resulting volatility estimate will react immediately following an unusually large return. Then the effect of this return in the EWMA gradually diminishes over time. The reaction of EWMA volatility estimates to market events therefore persists over time, and with a strength that is determined by the smoothing constant X. This is a number between 0 and 1. The larger the value of A, the more weight is placed on past observations and so the smoother the series becomes.

An -period exponentially weighted moving average of a time series x is defined as

x, i + Xx, 2 + 2 ( + + "~* , „ 1 + X + X2 + ... + x"-[

One does not need to define a look-back period {x,„,, . . ., x, „) with an exponentially weighted moving average. Since 0 < X < 1, X" 0 as n oo and the exponentially weighted moving average will eventually place no weight at all on observations far in the past. Also the denominator converges to 1/(1 - X) as n -> oo, so an infinite exponentially weighted moving average may be written

(i-r>]TV„

This is the formula that is used to calculate EWMA estimates of volatility and correlation. For volatility one first calculates an exponentially weighted variance estimate of squared returns:

? = (1 - X)

Then one converts this to annualized volatility in the usual way. For correlation, the covariance estimate

G\2,t =(l-X)2 r\,l-ir2,t-i i=l

is divided by the square root of the product of the two variance estimates with the same value of X. As with equally weighted moving averages, it is standard to square daily returns and cross products of daily returns, not in mean deviation form. These formulae may be rewritten in the form of recursions that are normally used for calculation:

&, ={\-X)rU+X&2 , (3.3)

CTi2,, = (1 - A,)ru, ! r2,, i + Xan.,-\ (3.4)

How should the smoothing constant X be interpreted? There are two terms on the right-hand side of (3.3). The first term, (1 - X)r] x, determines the intensity

Jan-95 Jul-95 Jan-96 Jul-96 Jan-97 Jul-97 Jan-98 Jul-98 Jan-99

-Lambda = 0.84 -Lambda = 0.96

Figure 3.6 EWMA volatilities of the CAC index.

of reaction of volatility to market events: the smaller is X the more the volatility reacts to the market information in yesterdays return. The second term, A,a2 ,, determines the persistence in volatility: irrespective of what happens in the market, if volatility was high yesterday it will be still be high today.

The effect of a single event diminishes because 0 < X < 1. The closer A, is to 1, the more persistent is volatility following a market shock. Thus a high X gives little reaction to actual market events, but great persistence in volatility, and a low X gives highly reactive volatilities that quickly die away. An unfortunate restriction of EWMA models is that the reaction and persistence parameters are not independent, because they sum to one. If this holds in any market, it is perhaps most likely to be the case in some foreign exchange markets, such as the major US dollar rates.

An unfortunate restriction of EWMA models is that the reaction and persistence parameters are not independent, because they sum to one

The effect of different X on EWMA volatility forecasts can be quite substantial, as shown in Figure 3.6. Which is the best value to use for the smoothing constant?4

As a rule of thumb EWMA volatility in most markets should take values of X between about 0.75 (volatility is highly reactive but has little persistence) and 0.98 (volatility is very persistent but not highly reactive). If you want to use the EWMA model for forecasting, very commonly lower values for X are used for short-term forecasts and higher values of X are used for long-term forecasts.

As with the equally weighted moving average, an EWMA is just an estimation method. Additional assumptions need to be made to turn it into a forecasting model. The recursion (3.3) is similar to a simple GARCH model. In fact an

In GARCH models there is no question ofhow we should estimate parameters, because maximum likelihood estimation is an optimal method that always gives consistent estimators (§4.3.2). Statistical methods may also be used for EWMA - for example. X could be chosen to minimize the root mean square forecasting error (§A.5.3) But given the problems with assessing the accuracy of volatility forecasts described in Chapter 5. statistical methods are not necessarily the best means of estimating X.

Jun-90 Jun-91 Jun-92 Jun-93 Jun-94 Jun-95 Jun-96 Jun-97 Jun-98 Figure 3.7 EWMA correlations of WTI spot and futures.

An EWMA is equivalent to an I-GARCH model, without a constant term

EWMA is equivalent to an I-GARCH model, without a constant term. In §4.2.3 it will be shown that term structure forecasts from I-GARCH models are constant. Similarly, term structure forecasts of volatility under any given EWMA model - that is, for a given choice of A, - will not converge. The current EWMA estimate of volatility is the volatility forecast for all risk horizons (§3.3).

An EWMA with a smoothing constant of 0.94 such as is used in the RiskMetrics data (§7.3) has a half-life of around 25 days. Thus it is similar to an equally weighted average of 20-30 days, and this is illustrated in Figure 3.7. This shows estimates of the same unconditional correlation parameter as that estimated in Figure 3.4 between WTI crude oil spot and futures. Figure 3.4 was based on an equally weighted moving average model, whereas Figure 3.7 is based on an EWMA model. The main difference between the estimates from these two models is evidenced following major movements in both markets at the same time: the equally weighted model will produce ghost effects of this event, as seen in Figure 3.4, whereas in the exponential model correlations will overshoot and then gradually decay.

When all is said and done, the smoothing constant is just a parameter in the estimation model. In §5.3 we shall see that single-point estimates of volatility and correlation can be very misleading if there is uncertainty surrounding the best choice for the model parameter. When an MtM value is based on volatility estimates one must be aware of the uncertainty that is introduced in the MtM value by the errors in volatility estimates. One should consider using not just one single value of X but instead some distribution over X. Then instead of a single MtM value, a whole distribution of MtM values will be obtained.5

5The same remark applies to all parameters in the valuation model, not just volatility but drifts, discount rates, betas, yields and so on.