65 60 55 50 45 40 35 30 25 20

Jan-98 May-98 Sep-98 Jan-99 May-99 Sep-99 Jan-00 May-00 Sep-00

-Market Price -----Simulated Price

98 May-98 Sep-98 Jan-99 May-99 Sep-99 Jan-00 May-00 Sep-00

-Volatility

-Simulated volatility from figure 6.14a -Simulated volatility from figure 6.14b

Figure 6.14 Using PCA to simulate a price history from (a) 150 and (b) 100 rece prices; (c) GARCH(1, 1) volatilities from market prices and simulated prices.

back that far) and performing a PCA of the period from 2 January 1998 to 2 March 2000. The first four principal components are taken and substituted in (6.12) to simulate a return series for the new stock over the same period. The results are illustrated in Figure 6.14a. For the purposes of this illustration I have only pretended that the stock is a new stock, but market prices were actually available.15 The actual market prices are shown on the figure so that the reader can compare them with the simulated prices.

Note that the returns that are simulated by (6.12) will be standardized, and will need to be multiplied by their standard deviation and added to their mean before transforming into a price series. Which mean and standard deviation should be used? I have found that good results are obtained by taking the standard deviation equal to the in-sample standard deviation of the new stock, but that for the mean it is better to use an average of the in-sample mean of the new stock and the mean of the seven other stock returns during the pre-sample period.

If an astute choice of the related stocks and the number of principal components has been made there should be a good fit between the actual market price of the new stock and the price that is simulated by the representation (6.12). The success of the next stage will depend on a successful calibration here, so it may be necessary to choose different stocks, possibly more stocks, and/or more principal components in the representation (6.12). The RMSE between the actual and simulated returns in this example was 0.0098, which is very low, so the model appears to fit very well.

Figure 6.14b shows that the results of a similar analysis, which only assumes that data on the new stock are available for 100 days, are not quite as good.16 Finally, Figure 6.14c compares the GARCH(1,1) volatility estimates that are generated by the actual market prices and both series of simulated market prices. Although the simulated volatilities both underestimate the effect of the September 1998 crash in equity markets, they are otherwise similar to the actual GARCH(1,1) volatility characteristics of the stock, even as far back as January 1998.

The returns that are simulated by (6.12) will be standardized, and will need to be multiplied by their standard deviation and added to their mean before transforming into a price series. Which mean and standard deviation should be used?

The methodology just described has applications beyond the simulation of missing data. In fact it may be applied to great effect in the construction of a hedging basket for a stock. If a basket of seven related stocks can be used to simulate a historical price series for a stock, then it could also be used to simulate a future price series. This means that a hedging basket can be based on a principal component representation such as (6.12). To illustrate this, return to the example of the banking stocks and consider the example of hedging a banking stock X, with seven other banking stocks X2, . . ., X%. Suppose that the hedging basket on 6 October 2000 will be based on data since 6 March

A hedging basket can be based on a principal component representation

15 The stock was in fact American Express.

16The model was not recalibrated to use different stocks so it is possible that better results would be obtained b\ choosing different stocks in the first stage.

2000, and so it is defined by (6.12), the principal component representation of X{ using four components in the eight-variable system. The weight for each stock X2, , X% in the hedging basket is obtained by substituting into (6.12) the representations of P\, P2, P and P4 in terms of these seven stock returns - that is, using (6.1) based on data only for X2, . . ., X$. Normally this second PCA will be based on a longer data period. Then rebalancing the hedge is straightforward: all that is necessary is for the weights in the hedging basket to be reset by recalculating the weights in (6.12). For example, if the hedge is to be rebalanced on 31 October, (6.12) should be revised using a PCA of all eight stocks from 31 March 2000 to 31 October 2000.

Another example of the use of PCA to fill in missing data is when a mark-to-market price is required for a portfolio that contains securities from different markets but some of these are closed, either because of a bank holiday or because of time differentials. Or perhaps trading is very thin in some markets, with stale quotes imparting a lack of variability in some returns. Of course, if derivative securities are being traded on an exchange that is currently open, the securities prices may be inferred from the current derivatives price. If this is not possible, however, one can use a principal components representation to infer prices, if their movements are correlated with the movements of the other security prices in the system.

The accuracy of these prices will depend on the strength of the correlation within a system and the number of securities that do have a current price. If the correlation is reasonably high then of course the system can be represented by just a few principal components. And if there are still many securities that do have a current price, these principal components can be updated to their current value using just those securities. Then the current value for the missing prices can be deduced by applying their factor weights to the current principal components. This type of marking-to-market is possible when there is a strong correlation between different market returns and it copes with the problem of market closures.