Forecasting Realized Volatility – Part 1

Entering The Realm

We’ll start with a quote from Richard Feynman, “Learn by trying to understand simple things in terms of other ideas.”  The things we will try to understand for the purposes of this article are the following. Do people invest (or trade) at different time horizons? Are there different views of risk amongst investors? And are there different views of where the same stock’s price will be one week, one month or one year from now?  These are the questions we will attempt to answer in a direct way.  From there we will be able to ask more subtle questions that will hopefully illuminate the RVI.

To the question, “Do people invest (or trade) at different time horizons,” the answer is clearly yes.  And though modern portfolio theory (MPT) assumed this fact away, please keep in mind that MPT is the grandfather of a lot of what followed in financial economics and financial mathematics, including steps forward like the RVI.  So though grandfather may have gotten some things wrong, he made some terrific pioneering efforts.

It is accepted by many, if not all, practitioners and by a growing number of academics that different market participants have different time horizons when it comes to their analysis of past events.  These different time horizons and resolutions affect their trading or investing goals.  The existence of heterogeneous trading behaviours has given rise to the hypothesis that the market itself is fractal.  By fractal we mean there is no single preferred time period or investment horizon in the market.  Not one day, two weeks, one month … nothing.  Why no one investment horizon?  Because the heterogeneous trading behaviours that make up the market can be characterised as different actor groups or components, each with their own frame of reference.  In other words, the differing time horizons of investors and traders are a key aspect of the market. To state it another way, it’s the variety of time horizons and their interaction that is a probably one of the most important factors that make up a market. Given the variety of timescales and frames of reference, each component could be modelled in form of an intrinsic or individual time for that component (this statement will be key in understanding the RVI later).

The fractal approach to analysing objects of different kinds, including financial data, can be stated as follows:

Objects are analysed on different scales, with different degrees of resolution, and the results are compared and interrelated.

Kind of sounds like the market as we know it, doesn’t it?

As to our next question, “Are there different views of risk amongst investors?” the answer is also clearly yes.  Studies by folks such as Olsen and Associates [1] and Lynch and Zumbach [2] have demonstrated that this difference in risk is also fractal, meaning that, as we noted in bold above, risk (or volatility) is analysed at different time scales with different degrees of resolution.  Obvious examples of this are people who trade volatility on a short-term basis, for example over a few days.  There are also other actors whose horizons are typically a month if not more (such as portfolio managers).  The sudden movement in the price of a stock could move volatility enough for a short-term trader that he will adjust his position accordingly that day or the next.  The portfolio manager, however, will more than likely view a one-day or even a few-day unexpected movement in price as a bump in the road and will probably not adjust her position or adjust it minimally.  What is also of interest in the works of Lynch and Zumbach is that risk (volatility) can cascade.  What the authors mean by this is that when the typical time horizons (or different scales and resolutions) collapse, it is a one way flow, always from the telescopic to the microscopic.  In other words, if enough of those participants with longer-term views of volatility begin to act like those with the shorter-term views, a cascade or a volatility cluster occurs, such as during that period of wretched months for the equity market in late 2008 and early 2009.

As to our final question, “Are there different views of where the same stock’s price will be one week, one month or one year from now?”, the answer is yes.  To restate our fractal argument above, there is no privileged time scale in the market.  The interaction of what we call components and their different time scales gives rise to endogenous events such as volatility clusters, trend persistence and the empirically demonstrated time lag between interest rate changes and foreign exchange rate adjustments.  As a result of such relationships, there can clearly be differing views on price.  And such differences in views are not a list of inefficiencies, as is often stated in economic texts.  It is, as we have demonstrated, a consequence of different time scales in the market.

REUTERS/Pascal Lauener

The Subtle Questions So if there is no privileged time scale in the market and if we want to model the individual time frame of each investor and trader, how do we do that?  This is where wavelets come in.  Wavelets allow us to take the original data – in our case a time series of daily volatility observations – and decompose it into the components or actor groups noted above.  Wavelets allow us to isolate the different timescales so we can examine separately the dynamics – the inner workings – of each component.  This is of great benefit to us as we will be able to say more about what is happening on the different scales and different resolutions and their affect on volatility.

While the mathematics of wavelets can be difficult to understand, there is a simple way to explain their ability to let us see or generate the timescales the market works on.  Multiresolution analysis (MRA) is a way of decomposing a time series into its components or actors.  The different components the MRA generates are the components of the original signal at different resolutions (which sounds like what we are looking for).  Via the use of MRA, we can generate components that span timescales from a few days to a few months.  So, with the MRA, we have a way of getting closer to our earlier statement about the market:

Objects are analysed on different scales, with different degrees of resolution, and the results are compared and interrelated.

Before we discuss how MRA results are used, we need to take a slight detour into so-called GARCH (generalised autoregressive conditional heteroskedasticity) models.  GARCH models are commonly used when modelling financial time series that exhibit time-varying volatility clustering: in other words, periods of high volatility interspersed with periods of relative calm. Over the past few years, GARCH modellers have also been looking at volatility over different timescales, typically by starting with tick-by-tick price data.  The results they have achieved are very impressive, especially if you look at a recent model called LM (for long memory) ARCH.  However, there’s a difference between the LM-ARCH approach and that of the RVI.   LM-ARCH models do not rely explicitly on an assumption that all time scales are equivalent(i.e., on using data spanning time periods of days to  a few months).  Instead, they focus on  intra-day (tick-by-tick) to daily data.  Another approach, called IGARCH, typically uses just one time period as input.  The RVI, by contrast, is based explicitly on the “no privileged time scale” argument.

With our components in hand, we have the raw material to make a volatility forecast.  As we want to forecast volatility, we need a guiding principle for constructing our forecasts.  I think a statement from Kevin Judd and Thomas Stemler [3] is appropriate here: “Forecasting: it is not about statistics, it is about dynamics.”  This is the title of their paper and a good guiding principle for us.  I say this because in our personal relationships, we understand (or try to understand) the dynamics that are at play between ourselves and our friends, boss and spouse.  And if you were asked to predict a friend’s reaction to something – a statement a colleague is thinking of saying, a change in events – you would use your understanding of your friend to predict what she might say or do.  Of course there is some wiggle room (called error bars in statistics) in what we predict.  Even then, more often than not we use our understanding of the person – our understanding of her internal workings and their relationship with us – to make our best forecast.  So dynamics play a dominant role in our lives and there is no reason to think that trying to forecast volatility–in other words, the interrelatedness of various people in the market –will be any different.