Conditional Volatility

[afterworkguinness]

Hello again,
I'm having a bit of trouble wrapping my head around the idea of conditional vs unconditional volatility.

In one of David's Youtube videos (and from other sources like Jorion) conditional volatility is a volatility estimate conditional on today's volatility and can change day over day; as we see in ARCH, GARCH and EWMA.

Given this, how can a volatility estimate not be conditional on today's estimate and not change day over day ?

 

[David Harper CFA FRM]

Hi afterwork,

I agree it's subtle. Volatility is a model-based construct. All of our (FRM) methods (i.e., MA, EWMA, GARCH, implied vol) do imply that the current volatility estimate changes (updates) each day based on new information. But conditional variance/volatility is not really a condition where our estimate updates; rather, conditional volatility is a volatility (and, really a model of volatility) that deliberately is informed by new information; i.e., "tomorrow's volatility estimate depends on (is conditional on) certain new information."

• So, while (eg) a 100-day moving average (historical standard deviation; our simplest approach) updates the estimate each day, it produces a current volatility estimate (e.g., 1% daily) that represents an average over the entire sample and, for the matter, an "unconditional" forecast of tomorow's or the day after's (we could re-sort the returns and get the same 1%). We run it today and get 1%, by which we mean "the volatility is unconditionally 1% over the sample and the forecast for tomorrow is unconditionally 1%." Tomorrow a new day gets added and we update the MA, upon which we have a new unconditional volatility
• While EWMA would vary based on the sort, to the extent an EWMA volatility makes a forecast, its forecast is the current volatility, a flat line. If the today's EWMA is 1%, the forecast is an unconditional 1%
• GBM assumes a constant volatility, even as we employ the constant volatility in Merton to predict default. This is "unconditional" in the sense that a 10% is not dependent on anything. We may update it tomorrow, so it changes, but it will still be unconditional within the model and to the extent it makes any forecast in Merton.
• Our most visible example of an unconditional variance is located in the first term of GARCH: given vol = omega + a*return(t-1) + b*variance(t-1), the unconditional variance (aka, long-run variance) = omega/(1-a-b) and it's a numerical value. So, notice that, armed with a GARCH model, the overall GARCH estimate will update each day (conditional on new information), but the unconditional variance (a value) will remain the same over the forecast period, while the GARCH estimate is conditional, the "long run variance" will remain unchanged (not dependent on any information) through the sample and forecast.
• In this way, by unconditional, we really mean "unconditional within the sample" or "unconditional within the forecast horizon" and, you can see how this reconciles an estimate that updates (changes) day-to-day but does not necessarily imply a conditional volatility model. (semantically, we can often re-define a conditional condition more narrowly into something "unconditional under narrow conditions". It is almost a matter of perspective definition!)

Here is from Carol Alexander who I really learned it from:

"CA MRA Vol II.4: Clearly, to understand a GARCH model we must clarify the distinction between the unconditional variance and the conditional variance of a time series of returns. The unconditional variance is just the variance of the unconditional returns distribution, which is assumed constant over the entire data period considered. It can be thought of as the long term average variance over that period. For instance, if the model is the simple ‘returns are i.i.d.’ model then we can forget about the ordering of the returns in the sample and just estimate the sample variance using an equally weighted average of squared returns, or mean deviations of returns. This gives an estimate of the unconditional variance of the i.i.d. model. Later we will show how to estimate the unconditional variance of a GARCH model.

The conditional variance, on the other hand, will change at every point in time because it depends on the history of returns up to that point. That is, we account for the dynamic properties of returns by regarding their distribution at any point in time as being conditional on all the information up to that point. The distribution of a return at timet regards all the past returns up to and including time t−1 as being non-stochastic. We denote the information set, which is the set containing all the past returns up to and including time t−1, by I(t−1). The information set contains all the prices and returns that we can observe, like the filtration set in continuous time. We write σ^2 to denote the conditional variance at time t. This is the variance at time t, conditional on the information set. That is, we assume that everything in the information set is not random because we have an observation on it. When the conditional distributions of returns at every point in time are all normal we write: r(t) | I(t-1) ~ N(0, σ^2)" -- Carol Alexander, MRA Vol II

 

[afterworkguinness]

Thanks for the very detailed reply. I think I get it when you put it as "conditional volatility is a volatility (and, really a model of volatility) that deliberately is informed by new information; i.e., "tomorrow's volatility estimate depends on (is conditional on) certain new information." So Moving Average isn't a conditional volatility because Volatility on T+1 isn't dependent on any new information, but GARCH is a conditional volatility because Volatility on T+1 takes Volatility on T and returns on T as an input ?

