday credit lines on the foreign exchange markets are free of interest.7 The spread can therefore be considered as a good measure of the amount of friction between different market participants, and thereby as a measure of market efficiency. The relatively high efficiency of the major FX spot markets is reflected by the small average size of s.

6 Some data sources have prices of minor markets in the spurious technical form of bid-ask pairs with bid = ask, presumably because only one of the two prices is available at a time. These quotes are not true bid-ask pairs.

7 A trader taking a forward position will, of course, have to pay the interest on his or her position between the trade and the settlement as well as the additional spread on the forward rate.

In markets with indicative quotes from different market makers, individual spreads s(tj) are often affected by individual preferences of market makers and by habits of the market (see Section 5.6.5). Therefore, a homogeneous time series of spreads s(tj) generated by interpolation contains a rather high level of noise. A more suitable alternative is to compute average spreads within time windows and to build a homogeneous time series of these average spreads.

3.2.6 Tick Frequency

The tick frequency at time tj, f(tj), is defined as

f(tj) = f(At;ti) = ~ N{x(tj) I tj - At < tj < ti] (3.15)

where N{x(tj)} is the counting function and At is the size of the time interval in which ticks are counted.

The "log tick frequency," log/(?,), has been found to be more relevant in Demos and Goodhart (1992). We can also define the average time interval between ticks, which is simply the inverse tick frequency, /~ (/,-). Tick frequency can also be computed by a time series operator as explained in Section 3.4.5.

The tick frequency is sometimes taken as a proxy for the transaction volume on the markets. As the name and location of the quoting banks are also available, the tick frequency is also sometimes disaggregated by banks or countries. However, equating tick frequency to transaction volume or using it as a proxy for both volume and strength of bank presence suffers from the following problems. First, although it takes only a few seconds to enter a price quotation in the terminal, if two market makers happen to simultaneously enter quotes, only one quote may appear on the data collectors screen; second, during periods of high activity, some operators may be too busy to enter the quote into the system; third, a bank may use an automatic system to publish prices to advertise itself on the market; fourth, the representation of the banks depends on the coverage of the market by data vendors such as Reuters or Bridge. This coverage is changing and does not entirely represent the whole market. For example, Asian market makers are not as well covered by Reuters as their European counterparts; they are more inclined to contribute to the local financial news agencies such as Minex. Big banks have many subsidiaries; they may use one subsidiary to quote prices made by a market maker in another subsidiary on another continent. Quotes from differently reliable and renowned sources have very different impacts on the market. For all these reasons, we should be cautious when drawing conclusions on volume or market share from tick frequency.

3.2.7 Other Variables

A set of other variables of interest are as follows:

The "realized skewness" of the return distribution. Roy (1952) evaluates the extreme downside risk in portfolio optimization in terms of the cubic

root of the third moment of returns. The skewness of returns can also be measured by a time series operator (see Section 3.3.12).

The volatility ratio, the ratio of two volatilities of different time resolutions: vam(mAt, n, p)/vam(At, mn, p), based on Equations 3.8 and 3.10, with an integer factor m > 1. This is a generalization of the variance ratio studied in Lo and MacKinlay (1988), Poterba and Summers (1988) and Campbell et al. (1997). The volatility ratio is around 1 for a Brownian motion of x, higher if x follows a trend, lower if x has mean-reverting noise. The volatility ratio (or an analogous volatility difference) is thus a tool to detect trending behavior.

The direction change indicator, counting the number of essential trend reversals within a time interval as defined by Guillaume et al. (1997).

3.2.8 Overlapping Returns

Some variables, notably returns, are related to time intervals, not only single time points. When statistically investigating these variables, we need many observations. The number of observations can be increased by choosing overlapping intervals. For returns, a modified version of Equation 3.7 is used:

where r, is again a regular sequence of time points (for any choice of the time scale), separated by intervals of size At. The interval of the return, however, is mAt, an integer multiple of the basic interval At. If ,- is considered for every i, we obtain a homogeneous series of overlapping returns with the overlap factor m. The corresponding series of nonoverlapping returns would be rm, 2 , ,

Figure 3.3 illustrates the concept of overlapping intervals. Does a statistical study gain anything from using overlapping as opposed to non-overlapping returns? The number of observations can be increased by using overlapping intervals with a growing overlap factor m, thereby keeping the return interval m At constant. At the same time, neighboring return observations become increasingly dependent. Thus a gain in statistical significance is not obvious. The problem was discussed in Hansen and Hodrick (1980), where a method of estimating parameters and their significance limits from overlapping observations has been developed and applied. In Dunis and Keller (1993), a "panel regression" technique is presented and applied where the overlapping observations are grouped in several nonoverlapping series with phase-shifted starting points.

Miiller (1993) has investigated this question under the simplifying assumption that x is drawn from an identical and independent distribution

= r(tt) - x(tj) - x(tj - m At) = Xj-Xj-m

(3.16)

ri = x(ti) - x(ti-i) 6 M(0,a2)

(3.17)