FIGURE 7.9 Lead-lag correlation of fine and coarse volatilities for four different impliei. forward rates derived from the Three-Month LIFFE Euromark, with a 3-hr grid in !?-timt. The sampling period is from April 1, 1992, to December 30, 1997. In the panels, a montr is represented by m.

Information Flow for the Volatility

2 4 6 8 10 12

Past Granularity

FIGURE 7.10 The correlation difference (Equation 7.1 I) between coarse and fine volatilities is explored for the USD-CHF FX rate. The asymmetry of the lead-lag correlation (at one lag of 4 weeks) is apparent around the diagonal, which naturally presents a correlation of I (and a difference of 0) because we are correlating a quantity with itself. The top half of the graph presents a positive difference in the lagged correlation whereas the bottom half presents the symmetric negative difference. The sampling period is from June I, 1987, to August I, 1997. (With permission of Gilles Zumbach.)

where v(t) is a measure of volatility calculated with the weekly variance of daily returns.7 Then triplets of similar volatility, v(t), are put into the same bin, and the autocorrelation of returns at lag At, conditional to v(t), is studied:

p{v) = p(r(t),r(t + At)\v(t)) (7.12)

Such an analysis has four parameters, At for the returns and the three parameters for the volatility as identified in Section 3.2.4 and in the Equation 3.8: At, n, and p.

7 In principle, any definition of volatility along the lines of Equation 3.8 can be chosen and its parameters varied until the conditional correlation reaches a maximum.

-0.4------:........:..........:-........ : ..... : I

-0.5-1---- -j-1--r- j i -i- I

0.02 0.04 0.06 0.08 0.10 0.12

Volatility

FIGURE 7.11 The conditional autocorrelation of weekly returns of USD-DEM as a function of the average absolute weekly return over 5 days. The sampling period is June I, 1973, to June I, 1994.

This function p(v) is presented for the FX rate USD-DEM on Figure 7.11. It is computed with a At of 1 week and a volatility definition that uses the mean weekly absolute returns over 5 weeks. In summary, the parameters for the graph are At = 1 week, n At = 5 weeks, and p = 2. The conditional correlation appears only for data at low frequency. The effect is quite strong for low volatility with a conditional correlation close to 0.3 at its maximum, decreasing down to negative values of -0.15 for high volatility. The computation is done with overlapping bins containing always the same number of observations to avoid changing the significance of the different results. From this figure, it appears that the "current state" of the market changes the price process behavior and the volatility plays an important role beyond its own dynamics. The results shown here for the most traded FX rates are also present in the other FX rates. It was also reported by LeBaron (1992b) for stock indices. Varying the parameters cause this effect to disappear for / smaller than 1 day. In the intraday region, influence of the heat wave effect becomes important and overshadows the findings.