volatility d can be compared with the current implied volatility level a, and then a trading strategy can be defined that depends on their difference.

The choice of strategy is an entirely subjective decision. It depends on how one proposes to implement the trades. For example, a simple trading strategy for an implied volatility forecast d that relates to a single threshold x, might be: buy one at-the-money straddle if d - a > x; otherwise do nothing. An alternative volatility strategy could be: buy one ATM straddle if d - a > x,; sell one ATM straddle if d - a < x2; otherwise do nothing. Or the strategy may go long or short several straddles, depending on various thresholds: buy nx straddles if d - > ,; sell w2 straddles if d - a<x2; otherwise do nothing.

The P&L results from this strategy will depend on many choices: the number of trades nx and w2, the thresholds xt and x2, the frequency of trades, the strike of the straddles and, of course, the underlying market conditions during the test, including the current level of implied volatility a. Clearly an evaluation strategy that is closest to the proposed trading strategy needs to be designed and the trader should be aware that the optimal forecasting model may very much depend on the design of the evaluation strategy.

When volatility and correlation forecasts are used for risk management the operational evaluation of volatility and correlation forecasts can be based on a standard risk measure such as value-at-risk. The general framework for backtesting of VaR models will be discussed in §9.5. But if the model performs poorly it may be for several reasons, such as non-normality in return distributions, and not just the inaccuracy of the volatility and correlation forecasts.

Alexander and Leigh (1997) perform a statistical evaluation of the three types of statistical volatility forecasts that are in standard use: historical (equally weighted moving averages), EWMAs and GARCH. Given the remarks just made, it is impossible to draw any firm conclusions about the relative effectiveness of any volatility forecasting method for an arbitrary portfolio. However, using data from the major equity indices and foreign exchange rates, some broad conclusions do appear. While EWMA methods perform well for predicting the centre of a normal distribution, the VaR model backtests indicate that GARCH and equally weighted moving average methods are more accurate for the tails prediction required by VaR models. These results seem relatively independent of the data period use.d.

GARCH forecasts are designed to capture the fat tails in return distributions, so VaR measures from GARCH models tend to be larger than those that assume normality. The ghost features of equally weighted averages that follow exceptional market moves have a similar effect on the historical VaR measures. Therefore it is to be expected that these two types of forecasts generate larger VaR measures for most data periods, and consequently better VaR backtesting results (§9.5.1).

5.2 Confidence Intervals for Volatility Forecasts

The standard error of the volatility forecast is not the square root of the standard error of the variance forecast

We have examined the ability of point forecasts of volatility to capture the constant volatility of a price process. These point forecasts are the expectation of the future volatility estimator distribution. Another important quality for volatility forecasts is that they have low standard errors. That is, there is relatively little uncertainty surrounding the forecast or, to put it another way, one has a high degree of confidence that the forecast is close to the true process volatility. In §A.5.1 it is shown how standard errors of statistical regression forecasts are used to generate confidence intervals for the true value of the underlying parameter. These principles may also be applied to create confidence intervals for the true volatility, or for the true variance if that is the underlying parameter of interest.

The statistical models described in chapters 3 and 4 are variance forecasting models.10 When the variance is forecast, the standard error of the forecast refers to the variance rather than to the volatility. Of course, the standard error of the volatility forecast is not the square root of the standard error of the variance forecast, but there is a simple transformation between the two. Since volatility is the square root of the variance, the density function of volatility is

g(a) = 2ah(a2)

for a > 0,

(5.2)

where h(a2) is the density function of variance." Relationship (5.2) may be used to transform results about predictions of variances to predictions of volatility.

