A question on volatility

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A question on volatility

Megh
Dear all, I was trying to understand the correlation among the volatilities in different financial market, however am in dilemma what could be the rightful and acceptable-to-everyone approach. I thought to estimate the volatilities of individual markets using some GARCH modeling, then just calculate the correlation coefficient on the estimated time series of estimated daily volatilities. 

Is it correct approach to understand the correlation? Can somebody point me any online paper or any idea on the same?

Thanks for your time.

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Re: A question on volatility

Paul Ringseth
Hi:

You really need to jointly estimate the correlations with the variances.  The easiest technique (but not the best) is Orthogonal GARCH from Carl Alexander's papers (http://www.carolalexander.org/publish/download/DiscussionPapers/OrthogonalGARCH_Primer.pdf ).  Recently Engle has recommended a factor DCC-GARCH variant using a heuristic, he calls the MacGyver technique, for large covariance matrices (http://papers.ssrn.com/sol3/papers.cfm?abstract_id=1293628 ).    Then Engle, Shephard and Sheppard came up with an exceptionally interesting technique for fitting all parameters in any large covariance matrix http://www.economics.ox.ac.uk/Research/wp/pdf/paper403.pdf -- the estimator is essentially the sum of the quasi-MLE's of all pairs.  Also you should check out Engle's new book -- Anticipating Correlations ( http://press.princeton.edu/titles/8768.html ).    

Whatever you end up doing, you should backtest and compare to published results, for example at Engle's volatility lab -- http://vlab.stern.nyu.edu/analysis .

But as long as the dimensionality of the desired correlation / covariance matrix is not too large ( <= 16 should be ok ), a straightforward DCC-GARCH fit should work.  Here's some R code:

http://www.r-project.org/conferences/useR-2008/slides/Nakatani.pdf 

Cheers -- Paul

-----Original Message-----
From: [hidden email] [mailto:[hidden email]] On Behalf Of Megh Dal
Sent: Wednesday, October 05, 2011 12:15 PM
To: [hidden email]
Subject: [R-SIG-Finance] A question on volatility

Dear all, I was trying to understand the correlation among the volatilities in different financial market, however am in dilemma what could be the rightful and acceptable-to-everyone approach. I thought to estimate the volatilities of individual markets using some GARCH modeling, then just calculate the correlation coefficient on the estimated time series of estimated daily volatilities. 

Is it correct approach to understand the correlation? Can somebody point me any online paper or any idea on the same?

Thanks for your time.

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-- Subscriber-posting only. If you want to post, subscribe first.
-- Also note that this is not the r-help list where general R questions should go.

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Re: A question on volatility

Patrick Burns-2
Paul,

If my understanding of Megh's question is correct,
then you've misinterpreted it.  I think the
correlations that are being sought are the correlations
between the volatilities of the assets, not the
correlations of the asset returns.

In any case, I'll attempt to give a bit of an answer
to the question as I understand it.

I'm uneasy about correlation of volatilities because
they are quite skewed.  Certainly favor rank correlations
over Pearson correlation.

Somewhere in Engle's body of work is a paper (or more)
on the transmission of volatility.  I don't recall
at all what the technique was, and vaguely remember
it being a mildly satisfying answer.

On 05/10/2011 21:10, Paul Ringseth wrote:

> Hi:
>
> You really need to jointly estimate the correlations with the variances.  The easiest technique (but not the best) is Orthogonal GARCH from Carl Alexander's papers (http://www.carolalexander.org/publish/download/DiscussionPapers/OrthogonalGARCH_Primer.pdf ).  Recently Engle has recommended a factor DCC-GARCH variant using a heuristic, he calls the MacGyver technique, for large covariance matrices (http://papers.ssrn.com/sol3/papers.cfm?abstract_id=1293628 ).    Then Engle, Shephard and Sheppard came up with an exceptionally interesting technique for fitting all parameters in any large covariance matrix http://www.economics.ox.ac.uk/Research/wp/pdf/paper403.pdf -- the estimator is essentially the sum of the quasi-MLE's of all pairs.  Also you should check out Engle's new book -- Anticipating Correlations ( http://press.princeton.edu/titles/8768.html ).
>
> Whatever you end up doing, you should backtest and compare to published results, for example at Engle's volatility lab -- http://vlab.stern.nyu.edu/analysis .
>
> But as long as the dimensionality of the desired correlation / covariance matrix is not too large (<= 16 should be ok ), a straightforward DCC-GARCH fit should work.  Here's some R code:
>
> http://www.r-project.org/conferences/useR-2008/slides/Nakatani.pdf
>
> Cheers -- Paul
>
> -----Original Message-----
> From: [hidden email] [mailto:[hidden email]] On Behalf Of Megh Dal
> Sent: Wednesday, October 05, 2011 12:15 PM
> To: [hidden email]
> Subject: [R-SIG-Finance] A question on volatility
>
> Dear all, I was trying to understand the correlation among the volatilities in different financial market, however am in dilemma what could be the rightful and acceptable-to-everyone approach. I thought to estimate the volatilities of individual markets using some GARCH modeling, then just calculate the correlation coefficient on the estimated time series of estimated daily volatilities.
>
> Is it correct approach to understand the correlation? Can somebody point me any online paper or any idea on the same?
>
> Thanks for your time.
>
> _______________________________________________
> [hidden email] mailing list https://stat.ethz.ch/mailman/listinfo/r-sig-finance
> -- Subscriber-posting only. If you want to post, subscribe first.
> -- Also note that this is not the r-help list where general R questions should go.
>
> _______________________________________________
> [hidden email] mailing list
> https://stat.ethz.ch/mailman/listinfo/r-sig-finance
> -- Subscriber-posting only. If you want to post, subscribe first.
> -- Also note that this is not the r-help list where general R questions should go.
>

