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30 realized volatility of Equation 3.8 as an L? norm, with r = n x (3.66) where the operator S computes the difference between successive values (see Section 3.4.3), r is the return interval, and r is the length of the moving sample. RTS[r; z] is an artificial regular time series, spaced by x, constructed from the irregular time series z. The construction of homogeneous time series was discussed in Sections 3.2.1 and 3.2.2; it is reformulated in Section 3.4.2 in terms of the RTS operator. Realized volatility based on artificially regularized data suffers from several drawbacks: For inhomogeneous time series, a synthetic regular time series must be created, which involves an interpolation scheme. The difference is computed with a pointwise difference. This implies some noise in the case of stochastic data. Only some values at regular time points are used. Information from other points of the series, between the regular sampling points, is thrown away. Resulting from this information loss, the estimator is less accurate than it could be. It is based on a rectangular weighting kernelthat is, all points have constant weights of either 1 / or 0 as soon as they are excluded from the sample. A continuous kernel with declining weights leads to a better, less disruptive, and less noisy behavior. By squaring the returns, this definition puts a large weight on large changes of z and therefore increases the impact of outliers and the tails of P(z). Also, as the fourth moment of the probability distribution of the returns might not be finite (Muller et al., 1998), the volatility of the volatility might not be finite either. In other words, this estimator is not very robust. These are reasons to prefer a realized volatility defined as an L1 norm: There are various ways to introduce better definitions for inhomogeneous time series. These definitions are variations of the following one: VolatilityLr, r; z] =  A[RTSLr; z]], with r = N x (3.67) Volatilityfr, r, p\ z] = MNorm[r/2, p; Air; z]] (3.68) where the moving norm MNorm is defined by Equation 3.60. For A, we can take the differential operator of Equation 3.61 or a similar operator. Let us emphasize that no homogeneous time series is needed, and that this definition can be computed
simply and efficiently for highfrequency data because it ultimately involves only EMAs. Note the division by 2 in the MNorm of range /2. This is to attain an equivalent of Equation 3.66, which is parametrized by the total sample size rather than the range of the (rectangular) kernel. The volatility defined by Equation 3.68 is still a realized volatility although it is now based on inhomogeneous data and operators. The kernel form of the differential operator A has a certain influence on the size of the resulting volatility. A "soft" kernel will lead to a lower mean value of volatility than a "hard" kernel whose positive and negative parts are close to delta functions. This has to be accounted for when applying operatorbased volatility. The variations of Equation 3.68 mainly include the following: Replacing the norm MNorm by a moving standard deviation MSD as defined by Equation 3.60. By this modification, the empirical sample mean is subtracted from all observations of [ ; ]. This leads to a formula analogous to Equation 3.11, whereas Equation 3.68 is analogous to Equation 3.8. Empirically, for most data in finance such as FX, the numerical difference between taking MNorm and MSD is very small. Replacing the differential by a /derivative D\x, ]. The advantage of using the gamma derivative is to remove the leading dependence, for example by directly computing the annualized volatility, independent of . An example is given by Figure 3.12. Let us emphasize that the realized volatility in Equations 3.66 through 3.68 depends on the two time ranges r and and, to be unambiguous, both time intervals must be given. Yet, for example, when talking about a daily volatility, the common language is rather ambiguous because only one time interval is specified. Usually, the emphasis is put on r. A daily volatility, for example, measures the average size of daily price changes (i.e., r = 1 day). The averaging time range is chosen as a multiple of , of the order > up to r = 1000r or more. Larger multiples lead to lower stochastic errors as they average over larger samples, but they are less local and dampen the time variations in the frequent case of nonconstant volatility. In empirical studies, we find that good compromises are in the range from r = 16 to = 32t. On other occasions, for example in risk management, one is interested in the conditional daily volatility. Given the prices up to today, we want to produce an estimate or forecast for the size of the price move from today to tomorrow (i.e., the volatility within a small sample of only one day). The actual value of this volatility can be measured one day later; it has = 1 day by definition. To measure this value with acceptable precision, we may choose a distinctly smaller r, perhaps = 1 hr. Clearly, when only one time parameter is given, there is no simple convention to remove the ambiguity.
0.60.5 >> 0.4 26.10 27.10 28.10 29.10 30.10 31.10 1.11 2.11 Date FIGURE 3.12 The annualized volatility computed as MNorm[r/2; D[r/32, = 0.5; xj] with = Ihr. The norm is computed with p = 2 and n = 8. The plotted volatility has five main maxima corresponding to the five working days of the example week. The Tuesday maximum is higher than the others, due to the stock market crash mentioned in the introductory part of Section 3.3. 3.3.12 Standardized Time Series, Moving Skewness, and Kurtosis From a time series z, we can derive a moving standardized time series: zMA[r;z] z[t]  (3.69) MSD[r;z] In finance, z stands for the price or alternatively for another variable such as the return. Having defined a standardized time series z[r], the definitions for the moving skewness, and moving kurtosis are straightforward: MSkewnesstr,, r2; z] = MA[r.; z[r2]3] (3.70) MKurtosisfri, r2; z]  MA[ri; z[t2]4] Instead of this kurtosis, the excess kurtosis is often used, whose value for a normal distribution is 0. We obtain the excess kurtosis by subtracting 3 from the MKurtosis value. The three quantities for our sample week are displayed in Figure 3.13. 3.3.13 Moving Correlation Several definitions of a moving correlation can be constructed for inhomogeneous time series. Generalizing from the statistics textbook definition, we can write two simple definitions:
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