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99

Modelling Non-normal Returns

Many financial assets have high-frequency returns with non-normal distributions; in particular, they often have fatter tails than the normal distribution. That is, the probability of extreme returns that are observed empirically is higher than the probability of extreme returns under the normal distribution. This feature is referred to as leptokurtosis, or simply fat tails.1 There are different approaches to modelling the non-normal characteristics of financial asset returns. One possibility is to use a stochastic volatility model, where conditional returns distributions are normal but their variance changes over time. The other approach, discussed in this chapter, is to model the unconditional distribution of returns using a non-normal density function.

Section 10.1 explains the statistical tests that are commonly employed for non-normal behaviour in financial return distributions. It is shown that non-normality tends to become more pronounced as the frequency of returns increases. Excess kurtosis in particular may be quite enormous in intra-day data. As long ago as 1963, Benoit Mandelbrot observed that financial returns may have distributions that are not normal.2 Section 10.2 describes some of the non-normal distributions that are commonly used in financial analysis A brief account of extreme value distributions (§10.2.1) and hyperbolic distributions (§10.2.2) is followed by an extensive discussion of the normal mixture distributions (§10.2.3).

As long ago as 1963, Benoit Mandelbrot observed that financial returns may have distributions that are not normal

Normal mixture distributions are a particular favourite of mine because many simple models based on normal returns may be modified quite easily to accommodate normal mixture density assumptions. Much of this chapter is focused on normal mixture density functions and their applications in finance. Covariance VaR measures that are based on normal mixture models for fat-tailed distributions are introduced in §10.3.1. The huge excess kurtosis in intra-day returns can present a serious problem for the pricing and hedging of

1 In fact leptokurtic literally means thin-arched or thin-centred. Fat-tailed distributions have thin centres (see Figure 10.1). Literally translated to the Greek, fat-tailed would be platyeschatic.

2 In fact Mandelbrot (1963) thought that distributions of financial asset returns would be similar to stable Pareto distributions. Stable distributions are so called because a sum of stable random variables is again a stable random variable. Thus the normal distribution is a special case of a stable distribution. Stable Pareto distributions are so fat-tailed that their kurtosis is infinite, and this implies that kurtosis estimates should increase with sample size; this hypothesis is not substantiated by empirical findings of kurtosis estimates that converge as the sample size increases. Kurtosis also tends to increase with sampling frequency, so the use of stable Pareto distributions has gone somewhat out of fashion for financial models, but more details may be found in Campbell et al. (1997).



Uncertainty in volatility can explain the volatility smile. We shall use normal mixture densities to model the uncertainty in volatility

options portfolios, and §10.3.2 shows how normal mixture densities can be applied to generate term structures of kurtosis for very high-frequency data.

At the end of Part I, in Chapter 5, we concluded that it is very difficult indeed to forecast volatility. Therefore, rather than asking how accurate is my volatility forecast?, we started to consider a different question: how much does it matter if my volatility forecast is wrong?. In §5.3 it was shown that uncertainty in volatility can have a big effect on option prices; in fact it can explain why OTM options have higher market prices than constant volatility model option prices. That is, uncertainty in volatility can explain the volatility smile. At the end of this chapter, in §10.3.3, we shall return again to this discussion, this time to use normal mixture densities to model the uncertainty in volatility. The option price bias for simple OTM options that was explained, using Taylor expansion, in (5.7) is made concrete with an empirical model based on normal mixture densities. The effect of uncertain volatility on delta hedged portfolios that was discussed in §5.3.2 will also be revisited.

10.1 Testing for Non-normality in Returns Distributions 10.1.1 Skewness and Excess Kurtosis

The skewness x is the standardized third moment of the distribution, and the kurtosis is the standardized fourth moment. Since the normal distribution has a kurtosis of 3 it is usual to subtract 3 from the kurtosis, so that both skewness and excess kurtosis will be zero for a normal distribution:

= [( -»)4]/ 4-3.

(10.1) (10.2)

Precise standard errors for skewness and excess kurtosis are difficult to compute without the assumption of normality

Positive excess kurtosis indicates more weight in both tails of the distribution than in the normal distribution. Hence the term fat-tailed for distributions with positive excess kurtosis, also called leptokurtic distributions, but either way it sounds as if they have some sort of unfortunate health problem.

If the underlying population is normal then standard errors for the sample estimates x and are approximately y/(6/n) and /(24/ ), where n is the sample size. That is, approximate standard errors for kurtosis estimates are twice the size of the standard errors of the skewness estimates. Precise standard errors for skewness and excess kurtosis are difficult to compute without the assumption of normality. Even a standard error for a variance estimate will involve computing the fourth central moment when the distribution is not normal.3

3Thc variance of the sample mean is always 2/«, where n is the sample size and 2 is the population variance; one does not need to assume normality. The variance of the sample variance s2 is 2 4/«, when the population is normal, but more generally it is (u4 - rj4)/n where 4 = £((.v - u)4) is the fourth moment about the mean. The computation of standard errors for sample estimates of (10.1) and (10.2) involves a complicated matrix product of central moments when the population is not normal (see Greene, 1993).



Table 10.1: Normality statistics for the return on the DEM-USD exchange rate

1-hour

6-hour

12-hour

1-day

1 -week

Skewness

0.289

0.233

0.200

0.090

0.340

Approx. s.e.

0.031

0.076

0.107

0.152

0.339

Excess kurtosis

8.34

3.39

1.54

0.63

0.22

Approx. s.e.

0.062

0.152

0.214

0.303

0.678

18245

54.77

4.62

1.11

The Jarque-Bera normality test is a form of Wald test (§A.2.5) where the null hypothesis is that the data are normal. The JB test statistic is defined in terms of sample estimates of skewness and excess kurtosis (x and ic) based on a sample size n:

JB = [( 2/6) + ( 2/24)] (10.3)

and it is asymptotically chi-squared with 2 degrees of freedom.

Table 10.1 reports the skewness and kurtosis estimates and Jarque-Bera statistics based on a year of tic data on the DEM-USD exchange rate that is bucketed into different sampling intervals. The number of observations taken to compute these statistics varies from 6264 for the hourly data down to 52 for the weekly data, so obviously their standard errors increase as the sampling frequency decreases.

Comparison of the sample estimates with their approximate standard errors (under the null hypothesis of normality) shows that there is evidence of skewness in the hourly and 6-hourly data, but not at any lower frequency. There is no evidence of leptokurtosis in the weekly data and not much in the daily data, but there is enormous excess kurtosis at the intra-day frequencies. The Jarque-Bera tests confirm this4 - the JB statistics are significantly different from zero on the intra-day data only, and they indicate that

> the hypothesis of a normal distribution can be maintained for the daily and weekly returns (but there are very few observations so the power of the test is rather low);

> intra-day returns distributions have excess kurtosis which increases with sampling frequency.

In most liquid financial markets there is highly significant excess kurtosis in intra-day returns, which increases with sampling frequency. This is one of the stylized facts of high-frequency financial returns, which is particularly pronounced in foreign exchange markets. However, the skew in high-frequency exchange rate returns is not as pronounced as it is in high-frequency equity returns (see Hseih, 1988; Baillie and Bollerslev, 1989b; Muller et al., 1990).

In most liquid financial markets there is highly significant excess kurtosis in intra-day returns, which increases with sampling frequency

4The 1% critical value of y}(2) is 9.21 and the 5% critical value of x2(2) is 5.99.



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