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58

Figure 6.7 Non-parallel shift in skew deviations as price moves up: (a) Yi > 0, y2 < 0-

y3 > 0; (b) yi > 0, y2 > 0, y3 > 0; ( ) y, > 0, y2 < 0, y3 < 0; (d) y, > 0, y2 > 0, y3 < 0.

coefficient y2 determines the tilt of the fixed strike deviations and y3 determines the convexity, so the four combinations shown represent stylized movements in the skew deviations.

Later on, estimates of the gamma coefficients will be used to generate empirical sensitivities for fixed strike deviations with respect to the index. However, for the present qualitative analysis, the important observation is that the sign of y2 will determine whether the range of the skew narrows or widens when the index moves up. When the range narrows, most of the movement will come from low strike volatilities. However, when the range widens, much of the movement also comes from the high strike volatilities.

To see this, consider the effect on fixed strike deviations from ATM volatility as the index moves up, depicted in Figure 6.7. A result of the upward movement in the underlying is that one of the high strike deviations, at strike

The sign o/y2 will determine whether the range of the skew narrows or widens when the index moves up. When the range narrows, most of the movement will come from low strike volatilities. However, when the range widens, much of the movement also comes from the high strike volatilities



aH = a, - c/H °H-"2-eH

"1

°

°L = °1 +

aH = a! - C/H

oH = o2-eH

Figure 6.8 Effect on fixed strike volatilities as price moves up: (a) y2 < 0; (b) y2 > 0.

K2 say, will change from a negative value to a value of zero because the ATM strike has moved from K{ to K2. Strikes above K2 will still have volatilities that are lower than the ATM volatility, strikes between Kt and K2 now have volatilities that are above ATM volatility, and strikes below K\ will still have volatilities above the ATM volatility. For the lowest strikes there will be little change: their volatility deviation from the new ATM volatility is about the same as it was before the movement in the underlying.

When Y2 is negative, as in Figures 6.7a and 6.7c, the range of the skew will narrow as the index moves up, and the deviation from ATM volatility of high strike options will decrease. On the figures the deviation at the high strike KH is denoted dH before the move and eH after the move, and when y2 is negative it is clear that eH < dH. This translates in Figure 6.8a to a narrowing of the range of the skew as the index moves up (and a widening as the index moves down), with most of the movement coming from low strike volatilities.

When Y2 is positive, as in Figures 6.7b and 6.7d, the range of the skew will widen as the index moves up, and the deviation from ATM volatility of low strike options will increase. On the figures the deviation at the high strike Kb is denoted dL before the move and eL after the move, and when y2 is positive it is clear that eL > dL. This translates in Figure 6.8b to a widening of the range of the skew as the index moves up (and a narrowing as the index moves down), with much the movement coming from high strike volatilities.

Let us now make some estimates of the gamma response coefficients for the principal components of the FTSE 100 skew deviations. The principal components have zero unconditional covariance; however, by (6.5) their conditional covariance is

cov,(P,-.„ Pj,) = jujjjV,,



where aj is the conditional variance of the index, V,(ASt). The time-varying gamma parameters are estimated using an exponentially weighted moving average model as an approximation to a bivariate GARCH/l, l).10 Thus

Y„ = ,( ,„ ASt)/Vt{ASt),

where the covariance and variance are estimated using an EWMA, as described in §3.2. For the sake of conformity with standard covariance calculations such as those in JP Morgan/Reuters RiskMetrics (§7.3) the smoothing constant X = 0.94 has been used.

EWMA estimates (with X - 0.94) of y,, y2 and y3 for each of the 1-, 2- and 3-month maturities have been calculated, and these are shown in Figure 6.9. The first point to note about all the graphs is that the estimate of y, is positive throughout. It is higher and more stable than the estimates of y2 and y3, and for the 1- and 2-month maturities it is generally declining over the sample period.

For the 2-month maturity the gamma estimates are shown in Figure 6.9b. The index seems to have little effect on the second and third principal components, in fact the estimates of y2 and y3 are close to zero for almost all the sample period. Therefore, it would be reasonable to apply parallel shift scenarios for fixed strike volatilities of 2-month maturities. In fact it is quite clear in Figure 6.10 that the skew deviations were moving parallel throughout the period, and so the fixed strike volatilities themselves were also moving parallel.

However, the 1- and 3-month volatilities shown in Figures 6.9a and 6.9c show that the estimate of y2 is often negative, particularly during the spring of 1998 and the spring of 1999. This is the range-narrowing regime defined above where most of the movement will be coming from the low strike volatilities. However, in the 1-month volatilities there are two notable periods, just before the beginning of the crash and during the market recovery, when the estimate of y2 was positive. On 14 July 1998, several days before the FTSE 100 price started to plummet, there was a dramatic increase in y2 and decrease in y3 so that y2 > 0 and y3 < 0. During this period the range of the 1-month skew will have narrowed, as the index fell. Then between 8 and 12 October 1998, the FTSE 100 jumped up 8% in 2 days trading, from 4803 to 5190. At the same time y2 jumped up and y3 jumped down, so that again y2 > 0 and y3 < 0, and the range of the 1-month skew will have widened as the index moved up.

The analysis of Figure 6.8b reveals that the narrowing of the range of the skew as the index fell, and the consequent widening again as the market recovered, would have been driven by movements in high strike volatilities. During this unusual period the high strike volatilities did indeed move much more than usual, as was evident in Figure 2.6.

It would be reasonable to apply parallel shift scenarios for fixed strike volatilities of 2-month maturities

,0This choice allows one to bypass the issue of parameterization of the bivariate GARCH which is a difficult issue in its own right (§4.5.2). It does of course introduce another issue, and that is which smoothing constant should be chosen for the exponentially weighted moving averages.



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