Parametric and non-parametric approaches to volatility

[sathyat]

Hi David,
Can you please throw some light on the classification - parametric and non-parametric ?

As per your answer for Question 4 for Quant A (http://forum.bionicturtle.com/viewthread/226), you say -

"non-parametric approaches do not depend on a distributional assumption "

What distribution assumptions are we making for EWMA and GARCH(1, 1) ? Are we not making the same assumptions for MDE ?

Thanks,
Sathya

 

[David Harper CFA FRM]

Hi Sathya,

Big topic. I got an email query a few days ago asking about L. Allen's figure 2.4 (the normal curve superimposed over the histogram of interest rates); this figure 2.4 relates to your question. The histogram of actual returns (any data!) is clunky, not smooth. A parametric approach uses the data as an excuse to find a nice-fitting curve - a curve described by a distribution with parameters (parametric distribution). It tries not to offend the data. Once found, the data is discarded. The point of a parametric normal approach is to get the distribution then discard the data. If you look at Allen's figure 2.4, the parametric normal is the bell shaped line. Remove the data, and you only need the variance and return to apply. But it will be rough and, especially, wrong in the tails.

So, the EMWA & GARCH(1,1) that we study are parametric because they assume returns are conditionally normal (or the standardized residuals are normal). If you look at the EWMA/GARCH (as classes of ARCH where the key assumption is normal returns), they are updating (new "innovations" = new returns) in order to improve the new variance - but it is still a variance based on the "parametric" assumption of normality. The whole benefit here is efficiency at the cost of accuracy. Also note: normal is but one parametric distribution. Normal implies parametric but parametric implies any of several distribution.

Non-parametric, as Kevin Dowd somewhere said, tries to avoid curves and let the data speak for itself. So the historical simulation has no a priori shape: it is what it is. Historical sim. is classically nonparametric because it cannot be summarized by a curve. There are in-between, semi-parametric methods, and frankly am not sure why Allen calls that version of MDE non-parametric rather than semi-parametric (but I am not an expert on MDE).

But, no matter, in the context of the L.Allen chapter non-parametric means "not parametric" and MDE (this version, there are variations on density estimation) fits because it does not assume normal/lognormal/any particular distribution that can be nicely summarized into a PDF/PMF. Rather, MDE tries to "let the data speak for itself" via the kernel function. Hopefully you can see that by weighting the historical returns according to a vector (e.g., the economic state is a conditioning variable. Does today's economy match the economy on date x/x/x? yes, okay, give that squared return a larger weight), this MDE is not behaving according to any distribution that we might mathematically summarize.

But sorry for the length (if i'd had more time, i would have written less!) and, on the MDE, it is a specialist area so I'll defer to a specialist...thanks, David

 

[rocky420]

Hi David,

Had a couple of questions w.r.t your spreadsheet.

For the variance on n and n-1 columns; why do we multiply the weights and the period return to get the variance. Is this from the EWMA formula? I am little confused.

Secondly, to get the lagged variance i.e. sigma squared of n-1 ; why do we need to sum the variances from n-2 to n-11. Confused again.

 

[David Harper CFA FRM]

Hi rocky420,

The weighted refers to either EWMA or GARCH(1,1).
Might be easier to start with idea of plain-old moving average (what Linda allen call STDEV; i.e., std deviation)
I think the weights are easier to follow if you see that standard deviation is weighted, too, but the weights just happen to be the same!

That is, STDEV = (1/n)Sigma[squared returns].
In words, the moving average (Allen's STDEV)is the average of the last (n) squared returns.
Implicitly, that is equal weight to each squared return; e.g., if 10 returns, then
10% * squared return of n-1 + 10% of squared return of n-2 + ....etc

So the EWMA just tweaks that by giving greater weight to most recent; e.g., if lambda = 90%, then
10% to (n-1) squared return, (10%)(90%) to (n-1) squared return

see how EWMA is STDEV but only the weights are changed to be exponentially declining?

okay, so to the second part:
you can see either EWMA/GARCH have infinitely declining weights.
Conveniently (See Linda Allen Ch 2), the infinite series can be reduced to the recursive series:

vol estimate = lambda(last variance) + (1-lambda)(last return^2)
but here, in order to get the last variance, I resort to the elaborate series. Which, btw, is rounded, I only take it back 10 or so data points, so it's not perfect.

this luckily is the mathematical reduction of the series with declining weights. You'll see on the XLS, there is a EWMA (recursive) and EMWA (elaborate)? This just shows they are equivalent.

Let me know if that doesn't makes sense...

David

 

[Kaiser]

Hi,

Could you confirm that GARCH(1,1) is not a parametric approach, specially when the question 324.1D confirms that EWMA is parametric and we know that EWMA is a special case of GARCH


323.1. B.
In regard to (C), normal GARCH(1,1) assumes conditional returns are normal but also, as a Monte Carlo Simulation, this is not a parametric approach.

Bionic Turtle FRM Practice Questions Reading 21 Allen, Understanding Market, Credit and Operational Risk: The Value at Risk Approach

323.1. Analyst Peter observes that conditional equity returns exhibit leptokurtosis (i.e., heavytail) and negative skewness. Due to time constraints, Peter must use a parametric (analytical) value at risk (VaR) model. Which of the following models is most likely able to model his conditional returns?
a) Normal VaR; i.e., basic so-called delta-normal VaR where portfolio return is a linear function of asset returns that are normal
b) Normal mixture VaR; i.e., portfolio return is a linear function of asset returns that are parametric but characterized by a mixture of normal ("normal mixture") densities
c) Normal GARCH (1,1) VaR model; i.e., Monte Carlo simulation with GARCH volatility
d) Student's t GARCH (1,1) VaR model; i.e., portfolio return is a linear function of asset returns that are characterized (parametrically) by student's t distribution

324.1. D. Implied volatility uses current prices. In regard to (A), parametric approaches tend to use the historical returns to inform (fit) the parameters, at least. In regard to (B), EWMA is parametric and hybrid is non-parametric


Rgds,

 

[David Harper CFA FRM]

@Kaiser duplicate, already responded to here @ https://forum.bionicturtle.com/threads/parametric-distribution.9048/#post-37704