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57

Linear parameterizations such as (6.3) and the consequent limitation of movements in volatility surfaces to parallel shifts alone is an oversimplification of what is happening in the data

A PCA of daily changes in the fixed strike volatilities shown in Figure 2.6 may not give very good results, because the data will be rather noisy as mentioned above. However, look at the deviations of the same fixed strike volatilities from the ATM volatility, shown in Figure 6.6. The fixed strike deviations display much less negative autocorrelation and they are even more highly correlated and ordered than the fixed strike volatilities themselves. A strong positive correlation with the index is very evident during the whole period. These stylized facts are also revealed in the 3-month FTSE 100 implied volatility data: compare the fixed strike volatilities shown in Figure 6.1 la with the fixed strike deviations from ATM volatility shown in Figure 6.11b.

The PCA of fixed strike deviations is based on the model

A(aK - aATM)

P\ + WK2P2 + WiaP3,

(6.4)

where the volatility maturity and the strike of the volatility are both fixed. Time series data on A(oK - aAXM) are used to estimate the time series of principal components Pu P2 and P3, and the constant factor weights wKi, wK2 and Wjq.

A PCA for 3-month implied volatility skew deviations based on the data shown in Figure 6.1 lb gives the output in Table 6.5. From Table 6.5a it is clear that the first principal component only explains 74% of the movement in the volatility surface and that the second principal component is rather important, as it explains an additional 12% of the variation over the period. It is interesting that the factor weights shown in Table 6.5b indicate the standard interpretation of the first three principal components in a term structure, as parallel shift, tilt and convexity components. Note that sparse trading in very out-of-the money options implies that the extreme low strike volatilities show less correlation with the rest of the system, and this is reflected by their lower factor weights on the first component.

Principal component analysis of A(oK - oATM)for a fixed maturity has given some excellent results

Principal component analysis of A(oK - aAXM) for a fixed maturity has given some excellent results. Alexander (2000b) shows that for fixed maturity volatility skews in the FTSE 100 index option market during most of 1998, 80-90% of the total variation in skew deviations can be explained by just three key risk factors: parallel shifts, tilts and curvature changes. The parallel shift component accounted for around 65-80% of the variation, the tilt component explained a further 5-15%, and the curvature component another 5% or so. The precise figures depend on the maturity of the volatility (1, 2 or 3 months) and the exact period in time that the principal components were measured.

The immediate conclusion must be that linear parameterizations such as (6.3) and the consequent limitation of movements in volatility surfaces to parallel shifts alone is an over-simplification of what is actually happening in the data. The next subsection explains how the principal component representation (6.4) provides a more general framework in which to analyse the dynamics of the skew.



Principal Component Analysis Table 6.5a: Eigenvalues of correlation matrix

Component Eigenvalue Cumulative R2

Pi 13.3574 0.742078

P2 2.257596 0.8675

P3 0.691317 0.905906

Table 6.5b: Eigenvectors of correlation matrix

4225

0.53906

0.74624

0.26712

4325

8.6436

0.7037

0.1862

4425

0.67858

0.58105

0.035155

4525

0.8194

0.48822

-0.03331

4625

0.84751

0.34675

-0.19671

4725

0.86724

0.1287

-0.41161

4825

0.86634

0.017412

-0.43254

4925

0.80957

-0.01649

-0.28777

5025

0.9408

-0.18548

0.068028

5125

0.92639

-0.22766

0.13049

5225

0.92764

-0.21065

0.12154

5325

0.93927

-0.22396

0.14343

5425

0.93046

-0.25167

0.16246

5525

0.90232

-0.20613

0.017523

5625

0.94478

-0.2214

0.073863

5725

0.94202

-0.22928

0.073997

5825

0.93583

-0.22818

0.074602

5925

0.90699

-0.22788

0.068758

6.3.2 The Dynamics of Fixed Strike Volatilities in Different Market Regimes

Which type of skew scenarios should accompany different scenarios on movements in the underlying? In Dermans models the skew always shifts parallel, whatever the underlying market regime, and it is only the reaction of ATM volatility to underlying price movements that depends on the regime, as specified in equations (2.4a)-(2.4c). However, the PCA in §6.3.1 indicates that the simple static or parallel shift scenarios for the volatility skew that are a consequence of these models may not be appropriate.9

Which type of skew-scenarios should accompany different scenarios on movements in the underlying?

In the model described below the movement in fixed strike volatilities as the underlying moves is determined by the way the principal components move as the underlying moves. It encompasses changes in the tilt and curvature of the volatility skew as well as a parallel shift. Therefore, the range of the skew can

9 However, if they are appropriate, is the volatility by strike static, so the volatility by moneyness has a parallel shift, or is volatility by moneyness static, which is equivalent to a parallel shift in volatility by strike1



- - oATM(-u)

Strike

Strike

widen or narrow as the underlying price moves up or down, and change convexity in the process. However, does the movement occur at in-the-money volatilities as much as out-of-the-money volatilities? And how do the magnitudes of these movements depend on the current market regime?

Each component P, (i= 1, 2, or 3) is assumed to have a linear relationship with daily changes AS in the underlying. A linear model with a time-varying parameter jUl is defined for each component:

(6.5)

where the e, are independent i.i.d. processes. Thus the movement in fixed strike volatilities in response to movements in the underlying will be determined by (a) the factor weights in the principal components representation (6.4) and (b) the gamma coefficients in (6.5).

Figure 6.7 depicts the movement in skew deviations as the index price moves up, according to the signs of y2 and - Note that y1 is always assumed to be positive, an assumption that is justified by the empirical analysis below. The



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