Jan Feb

Time

FIGURE 9.1 Top panel: Daily standard RiskMetrics USD-JPY volatility for January, 1999 to February 1999. Circles: Data sampled at 7 a.m. GMT. Diamonds: Data sampled at 5 p.m. GMT. Bottom panel: The USD-JPY price plotted against time.

very different volatility and thus risk levels for the same financial instrument, just because they live in different time zones. A difference can persist over weeks, as shown in Figure 9.1. This figure is just an example. The same surprisingly strong effect can also be found for other financial instruments, sampling periods, choices of daytime, and process equations.

Both deviating volatility values cannot be right at the same time; there must be an error in these values. This error is of stochastic nature; there is no systematic bias dependent on the daytime. In Figure 9.1, the difference between the two curves is neither always positive nor negative; it changes its sign.

Figure 9.1 demonstrates the large stochastic error of the RiskMetrics method. The large size of this error has two main reasons:

1. The rather small range of the kernel of about 16 business days. The number of independent observations is limited. We cannot essentially change this fact, because the choice of a short range is also motivated by the goal of fast adaptivity to new market events.

9.2 FORECASTING VOLATILITY FOR VALUE-AT-RISK

2. The results depend on only one observation per day, taken at a certain daytime. All the other information on prices of the day is thrown away. The value at that daytime may have little representation for the full day: it may be located on top of a short-Jived local peak of the price curve.

The second investigated volatility forecasting model was introduced by Miiller (2000). It follows RiskMetrics as closely as possible. There are only two innovative modifications:

The squared volatility a2(t) is computed at every available tick, not only once per business day.

Simple returns are replaced by operator-based, smoothed returns.

Nothing is changed otherwise; the sampling range of 15.67 business days and the business-daily nature of (smoothed) returns are preserved. The formula is again written with the help of time series operators:

again with At - 1 business day and r = 15.67 business days. Equation 9.4 is iteratively evaluated tick by tick. The iterated operator EMA[r, 4; x] is defined by Equation 3.53. As the simple EMA operator, it can be efficiently computed by using the iterative Equation 3.51.

The constant compensates for the fact that we use smoothed returns, x - EMA[A?,4;x], instead of the simple returns, x(t) - x(t - At). In the case of x following a Gaussian random walk, the theoretically correct value is = 128/93. Using this factor eliminates a systematic bias of the tick-by-tick volatility as compared to the RiskMetrics volatility.

Equation 9.4 is computed on a special business time scale defined as follows. The 49 weekend hours from Friday 8 p.m. GMT to Sunday 9 p.m. GMT are compressed to the equivalent of only 1 hr outside the weekend. This fully corresponds to the time scale of RiskMetrics, which omits the weekend days. A more sophisticated and appropriate choice of the business time scale would be the #-time of Chapter 6, but this is avoided here in order to keep the approach as close to RiskMetrics as possible.

The advantages of the tick-by-tick volatility forecast are demonstrated in Figure 1.3. The volatility as a function of time appears as one continuous, consistent curve. We obtain volatility values at any daytime now, not just once or twice a day. A risk manager in London measures the risk of the instrument on the same basis as a risk manager in East Asia, as should be expected. The variations of the volatility level over time are more moderate in Figure 1.3 than the corresponding variations of the RiskMetrics volatility, although the kernel range of 15.67 business days is the same.

The tick-by-tick volatility forecast is based on (almost) continuously overlapping returns. Overlapping returns lead to reduced stochastic noise of volatility measurements, as shown in Section 3.2.8. In addition to this, the tick-by-tick

(9.4)

volatility is based on smoothed rather than simple returns, which also leads to a reduction of stochastic noise.

The third volatility model is a multiple-horizon version of the second model:

a2 = * (9.5)

2=0 Jw

with

EMA

to ; (x-EMAA/o/tr, 4; xtf

(9.6)

where the partial volatility forecasts of Equation 9.6 are like the volatility forecasts of Equation 9.4. The weights /* of the partial volatility forecasts, their return intervals Ato / (, and their sampling ranges ro /* are in geometric sequences and can be flexibly chosen and optimized by setting the parameters n (the number of partial forecasts), fw, Ato, / ro, and fr.

The third volatility model (Equation 9.5) shares the advantages of the second one (Equation 9.4) and has the additional multiple-horizon property, which leads to superior volatility forecast quality. This is in analogy to the multiple-horizon EMA-HARCH process shown in Section 8.3.4, which is also superior to single-horizon processes such as GARCH.

9.2.2 Choosing the Best Volatility Forecasting Model

The quality of volatility forecasting models has to be measured in statistical tests, comparing the forecasts to the actual values of the target variable, which is a form of realized volatility here.

Tn out-of-sample tests of the three volatility forecasting models presented in Section 9.2.1, the tick-by-tick model of Equation 9.4 has distinctly better volatility forecasts than the RiskMetrics model of Equation 9.1 or 9.2. Equation 9.5 leads to even better volatility forecasts.

Testing the quality of volatility forecasts implies some technical difficulties. First, there are several quality measures to choose from. This is discussed in Sections 9.4.1 and 8.4.1, where volatility forecasts are derived from process equations and tested by several criteria.

A second difficulty lies in the bias of both realized volatility (the target variable) and volatility forecasts which appears if the return intervals chosen are too small. This bias is discussed in Section 3.2.4 and in Andersen et al. (2000). Volatility forecast tests are affected by this bias. A treatment of the bias is almost inevitable when designing volatility forecasting models and tests based on high-frequency returns over intervals of less than an hour. Corsi et al. (2001) propose a suitable bias correction method.

Due to these technical difficulties, there is no comprehensive study of high-frequency volatility forecasts and their qualities yet. The final goal is the development of a consistent methodology of risk analysis based on high-frequency data with superior forecasting quality: real-time risk assessment.