Blending Independent Components and
Principal Components Analysis
4.3 Time-varying volatility
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4.3 Time-varying
volatility
4.3.1 Introduction
Time-varying volatility is an example of a phenomenon that
cannot easily be modelled via an approach involving linear combination
mixtures. Instead, it can be thought of as an example a distributional
mixture as referred to in Section
2.2. This is because we can characterise a world which exhibits
time-varying volatility as one in which returns are coming from different
distributions (the distributions differing by reference to their variance)
depending on when the return occurs. If it occurs at a time when ‘volatility’
is high then, all other things being equal we would expect the return to be
more spread out than when ‘volatility’ is low. Another name for time-varying
volatility is heteroscedasticity.
It is widely accepted within the financial services industry
and within relevant academic circles that markets exhibit time-varying
volatility. Markets can, for possibly extended periods of time, appear to be
quite ‘quiet’, e.g. without many large daily movements, but then to move to a
different regime in which, say, daily movements are more significant. However,
volatility does not necessarily always appear to move in tandem across markets
or even across parts of the same market.
We set out below some ideas for how the blending of PCA and
ICA might be refined to cater for time-varying volatility and also some of the
challenges that might arise in practice.
There are several possible ways of catering for time-varying
volatility. One approach would be to assume that the market (or sub-components
of it) might be in two or more relatively stable discrete ‘regimes’, the
regimes being differentiated by some postulated underlying state variable that
reveals itself by reference to the level of volatility that a market is
exhibiting. Probabilities of movement between such the different regimes might
then be built up, most probably incorporating some sort of autoregressive
characteristics as in threshold autoregressive time series models.
A perhaps simpler approach is to assume that there is some
underlying (and relatively slowly changing) continuous variable
characterising the variability of the market or of a segment of it, which we
might estimate at any particular point in time by applying moving average
techniques to recent past observations. This sort of approach is in effect the
one used in Kemp
(2009) and Kemp
(2010). It is also the one implicit in GARCH models and their variants. We
also immediately recognise a link with the complexity pursuit variant of ICA
described in Section
2.7, which also focused on moving averages as a sign of ‘predictability’ of
a time-ordered series. It would be possible to use a moving average that applied
equal weights to observations within a fixed length window. However, as in Section
2.7 it might be preferable to focus on an exponentially damped moving
average, potentially allowing flexibility in the decay factor involved.
We should then bear in mind that there are several possible
ways in which we might define time-varying ‘variability’ (even in the context
of blind source separation when there is no obvious differentiator between
individual output signals, here return series). We can think of any particular
return series as possessing its own individual volatility which is
somehow evolving through time. The average of these individual time-evolving
volatilities, i.e. average volatility, might itself also be somehow
evolving through time, in a possibly more reliably predictable manner, given
the greater number of data points contributing to its calculation. However, we
can also characterise the ensemble of return series as exhibiting a potentially
time-varying cross-sectional volatility. Own/market average volatility
and cross-sectional volatility in effect characterise different parts of the
covariance matrix between different stocks. The former corresponds to the
elements along the leading diagonal or their average (the ‘variance’ terms),
whilst the latter corresponds to an average of the off-diagonal elements (the
‘covariance’ terms). When we talk about ‘average’ stock correlation some of the
same types of topics also arise, see e.g. Measuring
Average Stock Correlation.
4.3.2 Higher moments
If a partial parameterisation of the evolution of
‘variability’ through time can include elements bearing the hallmarks of the
structure of a covariance matrix then a more complete parameterisation might
introduce further elements akin to the higher moment structure of a
multidimensional probability distribution. This would probably involve a
rather sophisticated to model and that would lack parsimony and it may be
better to limit ourselves merely to models that incorporate one of three types
of time-varying volatility adjustments, namely ones involving:
(a) An exponentially
weighted moving average estimate of volatility for a given individual stock
(measured, say, by the standard deviation of past returns for that stock in
isolation, with greater weight given to more recent observations);
(b) An exponentially
weighted moving average estimate of volatility for the average of all stocks
(calculated, say, in a manner similar to (a) but applied to the average return
for the market as a whole); and
(c) An exponentially
weighted moving average estimate of cross-sectional volatility between stocks
(measured, say, by calculating for each time period the cross-sectional
standard deviation of returns across the stock universe and then determining a
suitable exponentially weighted moving average through time of these standard
deviations).
4.3.2 Contemporaneous
estimates of future volatility
It is also worth bearing in mind that there may be available
to us contemporaneous estimates of future volatility that may be more reliable
than exponentially weighted moving averages of past data. For example, we might
be able to source market implied volatilities (and correlations) from options
markets. The relevance of this sort of data to risk model design is discussed
further in Kemp
(2009) as is the more fundamental topic of whether it is better when trying
to measure risk to use ‘market implied’ probabilities of occurrence rather than
or in addition to estimated ‘real world’ probabilities of occurrence. An
introduction to how it is possible to calibrate probability distributions used
for risk measurement purposes to market implied data is given here.
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