5.2.1 Moving Average Models

A confidence interval for the variance a2 estimated by an equally weighted average may be obtained by a straightforward application of sampling theory. If a variance estimate is based on n normally distributed returns with an assumed mean of zero, then 2/ 2 has a chi-squared distribution with n degrees of freedom.12 From §A.2, a 100(1 - oc)% two-sided confidence interval for na2/a2 would therefore take the form (x2\ , X2 / ) ard a straightforward calculation gives the associated confidence interval for the variance ct2 as:

(n°2/xl,a/2, na2/xla/2)- (5-3)

For example a 95% confidence interval for an equally weighted variance forecast based on 30 observations is obtained using the upper and lower chi-squared critical values:

10The volatility forecast is taken to be the square root of the variance forecast, even though ) / ( 2).

"if v is a (monotonic and differentiable) function of then their probability densities #() and h(-) are related by g{yj= \dx/dy\h(x). So if .i- = „/.v. k/v < - = 2v.

,2The usual degrees-of-freedom correction does not apply since we have assumed throughout that returns have zero mean.

30 025 = 46.979 and . .975 = 16.791.

So the confidence interval is (0.6386a2, 1.7867a2) and exact values are obtained by substituting in the value of the variance estimate.

Assuming normality,13 the standard error of an equally weighted average variance estimator based on n (zero mean) squared returns is [*J(2/n)]a2. Therefore, as a percentage of the variance, the standard error of the variance estimator is 20% when 50 observations are used in the estimate, and 10% when 200 observations are used in the estimate.

The (infinite) EWMA variance estimator given by (3.3) has variance14

A-X

l + X

Therefore, as a percentage of the variance, the standard error of the EWMA variance estimator is about 5% when X = 0.95, 10.5% when X = 0.9, and 16.2% when X = 0.85.

To obtain error bounds for the corresponding volatility estimates, it is of course not appropriate to take the square root of the error bounds for the variance estimate. However, it can be shown that15

V(&2) % (2a)2 F(d).

The standard error of the volatility estimator (as a percentage of volatility) is therefore approximately one-half the size of the variance standard error (as a percentage of the variance).16 As a percentage of the volatility, the standard error of the equally weighted volatility estimator is approximately 10% when 50 observations are used in the estimate, and 5% when 200 observations are used in the estimate; the standard error of the EWMA volatility estimator is about 2.5% when X = 0.95, 5.3% when X = 0.9, and 8.1% when X = 0.85.

The standard error of the volatility estimator (as a percentage of volatility) is therefore approximately one-half the size of the standard error of the variance estimator (as a percentage of the variance)

The standard errors on equally weighted moving average volatility estimates become very large when only a few observations are used. This is one reason

,3It follows from footnote 11 that if X, are independent random variables (/- 1, . . ., n) then f(X,) are also independent for any monotonic differentiable function/(). Moving average models already assume that returns are i.i.d. Now assuming normality too, so that the returns are NID(0, ct2), we apply the variance operator to r}2 = "=] r] Jn. Since the squared returns are independent V(u2) = Vl=l ( 2 ,.)/ 2. Now V(r]) = E(rf) - [ ( 2)] = 3a4 - a4 = 2a4 by normality, and it follows that the variance of the equally weighted moving average variance estimator is 2< 4/ .

"Applying the variance operator to (3.3): Via1) = [(1 - X)2/(l - X2)]V(r2 ) = [(1 - X)/(\ + )]2 *.

,5Taking a second-order Taylor expansion of fix) about u, the mean of X, and taking expectations gives E(f(X))*f(v.) + { ( )/ Similarly, E(f(Xf) ~/(u)2 + V(X)[f(n)2 - )/"( )1, again ignoring higher-order terms. Thus V(f(X)) »f(\i)2V(X).

,6For the equally weighted average of length n, the variance of the volatility estimator is (2a4/«)(l/2a)" = a2/(2/j). so the standard error of the volatility estimator as a percentage of volatility is 1 /V(2n); For the EWMA. the variance of the volatility estimator is (2[(1 - X)/(\ + )] 4)(1 /2 )2 = [(1 - X)/(l + )] 2/2, so the standard error of the volatility estimator as a percentage of volatility is /[(1 - X)/2(\ + X)].