smiles and skew with PCA have almost invariably used daily changes in implied volatilities, by strike or by moneyness, as input. Derman and Kamal (1997) analyse S&P 500 and Nikkei 225 index options where the volatility surface is specified by delta and maturity. Skiadopoulos et al. (1998) apply PCA to log differences of implied volatilities for fixed maturity buckets. Fengler et al. (2000) employ a common PCA that allows options on equities in the DAX of different maturities to be analysed simultaneously.

There is an important difference between the research just cited and the approach taken in this section. Instead of applying PCA to daily changes in implied volatilities themselves, a PCA is applied to daily changes in the deviations from at-the-money volatility. The advantages of this approach are both empirical and theoretical. On the empirical front, time series data on fixed strike or fixed delta volatilities often display very much negative autocorrelation, possibly because markets overreact, so the noise in daily changes of fixed strike volatilities is a problem for PCA. However, the daily variations in fixed strike deviations from ATM volatility, A(aK - oAXM), are much less noisy than the daily changes in fixed strike (or moneyness) volatilities. Consequently, the application of PCA to fixed strike deviations gives results that are very much more robust than the results cited above.

Instead of applying PCA to daily changes in implied volatilities themselves, a PCA is applied to daily changes in the deviations from at-the-money volatility

On the theoretical level, it was shown in §2.3 that Dermans models of the skew in equity markets can be expressed in the form (2.5), where the particular market regime will be determined by the different behaviour of the ATM volatility specified by equations (2.4). There it was shown that in every market regime, whether it is trending, range-bounded or jumpy, the relationship between fixed strike deviations and the underlying price is always the same. In fact for any maturity t there is a linear relationship between the deviation of a fixed strike (K) volatility from ATM volatility and the underlying price, given by

J ATM

= -b(K-S).

(6.3)

For any given maturity x, the deviations of all fixed strike volatilities from ATM volatility will change by the same amount b(x) as the index level changes, as shown in Figure 6.5a. Four strikes are marked on this figure: a low strike KL, the initial ATM strike Ku the new ATM strike after the index level moves up K2, and a high strike KH. The volatilities at each of these strikes are shown in Figure 6.5b, before and after an assumed unit rise in index level (AS = 1). In each of the three market regimes the range of the skew between KL and KH, aL - , will be the same after the rise in index level. Thus as the underlying price moves, the fixed strike volatilities will undergo a parallel shift, and the range of the skew will remain constant. The direction of the movement in fixed strike volatilities depends on the relationship between the original ATM volatility C] and the new ATM volatility c2-

Recall from §2.3 that the movement in ATM volatility as the underlying price moves depends on the current market regime:

▲

Strike

°2 + dl + b(i)

aL = -, + dL

02 + + £>(t)

o2 + cl + ( )

°2 - dh + b(i) i a2 = o-, - b(x)

oH = , - dh

! a2-dh + b(%)

Trending Range-bounded Jumpy

Figure 6.5 Parallel shifts in (a) skew deviations and (b) fixed strike volatilities, as price moves up.

> In a range-bounded market a2 = o\ - b, but fixed strike volatilities have all increased by the same amount b, so a static scenario for the skew by strike should be applied.

>- When the market is stable and trending, c2 = ci and there is an upward

shift of b in all fixed strike volatilities. > In a jumpy market c2 = csx - 2b, so a parallel downward shift of b in the

skew by strike should be applied.

These observations suggest a method for the empirical validation of the sticky models for volatility regimes in §2.3. The method consists of performing a PCA of A(oK - oAXM) and examining the significance of the second and higher principal components. The sticky models imply that only the first principal component should be significant, but if it is found that the second or higher principal components are significant factors for determining movements in A(aK - oATM), then the parallel shifts in the skew that are implied by the sticky models will not be justified in practice. Instead, movements in the skew will be

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

Mar-99

-4025 -4325 4625 4925

-4075 -4375 4675 4975

. 5225 -----5275

- 5525 -5575

5825 ---5875

-6125 6175

6425 -6475

6725 6775

-FTSE100

4125 4425 4725 5025 -5325 5625 5925 6225 6525 6825

4175 4475 4775 5075 -5375 -5675 -5975 6275 -6575 6875

- 4225 - 4525

- 4825 -5125

- 5425 - 5725 -

- 6025 ---6325

6625 - -

6925 --

-4275 4575 4875 5175 -5475 -5775 -6075 6375 6675 6975

Figure 6.6 Deviations of fixed strike volatility from (1-month) at-the-money volatility.

characterized by a widening or narrowing of the range, depending on the current market conditions.

The principal component model described in this section extends Dermans linear models to allow non-linear movements in fixed strike implied volatilities as the underlying price changes. The data used to describe the model are FTSE 100 index options, but it should be noted that this framework is quite general and has applications to other types of financial assets.

6.3.1 PCA of Deviations from ATM Volatility

Time series data on implied volatilities, such as the 1 -month volatilities for the FTSE 100 shown in Figure 2.6, should contain all the information necessary to parameterize the skew. However, there are around 60 different strikes represented there, and their implied volatilities form a correlated, ordered system that is similar to a term structure. It is therefore natural to consider using PCA to identify the uncorrelated sources of information. Both analytic simplicity and computational efficiency will result from a model that is based on only a few key risk factors.

Both analytic simplicity and computational efficiency will result from a model that is based on only a few key risk factors