h(x) = (2 8))~] exp(-a(8z + x1)11).

2W24

(10.9)

The general hyperbolic density function has four parameters, two that determine its shape and two that determine scale and location.11

Applications of hyperbolic distributions are more wide-ranging than those of extreme value distributions. An empirical analysis of DAX stocks in Eberlein and Keller (1995) illustrates how well hyperbolic densities capture the skew and fat-tailed observed distributions of stock returns. If stock returns are assumed to be generated by hyperbolic distributions, then option pricing models should be based on hyperbolic Levy motion rather than standard Brownian motion.12 Eberlein et al. (1998) use the hyperbolic distribution to explain volatility smiles and the mispricing of options by Black-Scholes type models. They conclude that, compared with prices derived from the hyperbolic model, Black-Scholes prices are much too high for ATM options and too low for OTM and ITM options. This should be compared with the conclusions that we shall draw in §10.3.3, where Black-Scholes and normal mixture options prices are approximately the same for ATM options. In fact, we shall conclude that the Black-Scholes model only misprices OTM and ITM options.

Applications of hyperbolic distributions are more wide-ranging than those of extreme value distributions

10.2.3 Normal Mixture Distributions

Financial markets are characterized by jumps that often precede periods of a normal mixture density high volatility. Thus a mixture of two normal densities, one with a low is a probability-weighted volatility and the other with a high volatility, has an intuitive interpretation in sum of normal density financial markets. A normal mixture distribution can be thought of as a model functions for the unconditional returns distribution when the probability of the high-volatility normal in the mixture is the probability of a jump in the market.

Another intuitive reason for modelling high-frequency financial returns with normal mixture densities is that different agents may have different expectations of very short-term volatility, although their long-term expectations are more likely to concur. This model is examined in §10.3.2 when it is used to motivate the term structure of kurtosis in very high-frequency data.

Of course an excellent empirical reason for using normal mixtures instead of normal distributions for high-frequency returns is that they are very leptokurtic. Even simple normal mixtures, for example a mixture of just two zero-mean normal densities, will fit this type of financial data much better than a normal distribution would. A normal mixture density is a probability-weighted sum of normal density functions.13 For example, a mixture of two

11 These parameters may be estimated by maximum likelihood, as described in Eberlein and Keller (1995). l2The hyperbolic Levy process is purely discontinuous. That is, it changes its value only by jumps. 13 Note the probability weighted sum is of the density functions, not of the normal variates.

normal densities 1 ) = ( ; .,, 2) and 2( ) - ( ; 2, 2) is the density function14

zero means the variance of the mixture is just the probability-weighted sum of the individual variances

g(x) = ,( ) + (1 -/>) 2( ).

Note that there is only one random variable so it would be misleading to call the densities , and 2 independent. More generally, a mixture of n normal densities ,( ), . . ., „( ) is the density

g(x) = px ! (X) + . . . + P„<$>„(X),

(10.10)

where + ... + p„ - 1. The mean of a mixture distribution is just the average of the individual means,

(10.11)

where £,(x) - J ,( ) / .15 Similarly, the variance of the mixture consists of two parts, the average of the variances and the variance of the means - that is,16

= £ , +{£ - 2- >, j. (io.i2)

The means and variances of the individual normal densities determine the shape and scale of the normal mixture, as illustrated in Figure 10.6. In Figure 10.6a the mixture of two normal densities with zero mean but different variances produces a density with zero skew but a fat tail. The normal distribution ) has standard deviation 3 and the normal distribution 2( ) has standard deviation 2, so using (10.14) below the mixture density has excess kurtosis 3.53. Figure 10.6b shows a mixture of two normal densities with different means as well as variances. The mixture has a bimodal density with positive skew (more weight in the upper tail) and negative excess kurtosis, so it is thin-tailed.

It facilitates the analysis greatly if one assumes that £,(x) = 0 for every /. Nonzero means are only important for low-frequency returns, and low-frequency returns do not display significant departures from non-normality, in general. The use of normal mixture densities is most relevant for high-frequency returns, and therefore it is not very constraining to assume zero means.

In this case the second term on the right hand side of (10.12) is zero and then the variance of the mixture is just the probability-weighted sum of the individual variances. So the normal mixture density g(x) has volatility given by:

(10.13)

14That is g(x) =p[(2)-"2 exp(-(.v - u, )2/2a2)] + (1 - p) [(2. "! exp(-(* - 2)2/2 2)]. 5Eg(x) = \xg(x)dx = Y.p, / - -.

16 Vs(x) = lx2g(x)dx - (Jxg(x)dx)2. The first term is Hp,Vi(x) + TZp,E,(x)2 and the second term is £ : ,( )) .

0.22

- ) -- 2( ) - -Mixture (p= 0.5)

- ) - 2( ) - -Mixture (p = 0.5)

Figure 10.6 Mixture of two normal densities: (a) with zero mean; (b) with different means.

where rj, is the volatility of ,(*)-

Mixtures of zero-mean normal densities will have no skewness. However, it can be shown17 that the excess kurtosis of a zero-mean normal mixture density is

£*°?/{£a°?}2)-i

(10.14)

where pt is the weight on the zero-mean normal ,(*) with volatility rj,.

A simple example shows how a normal mixture can have positive excess kurtosis - that is, fatter tails than the normal density with the same volatility. Consider a mixture g(x) of two zero-mean normal densities: ]( ) with

17 This can be derived directly ( = EplEl(x4)/(EplEi(x2))2 - 3) or using the moment generating function of a mixture of n zero-mean normal densities; this is , exp(j(2ct2). Expanding this as a power series in (. the rth moment is the coefficient of f jr\.