Table 10.2: Normal VaR and normal mixture VaR

VaR is used in daily risk management to calculate capital reserves, to set trading limits and to assess the riskiness of portfolios under different scenarios (§9.1.3, §9.2.2 and §9.6). It may seem acceptable that the assumption of normality be used for the normal market circumstances VaR estimates that form the basis of market risk capital requirements, because excess kurtosis is usually small over a 10-day horizon. This is attractive, because the market risk requirement will always be lower under the assumption of normality than under the assumption of fat tails. Nevertheless normal mixture VaR estimates can be very useful for internal risk management purposes, particularly in the scenario analysis and stress testing that form the basis of limit setting, or to calculate daily VaR.

10.3.2 Term Structure Forecasts of Excess Kurtosis

As the sampling interval shortens, estimates and forecasts of both kurtosis and volatility tend to increase. We have already seen that leptokurtosis can be very pronounced in intra-day returns and intra-day volatility is well known to exceed volatility measured at daily intervals (§4.2.5, §13.1.3). However, excess kurtosis and increased volatility do not have to go hand in hand. In fact this section will show that there can be excess kurtosis in very short-term forecasts of returns even when volatility term structures are constant.

Term structure forecasts for volatility can be converted into term structure forecasts for kurtosis

The main cause of the term structure of kurtosis is nothing to do with a volatility term structure. It is a consequence of the central limit theorem.18 If very high-frequency (log) returns over one period are denoted X, at time / then the sum ST - J2j=i %t is a -period return. If 1-period returns are i.i.d., then the kurtosis of -period returns will always be less than the kurtosis of 1-period returns; in fact the excess kurtosis of ST is / , where is the excess kurtosis of X,

If returns are fat-tailed because they are generated by a normal mixture density, term structure forecasts for volatility can be converted into term

18 The central limit theorem implies that the sum of non-normal variables tends towards a normal variable: if , have i.i.d. distributions with mean 0, variance a2 and excess kurtosis then Y - {X{ + ... + X„) has a distribution with mean 0, variance no2 and kurtosis 3 + / , so the kurtosis approaches 3 and the excess kurtosis approaches zero as n increases.

This type of term structure could arise if there are two types of traders in the market. Type 1 traders have expectations of volatility given by the term structure vol\ and type 2 traders are trading on the basis of vol2

19 Note the distinction between this assumption and that made in the next subsection. Each market participant is certain about their own view of volatility, although different participants have different views. In the next subsection we shall consider a different case, where each individual is uncertain about their own view of volatility.

structure forecasts for kurtosis. Formula (10.15) has been used to construct the term structure forecast of kurtosis shown in Figure 10.8. It shows a market volatility term structure of a type that is not unusual in financial markets. It is relatively flat: average volatility over the next 200 hours is not much different from the average volatility over the next few hours, indicating that the current period is not particularly volatile. This type of term structure could arise if there are two types of traders in the market. Type 1 traders have expectations of volatility given by the term structure vol/ and type 2 traders are trading on the basis of vol2. There is not much difference between their volatility expectations over the next one or two weeks, but over the course of the next day or so there are substantial differences. Type 1 traders expect volatility to be around 17% in the next few hours and to decline over the next week or two to average about 12%. Type 2 traders think that volatility will be around 8% in the next few hours, although the weekly average will be approaching 10%.

Suppose that traders are using zero-mean normal distributions for the underlying, and 20% of the traders are of type 1 and 80% are of type 2. Then the volatility that is observed in the market can be assumed to be generated by a mixture of two zero-mean normal densities, with a probability of 0.2 on vol, and a probability of 0.8 on vol2. The market volatility term structure in the figure has been constructed by taking vT0.2(vol,)2 + 0.8(vol2)2] at every point in the term structure, as in (10.13).

The numbers in this example have been chosen in such a way that the combined expectations of volatility in the market are more or less constant. The volatility forecast is a little more than 10% for all time horizons. That is, the average volatility over the next hour, over the next week, or over more than a week is forecast to be 10%. At the same time the term structure of excess kurtosis, computed using (10.14), has been plotted on the right-hand scale. Note that excess kurtosis is around 2, much greater than 0 for the very short-term forecasts, but beyond the horizon of a few hours there is no leptokurtosis to speak of. Thus kurtosis forecasts can be very high over the next hour or so, but kurtosis term structures decrease rapidly to 0.

This subsection has shown how normal mixtures can provide a behavioural model for the rapid decline in kurtosis as the interval of time increases. Different market participants have similar expectations of volatility over longer-term horizons, although their views on volatility over the next few hours may be quite different.19 The normal mixture model used to illustrate this point in Figure 10.7 had just two types of traders in the market. If there were many different agents, with substantially different views about volatility over the next few hours, the very short-term kurtosis forecasts might be even higher.

- 1.5

193 Hours

-VoL ---Vol-, - - Market vol

Mixture kurtosis

Figure 10.8 Volatility and excess kurtosis term structure forecasts.

10.3.3 Applications of Normal Mixtures to Option Pricing and Hedging

When trying to price and hedge an options book on a daily or intra-day basis, excess kurtosis will be present in the underlying market returns. However, excess kurtosis disappears when returns are measured over more than a few days (§13.1) and so has no effect on the pricing of options with maturity more than a few days. Normal mixture densities that capture the observed daily excess kurtosis may be used to price very short-term options; an example of this is given in this subsection.

Normal mixture densities may also be used to price longer-term options, and the methodology is similar to that used for very short-term options. However, the behavioural rationale is not the same. For very short-term options, where price will be affected by excess kurtosis, the behavioural model is one where different market participants have different views on volatility, but individually they are certain about their own view, as in §10.3.2. For longer-term options the excess kurtosis is negligible and will not, therefore, affect the price. What does, however, affect the price of the option is uncertainty that one individual has in his view of volatility. An option pricer may be very uncertain about volatility over a period of a few months or more; it is very difficult to predict volatility, as we have seen in Chapter 5.20 Black-Scholes option prices will therefore need to be adjusted to account for uncertainty in volatility; in fact we have already seen that uncertainty in volatility is one of the reasons why the Black-Scholes

20 In fact, experience shows that a 30% relative error in a 1-month volatility forecast is not uncommon: this is the shift figure recommended by Capital Adequacy Directive regulators to take into account volatility risk in scenario-based models (§9.6.1).