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60

was positive during the pre- and post-crash periods, the range of the skew actually narrows as the index moves down and widens as the index moves up. Both cases are characterized by movements the volatilities of all strikes, including the high strikes, which in normal circumstances do not move as much as the low strike volatilities.

6.3.3 Parameterization of the Volatility Surface and Quantification of da/dS

The principal component model of fixed strike implied volatilities has, so far, been based on their deviations from ATM volatility. Thus it is not complete without a model for the behaviour of ATM volatility as the index moves. Following (2.6), we assume for a fixed maturity of 1, 2 or 3 months that

° atm = oc + PAS + £.

(6.6)

To give some idea of the orders of magnitudes of oc and (3, a simple ordinary least squares (OLS) regression over the whole period gives the following estimates (with /-statistics in parentheses):

Maturity (months)

0.044 (0.64)

-0.019 (-

-22.14)

0.036 (0.68)

-0.015 (-

-23.47)

0.032 (0.65)

-0.013 (-

-21.74)

Though larger than the estimates of (3, the estimates of oc are very insignificant, so it seems reasonable to assume that oc is zero. However, the estimates of (3 are all highly significant. If we are to capture the dependence of ATM volatility sensitivity on the current market conditions it is, of course, best to estimate a time-varying parameter (3, instead of a constant parameter. A sensible approach, which complements the estimation of the gamma response coefficients for the principal components in (6.5), is to estimate (3, with an EWMA (again with X = 0.94). Hence

(3, = cov,(AaATM,(, AS,)/V,(AS,).

These estimates are shown in Figure 6.12. As expected, the sensitivity of ATM volatility to changes in the FTSE is greater in 1-month than in 2-month options, which in turn have greater sensitivity than 3-month options. There is a striking pattern in Figure 6.12a, that is verified in Figure 6.12b when the level of the FTSE is superimposed on these sensitivities. It is very clear indeed that the sensitivity of ATM volatility moves with the level of the index. It does not jump unless the index jumps.

is very clear indeed that the sensitivity of ATM volatility moves with the level of the index. It does not jump unless the index jumps

In the Derman models presented in §2.3.1, ATM volatility sensitivity jumped between three levels, according to the current market regime. This assumption led to a linear parameterization of the volatility surface:



Jan-98 Mar-98 May-98 Jul-98 Sep-98 Nov-98 Jan-99 Mar-99

- 1 month -2 month - 3 month

-0.035

Jan-98 Mar-98 May-98 Jul-98 Sep-98 Nov-98 Jan-99 Mar-99 -1 month -2 month -3 month -FTSE100[

Figure 6.12 (a) ATM volatility sensitivity to the FTSE 100 index; (b) with the level of the index superimposed.

a(S, t) = b(t)S + c(t),

where the coefficient b(t) jumps between three different levels (0, b and -b) as the market moves between different regimes. However Figure 6.12 suggests that the ATM volatility sensitivity changes over time because the level of the index changes over time. It seems more reasonable to suppose, therefore, that

da/dS= 2a(t)S + b(t),

where a(t) < 0, and a(t) and b{t) may be calibrated from the estimates of 3, for different maturity options. Now this gives a quadratic parameterization of the volatility surface:

a(S, t) = a(t)S2 + b{t)S+c{t).



The finding of time-varying ATM volatility sensitivity that depends on the index level is therefore equivalent to a second-order Taylor approximation to the volatility surface.

The discrete-time framework of PCA also allows one to measure the da/dS term in (2.10) for the delta hedging of options with different strikes at any point in time, when the volatility surface is not constant. In fact, combining (6.4), (6.5) and (6.6) yields a model that shows how to change all fixed strike volatilities as the index moves:

*,, * VKAst>

where

PJC,, = P,+ SwJU7,,l. (6.7)

The fixed strike volatility sensitivities to the underlying moves can therefore be obtained as a simple sum of the coefficient estimates in the models (6.4), (6.5) and (6.6). Figure 6.13 illustrates the estimates of $Kl for 1-month maturity options for strikes between 4675 and 5875. These lowest and highest strikes are picked out in black and grey. The index sensitivities of all fixed strike volatilities are negative, so they move up as the index falls but by different amounts. During the crash period the sensitivities of all volatilities are greater and the change in the 5875 strike volatility sensitivity is very pronounced at this time. Before the crash it ranged between -0.005 and -0.01, indicating an increase of between 0.5 and 1 basis points for every FTSE point decrease. At

o.ooo

-0.030 -I-1-1-1-1-1-1-

Feb-98 Apr-98 Jun-98 Aug-98 Oct-98 Dec-98 Feb-99

---4725

4775

4825

-4875 -

- 4925

-4975 -

-5025 -

5075

5125

5175

5225

5275

5325

5375

5425 -

-5475

5525

5575

5625

5675

5725

- 5775 -

5625

-4675™

- 5875

Figure 6.13 Change in 1-month fixed strike volatility per unit increase in index.



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