probability p = 0.6 has volatility o", = 5% and 2( ) with probability 1 -p = 0A has volatility tj2 = 14.58%. By (10.13) the mixture density g(x) has volatility 10%, since 0.6 x 52 + 0.4 x 14.582 = 100. But the normal mixture has positive excess kurtosis, that is, it has fatter tails than <\>(x), the zero-mean normal density with volatility a - 10%. In fact by (10.14) the excess kurtosis is 2.535, much greater than zero.

Any mixture of normal densities with zero means will have leptokurtosis. That is, the tail probabilities will be larger than the tail probabilities of a normal with the same variance. This follows from (10.14) and the fact that "Epiof > (£/?,rj2)2.

The following illustration of the leptokurtosis of normal mixtures will form the basis of covariance VaR measures for normal mixture density functions in §10.3.1. Let g{x) =A<t>iW + • • • +pA„(x), where <\>{(x), . . ., „( ) are zero-mean normal densities, and let ( ) be a normal density with the same volatility as g(x). Suppose this volatility is a. Then, under ( ),

where Z is a standard normal variate.

Figure 10.7 shows the probabilities calculated by (10.15) compared with those from (10.16) for different values of between 1 and 26. We have used the same parameters as in the first example, that is p = 0.6, , = 5% and rj2 = 14.48%. Note that when is large the probabilities under the normal mixture g(x) are greater than the probabilities under the normal density ( ). For example, when = 20 the tail probabilities under g(x) and ( ) are 0.034 and 0.0228, respectively. But the probabilities nearer the centre of the distribution, for low values of c, are higher under the normal density: for example, with = 10 the probabilities under g(x) and ( ) are 0.1122 and 0.1587, respectively.

The parameters of a normal mixture density can be estimated by standard distribution-fitting methods. For example, Hull and White (1998) consider a mixture of two zero-mean normal densities with standard deviation ucj and vrj, where > 1 and v < 1 with probability p and 1 - p, respectively. For the weighted sum of the two variances to be equal to o-2 the parameters p, and v are subject to the constraint pu2 + (1 - p)v2 = 1. They are estimated by equating the theoretical and empirical densities of returns quartile by quartile.

When there are only two zero-mean normal densities in the mixture it is natural to estimate parameters by the method of moments. Examples may be generated using the normal mixture estimation spreadsheet on the CD. In the method of moments the first few sample moments are equated to the first few non-zero

Prob(X < -c) = Prob(Z < -c/tj),

(10.15)

and under g(x),

Prob(A < -c) = g(x)dx = £~Pi df>i(x)dx

(10.16)

0.35-

0.15-

0.05-

0.25

0.45-

0.4-

0.2-

0.1 -

1 2 3 4 5 6 7 8 9 1011121314151617181920212223242526 □ Normal Normal mixture

Figure 10.7 Probabilities under normal and under normal mixture distributions with the same variance.

population moments until enough equations are obtained to solve for the distribution parameters. Since there are only three parameters in a mixture of two zero-mean normal densities, only three equations are needed. For example, one might equate the second, fourth and sixth theoretical moments to their sample values based on a fixed return frequency. The second and fourth moments are easy to derive and, with a little algebra, the sixth moment can also be calculated using the moment generating function.

The method of moments may also be applied to estimate parameters of a normal mixture density function that are consistent with the observed term structures of volatility and kurtosis in very high-frequency data (§10.3.2). The three equations required for the method of moments estimation may be obtained from the second and fourth moments based on one frequency and the second or fourth moment based on another frequency (the other moment equation should be used as a check for consistency).

If more than two zero-mean normal densities are used in the mixture more sophisticated methods for estimating parameters may be suitable. Any number of normal densities, with or without zero means, could be used if a neural network is available to fit the model parameters (§14.2). Alexander and Williams (1997) find that it is optimal to use a mixture of three zero-mean normal densities to forecast excess kurtosis in foreign exchange markets. They find that the kurtosis term structure forecasts that are generated by these normal mixtures are closer to the empirical values than GARCH(1,1) kurtosis forecasts (these are described in Bollerslev, 1986). Their findings show that the kurtosis in a normal GARCH(1,1) model appears to be less than is empirically observed.

10.3 Applications of Normal-Mixture Distributions

Normality of returns assumptions underpin most pricing and hedging models because they are very convenient; but often these assumptions are not justified.

It is quite simple to change normal models into fat-tailed models when the fat-tailed densities are described by a normal mixture

as we have seen in §10.1. However, it is quite simple to change normal models into fat-tailed models when the fat-tailed densities are described by a normal mixture. This section shows how some basic pricing models and risk measures can be adapted when the underlying distribution is assumed to be generated by a mixture of zero-mean normal densities.

10.3.1 Covariance VaR Measures

Recall from §9.3 that the covariance VaRa estimate is given by Zaa where Za is the critical value of a standard normal variate and is the P&L volatility. In this model the P&L variations are assumed to be normal, and so a simple analytic formula for VaR can be derived. If we now assume that P&L variations have normal mixture distributions there is no analytic formula for VaR. However, it is a simple matter to use a numerical algorithm to obtain a normal mixture covariance VaR estimate. Readers may use the normal mixture VaR spreadsheet to do this.

When P&L is distributed as a mixture of n zero-mean normal densities the covariance VaR is based on (10.16), giving the formula

?roh(X < -c) = ]T/>,-Prob(Z < -c/tj,) = a. (10.17)

Putting in a significance level a and the probabilities and volatilities of the normal mixture, we can solve for using numerical methods. In much the same way as we back out implied volatilities in Chapter 2, we can back out the VaR number that is appropriate if P&Ls have fat-tailed distributions.

In much the same way as we back out implied volatilities in Chapter 2, we can back out the VaR number that is appropriate ifP&Ls have fat-tailed distributions

Table 10.2 compares the covariance VaR obtained when P&Ls are assumed to be normal and when they are assumed to be generated by a mixture of two zero-mean normal densities. The P&Ls are assumed to have the same volatility in each case, but in the second case they will be more leptokurtic. First a volatility for the normal mixture density is calculated, depending on the parameters p, rj, and a2, as the square root of the normal mixture variance (10.13). The normal covariance VaR for this volatility is shown. Beneath this is the normal mixture covariance VaR that is obtained by backing out from (10.17). In each case, a = 0.01.

Since normal mixtures have fatter tails than normal densities of the same volatility, these normal mixture VaR estimates are greater than normal VaR estimates. But how much greater depends on the parameters of the normal mixture. The largest difference in Table 10.2 occurs in the last column, corresponding to the case where 90 per cent of the time the volatility is 10% but 10 per cent of the time the volatility is 100%. The average volatility is therefore 33% and the normal VaR with this volatility is 76.8. However, the distribution is very fat-tailed in this case, so the normal VaR is very misleading and the normal mixture VaR of 128.16 is more representative.