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What makes GARCH(1,1) so prevalent in modeling volatility, especially in academia?
What does this model offer that makes it significantly better than the others?

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    $\begingroup$ simple, easy to use and fast $\endgroup$ – pyCthon Feb 13 '13 at 3:36
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    $\begingroup$ Freddy is right .. and I think we have to change the wording of the question ... I don't dare to edit it, but would you Wang Weinan? GARCH does not predict prices but (if at all) volatility. $\endgroup$ – Ric Feb 13 '13 at 12:54
  • $\begingroup$ See quant.stackexchange.com/questions/9351/… $\endgroup$ – JoleT Jan 5 '16 at 5:44
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First, Garch models stochastic volatility. Thus its use should be limited to estimating the volatility component. The difference in some of the volatility models is the assumption made of the random variance process components.

I believe it has been popular because it is an extension of the ARCH family of models and it is relatively easy to setup and calibrate because it relies on past observations. Think of it this way: If you are to pinpoint your PhD dissertation topic would you take the risk to delve into deriving a new model, taking the risk you utterly fail and get nowhere over your x years of research or are you more likely to work on extensions or improvements of what currently exists? The same applies here GARCH is an extension of ARCH and there are numerous extensions of GARCH as well, such as GARCH-M, IGARCH, NGARCH...

I disagree with cdcaveman that it is the best model out there because it suffers from major deficiencies. Every model makes assumptions but there are better models out there for sure which is why I do not know of too many volatility traders that rely primarily on GARCH models in their quest to forecast volatility.

Deficiencies:

  • It depends heavily on past variances
  • The definition of "long-term variance" is at best arbitrary
  • making the assumption of the randomness originating from a normal distribution
  • The weights are just a result of optimization (MLE or other optimizers) of past data and make up the bulk of the calibration process. Volatility dynamics are changing in the same way as most other inputs to asset prices are dynamic thus making the assumption that an optimization of past variances, which results in the weights that make up the bulk of the current variance estimate, will yield anything that produces excess returns is a horrible assumption, imho.
  • Though most multivariate models can get quickly complex, multivariate GARCH can be tricky in regards to specifying the covariances (VECH or BEKK come to mind). (credit to Bob Jansen for pointing out this aspect of GARCH).

Volatility models that are originating from trading desks and that are rarely to be found in academic paper or the public domain often

  • do not make a normal distribution assumption of the variance dynamics
  • heavily incorporate regime shifts
  • rarely rely on functions of linear nature
  • incorporate correlation structures with other asset classes and even non-price return related inputs.

In summary, its a neat model to output something to show off within minutes. Whether the results are usable is an entirely different question and again I do not know of too many pure index vol traders who embrace GARCH.

Edit:

A look at the SABR model (or dynamic SABR) might be beneficial when searching for better models, though the "backbone" dynamics of the SABR model are more applicable for some derivatives than others.

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  • $\begingroup$ Also, I would like to know why are people tend to use GARCH(1,1) but not other values for p,q? $\endgroup$ – Jack Feb 13 '13 at 4:54
  • $\begingroup$ Erroneous simplicity assumptions? After all, we all think most recent events are more relevant than the ones longer time ago. $\endgroup$ – Matt Feb 13 '13 at 6:02
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    $\begingroup$ Another drawback of GARCH is that it is impractical to estimate multivariate GARCH so you can't really use it with more than a small number of assets. $\endgroup$ – Bob Jansen Feb 13 '13 at 7:23
  • $\begingroup$ Thanks! great response..... i don't use Garch to trade... think about what garch will tell you if you looked at it right before earnings.. I like how you talk about relative pricing.. a trader or maket maker would look for relative pricing anomalies.. i think if foward vol in the back of the term structure is priced the best .. that is best to use as a basis.. buck vega or root vega.. $\endgroup$ – cdcaveman Feb 13 '13 at 7:27
  • $\begingroup$ @BobJansen, agree, though most multivariate models get quickly very complex. If you dont mind then I will add it to the list of shortcomings though. $\endgroup$ – Matt Feb 13 '13 at 7:58
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Let me start with a disclaimer that I have no interest in promoting GARCH models. However, I am aware of their history, their capabilities and some practical aspects of using them. That helps me come up with a few points to answer your question (Why is GARCH(1,1) so popular?):

