under study is the logarithmic middle price x. At time tj, it is denned as

X(tj) =

logPbid(Q) +logPask(Q)

log JPbid(tj) Pask(tj) (3.4)

where tj is the inhomogeneous sequence of the tick recording times. The variable x may simply be called the price in a context where its logarithmic nature is obvious and not explicitly relevant. It is based on the (geometric) average of the bid and ask price rather than either the bid or the ask price alone; this is a better approximation of the true price. The best choice, even better than Equation 3.4, might be a so-called effective price as discussed at the end of this Section 3.2.2.

In the foreign exchange (FX) market, a further advantage of Equation 3.4 is obvious. FX prices can be seen from two sides, the value of the U.S. Dollar (USD) in Japanese Yen (JPY) and the value of the JPY in USD. Equation 3.4 is perfectly antisymmetric: if x is the USD-JPY price, the JPY-USD price is simply - x. Statistical results based on absolute differences of x (or volatility) are identical for USD-JPY and JPY-USD. This is a desired property because USD-JPY and JPY-USD are the same market. If the logarithmic transformation was avoided or the logarithm of the arithmetic average of bid and ask was taken instead of Equation 3.4, the antisymmetry would be violated and the statistical results of USD-JPY and JPY-USD would differ. The logarithmic transformation has the additional advantage of making returns (differences of x) dimensionless-that is, independent of the original units in which the price is measured.

In the case of transaction or middle prices, Equation 3.4 is obviously replaced by x(tj) = logptransact(O) or lo8PmiddleCO)- For certain data types, the logarithmic transformation is less suitable; it is avoided or made in a mathematically different form. For spot interest rates as discussed in Section 2.3.1, the logarithm of the capital increase factor can be taken:

and analogous for the ask quote, where the interest rate i is inserted as a plain value, not in percent (e.g., 0.05 instead of 5%). Alternative definitions of x are explained whenever they are applied in this book.

The inhomogeneous series x(tj) can be transformed to a homogeneous time series by using an interpolation method as explained in Section 3.2.1, using Equation 3.2 or 3.3. For the homogeneous series of prices, we use the index i:

where f, is the homogeneous sequence of times regularly spaced by time intervals of size . As already mentioned, t and At may refer to any definition of the time scale, not only physical time.

In some markets such as the FX spot market, bid and ask prices are just indicative quotes produced by market makers who are often interested primarily

XbidUj) = log[l -Hbid((/)]

(3.5)

( ) = x{Al,tt) =

log Ibid (fr) +logPask(fr)

(3.6)

in either the bid or the ask price; the other price acts as a noncompetitive dummy value. This leads to a small error that affects Equation 3.4. Moreover, the quoted spread (ask minus bid price) does not exactly reflect the real spread, which is usually smaller as reported in Goodhart et al. (1995).3 Furthermore, because of transmission delays, it may be, for example, that market maker enters a quote after market maker A, but that the quote of market maker is the first to appear on a multi-contributor data feed. Data gaps due to transmission breakdowns become more significant at high frequencies.

All these effects can be modeled in the form of an effective price which is closer to the transaction price than the price x of Equation 3.4. In the absence of real transaction prices, we may define an effective price algorithm by looking at the properties of the prices and the market organization. All quotes have a finite lifetime, which is roughly around two minutes during periods of average FX market activity and can strongly vary depending on the market and its state. We can define the effective price as consisting of the best bid and ask quotes available (or the averages of bid and ask) in a time window of the size of a quote lifetime. Another idea for such an algorithm would be to eliminate the negative first-order autocorrelation of the returns present at very high frequencies (see Section 5.2.1). An example of an algorithm for the computation of effective price is given in Bollerslev and Domowitz (1993) where the trade-matching algorithm of the interbank market system "Reuters Dealing 2000" is used. Interestingly, the prices generated by this algorithm exhibit a positive rather than negative first-order autocorrelation. In contrast, Goodhart et al. (1995) still obtain a negative first-order autocorrelation, though less pronounced, in their analysis of the Dealing 2000-2 system. In this book, no definition of an effective price is given, but the behavior of prices in the very short term (seconds to minutes) is discussed in several aspects at several places.

3.2.3 Return

The return at time f,-, r(f,), is defined as

r(u) = r(At;ti) = x(ti) - x(ti - At) (3.7)

where x(tj) is a homogeneous sequence of logarithmic prices as defined by Equation 3.6, and At is a time interval of fixed size. In the normal case, At is the interval of the homogeneous series, and r(f,) is the series of the first differences of x(tj). If the return interval is chosen to be a multiple of the series interval, we obtain overlapping intervals as discussed in Section 3.2.8. Returns are sometimes also called price changes.

The return is usually a more suitable variable of analysis than the price, for several reasons. It is the variable of interest for traders who use it as a direct measure of the success of an investment. Furthermore, the distribution of returns

3 In their one-day study of real transaction prices, Goodhart et al. (1995) found that although the actual spread is usually within the quoted spread, it could be larger in highly volatile periods.

is more symmetric and stable over time than the distribution of prices. The return process is close to stationary whereas the price process is not.

3.2.4 Realized Volatility

The realized volatility v(tj) at time ,- is computed from historical data and it is also called historical volatility . It is defined as

v(ti) = v(At, n, p; ti)

1 "

- Y\r(At;ti-n+j)\p

(3.8)

where the regularly spaced returns r are defined by Equation 3.7, and n is the number of return observations. There are two time intervals, which are the return interval At, and the size of the total sample, nAt. The exponent p is often set to 2 so that v2 is the variance of the returns about zero. In many cases, a value of p - 1 is preferred, although p may also be a fraction, p > 0. The choice of p is further discussed below.

In order to compute realized volatility, the return interval, At and a sample of length nAt need to be chosen. By inserting At = 10 min in Equation 3.8, one can compute the volatility of regularly spaced 10-min returns. One important issue is that many users want their realized volatility in scaled form. Although the volatility may be computed from 10-min returns, the expected volatility over another time interval (e.g., 1 hr or 1 year) may also be calculated. Through a Gaussian scaling law, v2 oc At, the following definition of scaled volatility is obtained:

Scaled = yj ? 1 V (3.9)

The most popular choice of the scalingreference interval Afscale is Ascaie = 1 -If this is chosen, uSCaied is called an annualized volatility, fann

Uann = \JV ( 0)

Practitioners often use annualized volatility in percent (multiplying Uann by 100%). Typical annualized volatility values for some FX rates are around 10%.

In practice, various volatility definitions may lead to confusion. Terms such as "one-day historical volatility" should be avoided because they do not express whether "one day" refers to the return measurement interval At, the sample size nAt or the scaling reference interval Ascaje- In order to clarify this, we give a detailed recipe to compute realized volatility in practice:

Consider and choose three time intervals:

- The time interval of return observations, At