with / > 0. Therefore, as r(nAt) is of the form Y11=\ £ we can expect r,(nAt) to be of the order an (i.e., of the order 1/"), for large values of n. As a result, the dominating term in Equation 5.13 will be p 1 ( )/ , such that the relationship between Equation 5.13 and 1 ( ) will be linear with slope p/a for all distributions that satisfy a generalized central limit law.

Theorem: Let {£r}Z\ denote a sequence of i.i.d. random variables with common distribution F(.). Let F(.) belong to the domain of attraction of a stable law with index a. Let p be such that 0 < p < a for a < 2 and 0 < p < 2 for a - 2. Then

\n{E[\r{nAt)\P])-\n(E[\r(At)\P]) p

hm--- = - (5.14)

noo \n(n) a

Let us now consider the case p = 2. Following the theorem, the class of distributions of s(t) must now be restricted to the distribution with finite variance. Using the independence assumption, we can easily obtain the following result:

E[\r(nAt)\l] = E

]Pe(? - i At)

YE[e2{t -iAt)] = no2 /=0

(5.15)

Consequently, the numerator in Equation 5.14 reduces to 1 ( ) and the fraction to 1. This is in accordance with the theorem, as it follows from the standard limit theory that distributions with finite variance lie in the domain of attraction of the normal distribution, which has an a - 2, which, in this particular case, should not be confused with the tail index a of Section 5.4.2. In our study of tail indices, the conclusion was that the second moment of the distribution was finite (our tail indices were all largely above 2). Also, the results lead to a value of 0.5 (Table 5.8), which is the value one should theoretically obtain for p = 2 (as we were studying the square root of the second moment). There is, though, a difference between our empirical results and this theoretical result for p = 1 and for the interquartile range. There are at least three explanations for this. The first is that the distribution F(.) of the random variable is not really common, which can be caused by heteroskedasticity. The second explanation is that the theorem is an asymptotic result and we might not have converged with our data to the limit. The third explanation is related to the i.i.d. assumption for the innovations under which the theoretical result is obtained. A drift exponent for £[r], which is different from 0.5, could be an indication that there is a certain dependence between consecutive prices. We have already seen some of these dependencies for the very short-term (negative autocorrelation of returns) and we shall see some more in this and in the next chapter.

5.5.3 A Simple Model of the Market Maker Bias

In Section 5.5.1, we reported the findings published in Muller et al. (1990) about the scaling law of volatility measures. The parameters of this law seem very stable (see Guillaume et al, 1997) but depend on the way the statistical quantities are computed and on the errors that enter the evaluation through the least square fit of the scaling law parameters. Tn Muller et al. (1990), we briefly mention the problem but in order to help people reproduce our results, we give here the full derivation of the error, which consists of a stochastic component and a market maker bias.

When making statistical studies of returns, researchers only considerthe usual statistical error due to the limited number of observations. This error is clearly dominant when the return is measured over time intervals of a day or more. When the time interval is reduced to a few minutes, however, the uncertainty of the price definition due to the spread must be also considered. The market makers are biased toward one of the two prices, either the bid or the ask, thus introducing a bouncing effect that reflects in a negative autocorrelation of the returns in the very short-term (see Section 5.2.1, Goodhart and Figliuoli, 1991; Guillaume era/,, 1997). The true market price is between the bid and the ask quotes but not necessarily in their exact midpoint.21 This uncertainty can be assessed to a considerable fraction of the nominal bid-ask spread. For short horizons, the amplitudes of price movements become comparable to the size of the spread. If spreads are large (especially minor FX rates), the uncertainty implies an important measurement error. For bid-ask data from electronic exchanges and transaction price data, the measurement error due to uncertainty is smaller or even negligible.

The purpose of this section is to derive the error on the statistical quantity entering the scaling law computation when the measurement error is also taken into account. We derived this model independently (Muller et al, 1995) but it turns out that it is very similar to the model developed by Roll (1984) for equity prices.

We rely here on the definitions provided in Chapter 3 for the quantities we are going to use. The scaling law (see Equation 5.10) is empirically computed by fitting its logarithmic form,

The law becomes linear in this form. For the linear regression, we need to know the errors of log Ax. We saw that there is a similar scaling law for (( 2))1/2:

The problem is to find the error on Ax knowing that we have an uncertainty related to the price definition and to the spread. Expressions with absolute values

log(Ax) = DlogAt + log

(5.16)

(5.17)

21 For a discussion of this point, see Muller et al. (1990); Bollerslev and Domowitz (1993); Goodhart and Payne (1996).

such as I Ax I are known for their poor analytical tractability. Therefore, the whole error computation is done for the analogous case of (( 2))/2. Following the arguments given in the introduction, let us assume that x* is the series of true logarithmic market prices whereas the observed middle values x/ as defined by Equation 3.6 are subject to an additional market maker bias e,:

Xj = x* + si (5.18)

The true return is defined analogous to Equation 3.7:

r* = *{ ) = xf - x* x (5.19)

Its relation to the observed return follows from Equations 3.7 and 5.18,

= r* + si - £, , (5.20)

To compute the error, a minimum knowledge on the distribution of the stochastic quantities is required. We know that the returns r, and r* follow a Gaussian distribution only as a crude approximation (sec Section 5.4), and the market maker bias Si might also be nonnormally distributed. Nevertheless, we shall assume Gaussian distributions (the same assumption is used in Roll, 1984) as approximations to make the problem analytically tractable:

r* e Af(o,Q*2} (5.21)

Bi e m(o,y) (5-22)

where /"( , cr2) is the Gaussian probability distribution with mean jjl and variance ct2. The maximum deviation of the market maker bias is the distance between the middle price and the bid price (or ask price), that is half the spread. As a coarse approximation, we assume that the standard deviation is half that value, that is one-fourth of the typical value of the relative spread (Equation 3.12). This means that rj2 is assumed to be one-eighth of the squared relative spread.22 Studies with transaction prices have shown that the "true" spread is very different from the quoted spread (see Section 5.2.2). This quantity is a kind of convention; the market maker is really interested in one of the bid or ask prices and adds or subtracts a canonical value to the price she or he wants to use. In normal market conditions, the price is settled with an equal distribution of buyers and sellers (same assumption as in Roll, 1984). Thus, in a first approximation, the two random variables, the true returns and the market maker bias as it appears in quoted prices can well be assumed to be independent. The choice of the market maker bias is somewhat

We neglect spread changes.