  1. GARCH is inspired by ARMA, a classic time series model, and most likely the most popular one. ARMA being so heavily researched, well understood (both from theoretical and computational points of view) and broadly used paved a solid way for the advent of GARCH.
  2. GARCH(1,1) is very simple, yet it delivers good fit and accurate predictions; while this may not be immediately obvious by looking at $R^2$ values, it is actually the case; see Andersen & Bollerslev "Answering the skeptics: Yes, standard volatility models do provide accurate forecasts" (1998).
  3. GARCH is able to reproduce some of the stylized facts of asset returns, especially volatility clustering and heavy tails.
  4. GARCH(1,1) is hard to beat; see Hansen & Lunde "A forecast comparison of volatility models: does anything beat a GARCH (1, 1)?" (2005). Or isn't it? See Alexios Ghalanos blog post "Does anything NOT beat the GARCH(1,1)?".

To respond to some of Matt Wolf's points:

  1. Garch models stochastic volatility
    Not 100% correct. GARCH specifies a deterministic equation for volatility; volatility at time $t$ is completely determined by information as of $t-1$. Compare to stochastic volatility models; some of them just add a stochastic term to the deterministic GARCH equation to make it stochastic.
  2. making the assumption of the randomness originating from a normal distribution
    There is no mandatory normality assumption, the choice of the distribution to be assumed is free. Check e.g. the variety of distributions available in "rugarch" package in R.
  3. Though most multivariate models can get quickly complex, multivariate GARCH can be tricky in regards to specifying the covariances (VECH or BEKK come to mind)
    While that is correct, the issue has been addressed and there are a number of multivariate GARCH models (such as DCC, to give just one example) that accomodate high-dimensional time series easily. See e.g. Bauwens et al. "Multivariate GARCH models: a survey" (2006) or Silvennoinen & Terasvirta "Multivariate GARCH models" (2009). For vast-dimensional case, see Engle et al. "Fitting vast dimensional time-varying covariance models" (2008).
  4. [Other models, but not GARCH(1,1)] heavily incorporate regime shifts
    This is possible with GARCH class of models in general, but specifically vanilla GARCH(1,1) fails here indeed.
  5. [Other models, but not GARCH(1,1)] incorporate correlation structures with other asset classes and even non-price return related inputs
    Again, this is possible with GARCH class of models, but specifically vanilla GARCH(1,1) fails here indeed.

Now, let me reiterate that I am not trying to promote GARCH models; I am just trying to show why they are so popular.

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    $\begingroup$ Richard, thanks for the added points. Re stochastic volatility, you are correct, plain Garch does not model stochastic volatility for lack of stochastic component. But it can be incorporated. Regarding your other points, I mostly agree with your notions, though those are all "add-ons" and not properties of the standard Garch model. $\endgroup$ – Matt Nov 9 '18 at 23:04
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Volatility tends to cluster and mean revert... garch incorporated the behavior the best..... Ema doesn't do that.. its a step beyond exponential smoothing and weighting to replicate vol behavior

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  • $\begingroup$ Garch by far does not incorporate the behavior the best. Thats a pretty bold statement at best. $\endgroup$ – Matt Feb 13 '13 at 4:09
  • $\begingroup$ Freddy is correct, there are better "general" models known to the academic communities and that is still far from what is state-of the art in the industry $\endgroup$ – pyCthon Feb 13 '13 at 4:22
  • $\begingroup$ great.. so tell the guy about the other models.. i was just speculating as to why the guy got that impression.. just trying to contribute :) $\endgroup$ – cdcaveman Feb 13 '13 at 7:32
  • $\begingroup$ Agree with @cdcaveman. unlike other models, GARCH is capable of catching persistency, which allows for prediction $\endgroup$ – Anya Mar 6 '13 at 10:20

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