Moving Average Models

Moving averages have many uses in financial data modelling. For example, in technical analysis, where they are called stochastics, the relationship between moving averages of different lengths can be used as a signal to trade. Simple moving averages have also been used to estimate and forecast unconditional volatility and correlation. The terms historic volatility and historic correlation are applied to the statistical forecasts that are based on equally weighted moving averages. This rather confusing terminology remains standard, even though several other types of statistical forecast based on historic data are now in common use.

In this chapter, §3.1 describes how historic volatility and correlation measures are obtained, and outlines their advantages and limitations. The equally weighted average is calculated on a fixed size data window that is rolled through time, each day adding the new return and taking off the oldest return. Another common form of weighted average measure for volatility and correlation is an exponentially weighted moving average; this is described in §3.2. If exponentially weighted moving averages are used correctly they can produce much more reasonable estimates of short-term volatility or correlation than equally weighted moving averages. In fact in some cases an exponentially weighted moving average correlation model may be preferred to a more sophisticated correlation model, including some of the GARCH correlation models that are described in §4.5. The reader should use the moving averages spreadsheet to see how these models work in practice.

Perceived changes in volatility and correlation can have important consequences, so it is essential to understand what is the source of variability in any particular model. Any weighted average model is only appropriate for estimating unconditional volatility or correlation (§1.3 and §1.4). As such it provides no reason for estimates to differ as the data window moves through time, except for sampling error, or noise. It is important to realize that the unconditional volatility or correlation of the time series is one number, a constant for the whole series, and so the variation observed as weighted average volatility or correlation estimates are moved through time can only be attributed to sampling errors.

In some cases an exponentially weighted moving average correlation model may be preferred to a GARCH correlation model

An important but subtle point, that will be returned to with a more detailed discussion in §10.3.3, is that in this chapter, and in Chapter 4, the fundamental

We can, of course, say that our current estimate of volatility or correlation is the forecast, and this is what is generally done

parameters that are being forecast are the variance and the covariance. These are the parameters that determine the returns distributions, and it is standard to base statistical forecasts on a model for these parameters. However, a forecast is an expectation, taken under some probability measure, and the expectation of a square root is not equal to the square root of an expectation, that is, E(o) /E(a2). Despite this observation, most statistical models for forecasting volatility are actually models for forecasting variance: they first forecast a variance, and then the volatility forecast is taken as the square root of the variance forecast (and if necessary annualized in the usual way, as in (1.2)).

It should also be understood that a weighted average is really just an estimation method. We can, of course, say that our current estimate of volatility or correlation is the forecast, and this is what is generally done, but there are more sophisticated models available. For example, GARCH models can provide forecasts of the whole term structure of volatility for any point in the future (§4.4.1).

3.1 Historical Volatility and Correlation 3.1.1 Definition and Application

Historical estimates of volatility and correlation are obtained in two stages. First one obtains unbiased estimates of the unconditional variance and covariance that are based on equally weighted averages of squared returns and cross products of returns. Then these are converted into volatility and correlation estimates by applying formulae (1.2) and (1.3), respectively.

An -day historic volatility estimate is often based on an equally weighted average of n squared daily returns. An unbiased estimate of unconditional variance at time /, using the n most recent daily returns and assuming the mean is zero, is

Using (1.2). the estimate at time t of an -day historic volatility is the square root of this estimate multiplied by the number of daily returns per year. Note that (3.1) gives an estimate of the variance, rj2. that is to say, the hat " should be written over cr and not just . It is not the square of the unbiased

it is usual to apph moving averages to squared returns rjU = 1. 2. 3..... ) rather than squared mean

deviations of returns ir. - rr. where r is the average return over the data window. Standard statistical estimates of variance are based on mean deviations, but empirical research on the accuracy of variance forecasts in financial markets has demonstrated little advantage in using mean deviations, except if the returns are very low-frequency. The volatility of monthlv returns mav be betier estimated using a non-zero value for r, in which case the unbiased estimate of variance will have n - 1 rather than n in the denominator. For analysing daily financial series it has become standard lo base variances on squared returns and covariances on cross products of returns (Figlewski, 1997; Alexander and Leigh. 19971.

estimate of the standard deviation. For typesetting reasons, however, the notation rj2 is used throughout this book to denote the variance estimate or forecast.

The current -day historic volatility estimate is sometimes taken as the forecast of future volatility to plug into the pricing model for an option that matures in n days. Historic volatility estimates are also commonly used in the -day covariance matrix for measuring portfolio risk. Sometimes one looks back exactly n days in order to forecast forward n days: only recent market conditions are thought to be more appropriate for short-term forecasts, but for long-term forecasts one should take a longer averaging period. It is, however, more common to use a historic period of more than n days for an -day forecast, especially if n is small. For example, for a 10-day forecast most practitioners would look back 30 days or more.

Long-term historic forecasts need not be based on daily returns, provided the number of observations remains sufficiently large. Weekly or monthly returns may be used in a similar way. For example, a historic forecast of volatility over the next six months that is based on weekly returns data could use the last 26 weekly returns in (3.1) and an annualizing constant of 52.

Similarly, -day historic correlations are calculated by dividing the equally weighted covariance estimate over the last n days by the square root of the product of the two -day variance estimates:2

If these estimates are based on a small sample size they will not be very precise. The larger the sample size the more accurate the estimate, because sampling errors are proportional to XjJn. Therefore a short moving average will be more variable than a long moving average. Put another way, short-term volatility or correlation estimates have much more variability over time than long-term estimates. For example a 30-day historic volatility (or correlation) will always be more variable than a 60-day historic volatility (or correlation) that is based on the same daily return data. It is important to realize that, whatever the length of the averaging period, and whenever the estimate is made, the historic estimates are always estimating the same parameter: the unconditional volatility (or correlation) that is a constant for the whole process. The variation in -day historic estimates can only be attributed to sampling error; there is nothing else in the model to explain it.

2Again, assuming zero means is simpler, and there is no convincing empirical evidence that this degrades the quality of correlation estimates and forecasts in financial time series.

The variation in n-day historic estimates can only he attributed to sampling error; there is nothing else in the model to explain it

p, =

(3.2)