 

[David Harper CFA FRM]

Yes, I agree with what you wrote. It is interesting how much depends on our definition. I happened to just check Linda Allen's Chapter 2 and I notice that she characterizes moving average (she refers to MA as STDEV) as a "conditional" model; i.e., page 42: "In figure 2.7 we compare RiskMetrics™ to STDEV. Recall the important commonalities of these methods • both methods are parametric; • both methods attempt to estimate conditional volatility; • both methods use recent historical data; • both methods apply a set of weights to past squared returns." But, to my knowledge, that's an imprecise usage. She means it in the (understandable) way that you meant it in your question: if the MA updates each day based on a new daily return, then the updated volatility estimate is "conditional" on the latest return.

.... but, technically, hers is not what we mean by conditional. IMO, we really mean something nearer to what you wrote, but i would tweak it to emphasize that volatility is not an observable metric, volatility (even and especially implied volatility) is an output of a model, so what we mean by a conditional volatility is our model has a built-in feature such that the volatility estimate is dependent (conditional on) some particular "information set." Under this view,

• If we add a day (return^2) to the MA, what we are really doing with the update is we are merely updating our sample in order to get a better estimate of the same "true" unconditional (population) variance as the day before. To the MA model, another day is just a sample update, MA doesn't really think that T+1 improves the estimate except that it is recent and/or increases the sample size. There is a sample and it estimates the true population; simple, not GARCH's notion that our true population volatility is different each day within the sample, with the implication that every day has a different true volatility!
• To GARCH, a new day is not merely a more recent sample, it is (by design) a update of the information set that produces the estimate

although i agree with your implication, i think, it's easier to distinguish in terms of how these models forecast: if we think about the estimate at today (t), and say a forecast of (t + 5) days, MA and EWMA both forecast a flat "unconditional" line (or don't really forecast depending on how you look at that): their estimate of (t+5) is today's (t); i.e., unconditional .... whereas GARCH() returns a volatility estimate that, at least, is conditional on (t). (But, again, GARCH's conditionality is more than the fact it's forecast is not flat: it does not know future returns, so it has a predictable forecast. But, as a model, it expects to incorporate those returns). Thanks,

 

[afterworkguinness]

Thanks again for the very detailed response, it really makes sense now.

 

[TurtleDaddy]

Hello, this does seem like a subtle difference that many articles/reading materials gloss over or take for granted. I have been thinking about this issue lately and was wondering whether another way to see the distinction between unconditional volatility/conditional volatility would be to couch the difference in terms of 'sample','population','estimation' and data generating processes. Do you think the following makes sense?

1.A return time series of length say 1:T is merely a sample (which we can observe) generated from a data generation process (which cannot be observed).

2.We wish to use sample data to understand population relations,come as close to the true unobservable data generation process as possible.

3.To that end we use regression/estimation techniques,compare differently specified models,find parameter estimates,test their significance and conclude the winner of the horse race to be one of these fitted models.

4.Every point in the observed sample time series can be thought of as a product/output of the true unobservable data generating process. If we assumed that the true population distribution governing the timeseries had constant variance over time, it is as if we are saying that each data point in the observed sample series is a random draw from a single distribution (mean=mu,variance=v) that does not change through time. If we assumed that the true distribution had time varying volatility,every point in the observed time series of returns would have been drawn from a population distribution that has variance (v1) at t1 and v2 at t2 and so on.

..sorry if i mangled the explanation a bit...do you guys think it makes sense?
Thank you.