1.5.2 Moving averages: UWMA, EWMA, GARCH

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Consider a sequence of spot prices {S

This formula provides the basis for the so-called uniformly weighted moving avergage (UWMA) and has been implemented in the

For small changes ln(S

and the small drift associated with the mean is generally neglected in comparison to the much larger fluctuating component. For a large number m»m-1 the formula for the variance can then be simplified to

The UWMA of drift, volatility and other quantities that can be estimated from time series, such as correlations, suffers from two main short-commings: the most recent events that are most significant, only carry the same uniform weight a

To tackle the first problem, the average window can be dropped in favour of a recursive or auto-regressive definition, producing an exponentially weighted moving average (EWMA) where the last known quantity is constantly updated with the most recent market increment

Insert (1.5.2#eq.5) back in itself and work through the recursion a few times to convince yourself that the weights, which were uniform in (1.5.2#eq.4), now are exponentially decaying with a ``forgetting rate'' a

The second issue is generally solved by writing the long term average as V=w/(1-a-b) and introducing a reversion term in a so-called generalized auto-regressive conditional heteroscedasticity model, using the p most recent increments and the q most recent volatility estimates in GARCH(p,q). The most commonly used is GARCH(1,1)

where a controls the sensitivity to most recent increments, b the forgetting rate and w=gV is linked with long term average. For consistency, the parameters must satisfy a+b+g=1 and to prevent negative long term average volatility, it is important that a+b<1. Clearly, the EWMA model is a particular case of GARCH(1,1), where w=0, a=1-l, b=l.

Qualitative arguments support models with features such as the exponential weighting (``forgetting''), a reversion mechanism (``long term average'') and the tendency to reproduce the auto-correlation of the market (``clustering'', i.e. large u

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