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74 FIGURE 7.2 Bias factors plotted versus time, for the FX rate USDJPY (upper panel) and the Japanese equity index Nikkei225 (lower panel). Deviations from I indicate a bias in realized volatility. The bias factor is the ratio of two mean realized volatilities over the same sample (see Equation 7.3). The investigated return measurement intervals At are as follows. Bold curves: At = I hr; middle curves: At = 5 min; thin curves: At  2 min. can be neglected more easily. Surprisingly, the biases have different signs. The bias of the foreign exchange (FX) rate is positive, whereas that of the equity index is negative (B(f,) < 1). The bias can be explained by microstructure effects, but these are obviously different for different markets. The microstructure effects of FX rates were discussed in Chapter 5, in particular the negative autocorrelation due to a bouncing effect within the bidask spread (Section 5.2.1). The bias due to this effect can be modeled as in Section 5.5.3 and in Corsi et al. (2001), where the influence of data gaps on the bias is also analyzed. There is ongoing research aiming at refined versions of this bias model. The negative bias of the equity index has to be explained differently. An equity index is a weighted average of some equity prices. Some of the individual equities play a leading role in price adjustments and establish small trends that the other equities follow. This mechanism causes a shortterm (few minutes) positive autocorrelation of the index returns and eventually a negative bias of realized volatility when a very short
interval Af is chosen. The bias factors moderately fluctuate over time, but there are no dramatic shifts. The overall levels are maintained even over the 10year sample of Figure 7.2 (upper panel). The bias can be avoided cither by taking large return intervals Af (with the disadvantage of large stochastic errors) or by introducing a bias correction for small intervals Af. Eliminating the bias seems to be a demanding task requiring a model of the microstructure effects. Section 5.5.3 has such a model for FX rates, but other markets such as equity indices need other models. Instead of developing bias models for each market, we suggest a simple bias correction method that needs no explicit model and only relies on the assumption that the biasgenerating mechanism is much more stable over time than the volatility itself. The limited size of bias fluctuations in Figure 7.2 justifies this assumption. The bias correction is simple. Each realized volatility observation is divided by the bias factor as measured in the past: v(At, n, 2; tj) vcorr(At,n,2;tj) = (7.4) o(tj) where B(f,) is defined by Equation 7.3. This bias correction can be computed in real time, because it is based on information fully available at time tj. Some variations of Equation 7.4 are possible, as suggested by Corsi et al. (2001). The bias correction factor can be computed by moving average operators as explained in Section 3.3 instead of the sums of Equation 7.3. Figure 7.3 probes the success of the simple bias correction. The bias factor corr of the already biascorrected realized volatility can be measured in the same way as the bias of the uncorrected volatility (Equation 7.3): D . . vcorr(At,mq,2;tj) ocorrUi) =  (./.>) v(Afref. m, 2; tj) A perfect bias correction implies Bcorr(tj) = 1. However, the bias correction is not perfect. Both the bias correction and its measurement in Equation 7.5 rely on a quantity u(Afref, m, 2; tj), which has a stochastic error. These imperfections are visible in the form of fluctuations of Bcon about 1 in Figure 7.3. Figure 7.2 and Figure 7.3 are based on the same samples and parameters and can directly be compared. Bcorr in Figure 7.3 is much closer to 1 than in Figure 7.2, in all cases. This fact demonstrates a successful bias correction for both markets, FX and the equity index. In spite of the success of Equation 7.4 as shown in Figure 7.3, the simple bias correction has some shortcomings, one of them being the multiplicative nature of the formula. Realized volatility values are corrected by a slowly varying correction factor, irrespective of the current volatility level. One can argue that an additive or nonlinear correction of realized volatility would reflect reality better than the multiplicative correction. (An additive correction may lead to impossible negative volatility values, though.) A fair judgment may be as follows. Equation 7.4
1998 1999 2000 FIGURE 7.3 Bias factors plotted versus time, for the FX rate USDJPY (upper panel) and the Japanese equity index Nikkei225 (lower panel), computed by Equation 7.5. The investigated realized volatility values have already been biascorrected by Equation 7.4, so the small deviations from I indicate imperfections of the bias correction. The investigated return measurement intervals At are as follows. Bold curves: At = I hr; middle curves: At = 5 min; thin curves: At = 2 min. The same scaling as in Figure 7.2 is used. succeeds in largely reducing the bias and is thus better than no bias correction. As soon as an appropriate model of the biasgenerating process for a particular market exists, the corresponding biascorrection method will be clearly superior to Equation 7.4. Bias correction is a means to compute realized volatility with smaller intervals Af and, for a given sample of size T = n Af, a smaller stochastic error. Unfortunately, the bias correction introduces an additional stochastic error due to the factor u(Afref, m, 2; f,) in Equation 7.3. Corsi et al. (2001) show that a biascorrected volatility with reasonable parameters has a total error that is still distinctly smaller than the error of uncorrected volatility. The following rough calculation also shows this. Uncorrected volatility requires a rather large Af of about 1 hr (with q = 24) to keep the bias at bay. The stochastic error is proportional to 1/24, and a bias of roughly of the same size adds to the error. Biascorrected volatility can have a
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