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18

shown how these correlations may depend on the current market regime. A joint distribution over all possible price and ATM volatility changes is generated, so that probabilistic scenario analysis methods can be employed (§9.6.2). The principal component model that is described in §6.3 will allow these scenarios to be extended to scenarios for the whole smile or skew.

In equity index options there is sometimes a clear negative correlation between ATM volatility and the underlying price, but at other times this correlation can seem very weak. Figure 2.8 shows, for three different 2-month periods during 1998, a scatter plot of the daily changes in 1-month ATM volatility against daily changes in index price for the FTSE 100 European option. The periods chosen were: (a) February and March 1998; (b) May and June 1998; and (c) August and September 1998.

Casual observation of these scatter plots indicates a significant negative correlation between the 1-month implied volatility and the index price, but the strength of this correlation depends on the data period. Period (a), when the UK equity market was very stable and trending, shows less correlation than period (b), when daily movements in the FTSE 100 index were limited to a normal range; but the negative correlation is most obvious during the mini-crash period (c) that followed the Russian crisis in July 1998. These observations are not peculiar to the 1-month ATM FTSE 100 volatilities, and not just during the periods shown: negative correlations, of more or less strength depending on the data period, are also evident in other fixed term ATM volatilities and in other equity markets.

Realistic scenarios for ATM volatility and index prices would, therefore, be for movements in ATM volatility to occur in the opposite direction to the index price movements. The volatility regime models of §2.3.1 imply that the size of the relative movements will depend on current market regime. In stable trending markets ATM volatility changes should be independent of changes in the index; in range-bounded markets there should be a negative correlation between ATM volatility changes and index price changes; and the magnitude of this correlation will double when the market enters a jumpy regime.

The following simple model for the joint density of daily price changes and daily ATM implied volatility changes, denoted ProbtAS and ), can be used to assess the current market conditions and quantify the correlation effect. By the theorem of conditional probability,

Prob(A5 and Act) = Prob(Acr AS) Prob(AS).

Now suppose that price changes are normally distributed,

AS ~ N(ii, a2), (2.6)

and that the conditional distribution ProbtAcr AS) is given by the linear model

= a + (5A5 + e,

(2.7)



-:

-200

-100

6-4-

.2*~

-2< -4--6 -8-G-1

.100

-4 -

-200

-200

-100

-2! -4 -6 -8--40-

•tJDO

-200

-40-8 6 4 "2

-100

-2 -4 -6 -8 -4

Figure 2.8 At-the-money volatility versus FTSE 100 (daily changes): (a) February a March 1998; (b) May and June 1998; (c) August and September 1998.



where e ~ N(0, <r2). Then AS ~ N(a. + (5A5, a2) and the joint density will be bivariate normal, that is, the product of two normal densities N(a + (5A5, a\) and 7Y(u, cr2).

Estimation of the parameters a, (5, 2, and a2 could be done using any type of distribution fitting method. A simple and transparent method is to estimate all parameters using equally weighted historic data over a prespecified time period. Thus the parameters in (2.6) are estimated by taking the sample mean and variance of all the daily price changes during the recent past. The parameter estimates in (2.7) can be obtained by ordinary least squares regression of Acs on AS over the same data window. It is important not to use too much historic data for the parameter estimation, so that the parameter estimates will capture only the most recent market behaviour. However, obviously there is a trade-off between accuracy in the estimates and their ability to isolate the current market regime.

Figure 2.9 shows a joint density for 31 July 1998 that has been constructed using this method. The parameter estimates in Table 2.3 were obtained using two months of daily data.

The data period in Table 2.3 corresponds to a relatively stable market, and this is evident from the joint density shown in Figure 2.9. There is a significantly negative but relatively small correlation between the changes of

Figure 2.9 Joint density for price and volatility, 31 July 1998.



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