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of GARCH conditional correlation estimates for the US dollar-sterling and the Japanese yen-sterling exchange rates between 1993 and 1996. These correlation estimates show large jumps during the data period. Several times they either double or halve in value, more or less overnight.

This type of instability is quite common in conditional correlation estimates, but is much less apparent in unconditional correlation estimates. The past does not have the same role in conditional models as it does in unconditional correlation models. In unconditional correlation models the variation in correlation estimates is only due to sampling error, and events that happened far in the past can still affect the sampling error as much as if they happened only yesterday, so sampling errors can be perpetuated over a long period of time. However, in conditional correlation models the variation in successive estimates is also due to variation in the process parameters. The parameters vary because conditional correlation measures the co-dependency of two returns at a particular instant in time, assuming that everything that has happened in the past is predetermined. Current estimates are still influenced by past data, because it is in the information set, but the long-term past is less relevant than it may be in an unconditional model.

1.5 Remarks on Implementing Volatility and Correlation Models

It is important to have some consistency between the data frequency and the time-varying characteristics of the model that is used. For example, a time-varying volatility model would normally be based on high-frequency returns - in fact it is standard to use at least daily if not intra-day data. Any lower frequency of data will not capture the volatility clustering that is a characteristic of most financial markets (§4.1.1). Any attempt to estimate a time-varying volatility model for daily variations using low-frequency data would not give very meaningful results. Even daily data cannot capture the huge market swings that are sometimes experienced during the course of the day. However, the management of intra-day databases poses enormous practical problems and many institutions would not have such data readily available.

Any attempt to estimate a time-varying volatility model for daily variations using low-frequency data would not give very meaningful results

On the other hand, if a volatility or correlation model has no need to account for short-term variations in volatility or correlation, and is only required to generate forecasts of the unconditional volatility or correlation over a long-term risk horizon, there is no need to use high-frequency data. In fact the excess variation in high-frequency data will only be attributed to noise or sampling errors, because there is nothing else in the model to explain it. Therefore, very high-frequency data can give misleading results when simple unconditional volatility or correlation estimates are all that is called for.

It is also important to employ a statistical volatility or correlation model that is consistent with the horizon of the forecast. To forecast a long-term average



volatility it makes little sense to use a high-frequency time-varying volatility model. On the other hand, little information about short-term variations in daily volatility would be forthcoming from a long-term moving average volatility model.

A common option maturity is 3 months, so should one use a time-varying volatility model or a constant volatility model to price such an option? The answer depends on the volatility characteristics of the underlying asset returns - in particular, on their volatility term structure (§2.2.2). If 3-month volatility is close to the long-term average volatility then it is satisfactory to use a constant volatility model with a long averaging period (Chapter 3). However, if 3-month volatility often differs considerably from the long-term average it is better to use the 3-month forecast from a time-varying volatility model (§4.4.1).

Forecasts of volatility and correlation may just not be available for distant time horizons. Often implied volatilities cannot be obtained because there is no real market for options of the appropriate maturity. Sometimes statistical forecasts cannot be made because the underlying market is only very recent. In this case, if pricing a long-term option is a necessity, it is important to decide how uncertainties in volatility will be taken into account (§5.3).

Missing data is a common problem that can affect both long-term and short-term volatility and correlation forecasting. Trading may be very thin, not just in long-term options, and if quotes are left stale for a long period of time then implied volatilities are never going to be very accurate. Lack of liquidity can also present problems when trying to obtain meaningful parameter estimates in statistical models. Short-term uncertainties will be impossible to quantify if there is currently no market in the underlying. However, if the underlying asset with missing or inaccurate data lies in a highly correlated system then much can be done to get around the problem by using principal component analysis (§6.4.2).

To obtain a statistical estimate or forecast of correlation, the historic data on the two asset returns need to be of the same frequency and measured at synchronous points in time. There may be a problem with obtaining data at exactly the same time for cross-market correlation estimates. If data on one series is measured before the data on the other, correlation estimates may be seriously biased. When it is impossible to obtain synchronous daily data, for example when two markets are never open at the same time, it may be better to move to a different frequency.

1.6 Summary

The aim of this chapter has been to introduce the reader to some of the important but complex concepts that will be discussed in Part I of the book. Volatility and correlation have been described as parameters of stochastic processes that are used to model variations in financial asset prices. Unlike

the underlying asset with missing or inaccurate data lies in a highly correlated system then much can be done to get around the problem by using principal component analysis



market prices, they are unobservable and can only be estimated within the context of a model. Therefore the analysis of volatility and correlation is a very complex subject. In this introductory chapter some basic distinctions have been drawn, in particular between:

5» implied and statistical volatility estimates and forecasts; >- constant parameter and time-varying parameter models for volatility and correlation.

It is necessary to understand these distinctions in order to motivate the exposition in the next four chapters.



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