have homogeneous data, for example when modeling financial data with ARMA or GARCH processes. Another example is the computation of empirical probability distributions. Such computations are done with a smooth version of the formula

Piz) = ~ f dtS(z-z(t)) (3.83)

1 Jo

with z{t) the (continuously interpolated) empirical data. In the time integral, a measure can be added, or the integral can be evaluated in business time to account for the seasonalities. The evaluation of the time integral is computationally heavy, and it is much simpler to generate a regular time series and to use the familiar binning procedure to obtain a histogram of z- Note also that a moving probability distribution can be defined by replacing the time integral by an MA operator (see the remark at the end of Section 3.3.7).

3.4.3 Microscopic Return, Difference, and Derivative

From the tick-by-tick price time series, the microscopic return for a quote is defined as rj = Xj - Xj-\. This return can be attributed to one quote, even if, strictly speaking, it is related to two subsequent quotes. Note that this is a "microscopic" definition that involves neither a time scale nor an interpolation scheme. Using the backward shift operator B, the return time series can be defined as

r=x-B[x\ = (l-B)x = 8x (3.84)

where the microscopic difference operator11 is 8 - (1 - B). The lag n difference operator is defined by 8[n] = (1 - Bn).

The microscopic derivative operator 3 is defined as

-, - x i-\ 8x

3\8t0\xj = J J

Sto + tj-tj-i 8t0 + 8t

(3.85)

The constant 8to regularizes the expression when - t\-\. A reasonable value of 8to must be small; the actual choice depends on the application. Similar to the macroscopic /-derivative, a microscopic /-derivative can be defined as

d[St0, y]x =--- (3.86)

The best parameters should follow a study yet to be done for the random process of x. The constant 8 to regularizes the expression when tj =tj-\.

These derivatives are potentially very noisy and can be averaged. In general, the macroscopic derivative D (Equation 3.64) seems more relevant for applications to random processes.

3.4 MICROSCOPIC OPERATORS 79 2-1-1

re > u

ui-11 1 i 1 1 1-i 1-1-1-i 1-1 1 i-1-1-1- ~ --r~i-1-r"~"1

26.10 27.10 28.10 29.10 30.10 31.10 1.11 2.11

Date

FIGURE 3.16 Microscopic volatility, computed with = 0.5, StQ = 0.001 seconds, the time interval expressed in years (annualized), and = I hr.

3.4.4 Microscopic Volatility

The microscopic volatility is defined as the norm of the microscopic derivative,

Microscopic volatility[r; z] = MNorm[r/2; 9z] (3.87)

which also depends on the implicit parameters Sto and of 3z. Let us emphasize that this definition does not require a regular time series (and that it is not an MA of the macroscopic definition of volatility). In a way, this definition uses all of the information available on the process z. The constant r controls the range on which the volatility is computed. The microscopic volatility for our standard example week is displayed in Figure 3.16.

3.4.5 Tick Frequency and Activity

The tick frequency f(tj) counts the number of ticks per time unit. One definition based on regular time intervals is already given by Equation 3.15 (see also Guillaume et al. (1997), for example). In general, the tick frequency at time tj is defined as

f\T](t) = ~N{tj \ tj e [t-T,t]} (3.88)

where N{tj] counts the number of elements in a set and where T is the sample time interval during which the counting is computed. The tick frequency has

The operator S should not be confused with the £ function used in Chapter 3.

E, 400-

<

200-

> 300-

600-

100-

0+- 26.10

l 28.10

27.10

29.10

30.10

31.10

1.11

2.11

Date

FIGURE 3.17 Tick activity A as defined by Equation 3.90, ticks per hour, computed with r = I hr. The five working days of the example week can be clearly seen.

the dimension of an inverse time and is expressed in units such as ticks/minute or ticks/day. This simple definition has some properties that may not always be desired:

The formula is computationally cumbersome when computed on a moving sample, especially for large T.

It is an average over a rectangular window. We often prefer moving averages whose kernel (= weighting function) fades more smoothly in the distant past.

If no quotes are in the interval spanned by T, this definition will give / = 0. A related problem is the unusable limit T ->- 0 if one wants to measure an instantaneous quote rate.

For these reasons, we prefer the definition in Equation 3.90. The tick rate is defined as

The tick rate has the same dimension as the tick frequency. This definition has the advantage of being related only to the time interval between two subsequent ticks. Following Equation 3.89, an activity can be attributed to one tick, analogous to a return that is attributed to the jth tick by rj = Xj - xj-\ = (8x)j. The activity A is the average tick rate during a time interval r:

A[i;z]=MA[r/2;fl[z]]

(3.90)