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the beginning of the crash the 5875 sensitivity increased to about 1.5 basis points, and since the FTSE fell by 1500 points during the crash, that corresponds to a 22.5% increase in 5875 volatility. Then, at the height of the crash between 1 and 9 October, the 5875 sensitivity became increasingly large and negative as the FTSE index reached a low of 4786 on 5 October. On 9 October the 5875 sensitivity was an impressive -0.028, indicating a 2.8 basis point increase in volatility for every point off the FTSE at that time.

The method outlined here explains how to use PCA to calculate the volatility sensitivity to underlying price changes directly, and for options with different strikes

An increase in low strike sensitivities is much less pronounced around the time of the crash. What is interesting about the 4675 volatility sensitivity is that it is often far greater (in absolute terms) than the high strike volatility sensitivities. Therefore, most of the movement will be coming from the low strikes as the range of the skew narrows when the index rises and widens when the index falls. Very approximately the 4675 volatility gains about 1 or 2 basis points for every point fall in the FTSE index during the period, although the sensitivity varies considerably over the period. At the end of the data period it is extraordinarily large, and it can be seen in Figure 6.6 that range narrowing of the skew was very considerable at this time.

6.3.4 Summary

This section has presented a new principal component model of fixed strike volatility deviation from ATM volatility. It has been used to estimate the change that should be made to any given fixed strike volatility per unit change in the underlying, that is /dS. This quantity is an important determinant of an option delta when volatility is not constant. Market traders often approximate this sensitivity by / but the method outlined here explains how to use PCA to calculate the volatility sensitivity to underlying price changes directly, and for options with different strikes.

Non-parallel movements in the volatility surface are particularly important for short-maturity volatilities

The model also shows how to construct scenarios for the volatility surface that should accompany given moves in the underlying price. A quadratic parameterization of the volatility surface is consistent with the model, and this allows non-parallel movements in the surface as the underlying price moves. Dermans sticky models that were discussed in §2.3 correspond to a local linear approximation for the volatility surface and only allow for parallel shifts. However, the principal component approach that has been developed here shows that non-parallel movements in the volatility surface are particularly important for short-maturity volatilities.

Empirical application of the model to the FTSE 100 index options has revealed two distinct regimes of volatility in equity markets. In the first regime the range of the skew narrows as the index moves up and widens as the index moves down. Most of the movement is in low strike volatilities; high strike volatilities remain relatively static as the underlying moves. The second regime corresponds to a market crash and recovery period, where the range of the



skew is quite static and the volatilities of all strikes will shift parallel and in line with the (considerable) changes in the ATM volatility.

During the very jumpy markets before and after the 1998 crash the very short-term volatilities behaved a little differently from the volatilities of 2- and 3-month maturities. The 1-month volatilities actually exhibited a narrowing in the range of the skew as the index fell during the initial period of the crash (mid-July to mid-August). In the recovery period (mid-October to mid-November) the skew range widened as the index moved up. At both times the high strike volatilities were moving more than usual, just like the 2- and 3-month high strike volatilities, but there was less movement in very short-term very low strike volatilities. It was not until the height of the crash that the low strike 1-month volatilities became more responsive to the index, so that the skew shifted parallel, in line with the ATM volatility changes.

Because the model admits non-linear movements in the volatility smile as the underlying moves, it has very general applications. It should be particularly useful in currency option markets, where smiles are non-linear, and a parameterization of the swaption skew would be useful for several interest rate option models.

6.4 Overcoming Data Problems Using PCA

Principal component analysis provides a means of coping with several common data problems. Data may be unavailable, or just difficult and time-consuming to gather on a regular basis. Even when data are available, they could be full of measurement errors, and this will bias parameter estimates (§A.4.2). This section shows how PCA can be used to fill in missing or erroneous data. It also shows how PCA can be used to overcome the difficulty with model specification that is often encountered when data on explanatory variables are highly collinear (§A.4.1).n

Section 6.4.1 shows how PCA provides a means of performing regressions on totally uncorrelated variables, so that collinearity problems in regression are minimized. Sometimes, however, collinearity of variables in a system can be an advantage. In §6.4.2 it is shown how PCA can be used to fill in observations on missing data; the method works very well when the missing data are from a reasonably highly correlated system. Otherwise the modeller will have to take some views on the likely correlation of the missing variable within a system.

Some of the models presented in this section will use principal components in time series regression analysis, and the reader should be aware of a tricky statistical issue surrounding this. In order to compute the principal components of a system, one

"if explanatory variables are highly collinear their r-statistics will be depressed, and this can augment the attenuation bias caused by data errors. More details are given in Appendix 4.



The value of a principal component at time t = 1 will actually depend on the future values of the input variables. Depending on how the model is used, this may or may not be acceptable

has to use a quantity of data to calculate the correlation matrix of the input variables. Therefore the value of a principal component at time t = 1 will actually depend on the future values of the input variables. Depending on how the model is used, this may or may not be acceptable. For example, it would not be acceptable if the model is used to predict the values of another variable at time t = 1, because it will be using data from the future.

The bottom line is really whether the model that is built on an historical data set will be able to perform in the same way in real time. If not, the principal components will have to be computed in a time series framework as follows: begin with a data set from the start of the data history that is long enough for the principal component specification to be reasonably robust.12 Say it runs from t = 1 to t = T. Perform a PCA on this data set, save the factor weights wit and then calculate the value of each component at time T+ 1 as

Pm - w\mXi + w2mX2 + ... + wkmXk,

where the data on the input variables Xu X2, . . ., Xk are taken at time T+l. Then roll the data period one time interval and again do a post-sample computation for the values of the principal components. Repeating in this way, a set of data on all principal components starting at time T+ 1 may be obtained, where each value of each component depends only on the past and not on the future.

6.4.1 Multicollinearity

When explanatory-variables are highly collinear then model parameter estimates and their standard errors will be affected

In Appendix 4 some of the problems with linear regression that are common to factor models, index tracking models and other linear portfolio models are outlined. The choice of explanatory variables in a linear model may pose a difficult problem. What stocks should be selected for a benchmark tracker, or what factors should be chosen in a multi-factor model? Theoretical considerations are important in determining this choice, but there is another practical aspect to the problem. When explanatory variables are highly collinear then model parameter estimates and their standard errors will be affected.

PCA is one of the most efficient means of dealing with multicollinearity. Other methods, such as changing the data, excluding one or more collinear variables, or employing a ridge estimator all have their drawbacks (§A.4.1). However, an orthogonalization of the variables using PCA is a simple way to obtain efficient parameter estimates for the original model with nothing more than OLS estimation.

The idea is straightforward. Let Xv

Xk be the explanatory variables for

the linear model with dependent variable Y. variables are assumed to be stationary.

Both Y and the explanatory

i2This will depend, of course, on how highly correlated the system is.



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