--
Patrick Burns
[hidden email]
http://www.burns-stat.com
http://www.portfolioprobe.com/blog
twitter: @portfolioprobe

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Re: A question on volatility

Eric Zivot
I agree with Pat. Time varying correlations in a multivariate GARCH model
are different from the correlations between volatility series. Because
volatility is "unobservable" (i.e, except for special cases like the VIX)
and derived measures like implied volatility are model based (e.g. derived
from Black-Scholes) it is not straightforward to define and measure
correlations between volatilities. One model-based approach in which
volatility is a random variable is the stochastic volatility model. One can
build multivariate models in which the correlation to volatility shocks is
parameterized (but this is not the correlation between volatilities). GARCH
models produce very noisy estimate of volatility and so the correlations
computed from GARCH volatilities are likely to be very  noisy as well. A
better approach would be to compute volatilities using intra-day high
frequency data (e.g. realized volatility) - see the realized package. This
would give you much more precise estimates of volatility. Then the problem
would be to model the correlation between the observed volatilities. For
example, simple EWMAs. One could even consider a simple vector
autoregressive model for a multi-variate time series of volatilities. This
is what Andersen, Bollerslev, Diebold and Labys did in their Econometrica
paper. One potential problem is that the realized volatility series tend to
be non-stationary. Just some thoughts.

-----Original Message-----
From: [hidden email]
[mailto:[hidden email]] On Behalf Of Patrick Burns
Sent: Wednesday, October 05, 2011 1:39 PM
To: [hidden email]
Subject: Re: [R-SIG-Finance] A question on volatility

Paul,

If my understanding of Megh's question is correct,
then you've misinterpreted it.  I think the
correlations that are being sought are the correlations
between the volatilities of the assets, not the
correlations of the asset returns.

In any case, I'll attempt to give a bit of an answer
to the question as I understand it.

I'm uneasy about correlation of volatilities because
they are quite skewed.  Certainly favor rank correlations
over Pearson correlation.

Somewhere in Engle's body of work is a paper (or more)
on the transmission of volatility.  I don't recall
at all what the technique was, and vaguely remember
it being a mildly satisfying answer.

On 05/10/2011 21:10, Paul Ringseth wrote:
> Hi:
>
> You really need to jointly estimate the correlations with the variances.
The easiest technique (but not the best) is Orthogonal GARCH from Carl
Alexander's papers
(http://www.carolalexander.org/publish/download/DiscussionPapers/OrthogonalG
ARCH_Primer.pdf ).  Recently Engle has recommended a factor DCC-GARCH
variant using a heuristic, he calls the MacGyver technique, for large
covariance matrices
(http://papers.ssrn.com/sol3/papers.cfm?abstract_id=1293628 ).    Then
Engle, Shephard and Sheppard came up with an exceptionally interesting
technique for fitting all parameters in any large covariance matrix
http://www.economics.ox.ac.uk/Research/wp/pdf/paper403.pdf -- the estimator
is essentially the sum of the quasi-MLE's of all pairs.  Also you should
check out Engle's new book -- Anticipating Correlations (
http://press.princeton.edu/titles/8768.html ).
>
> Whatever you end up doing, you should backtest and compare to published
results, for example at Engle's volatility lab --
http://vlab.stern.nyu.edu/analysis .
>
> But as long as the dimensionality of the desired correlation / covariance
matrix is not too large (<= 16 should be ok ), a straightforward DCC-GARCH
fit should work.  Here's some R code:
>
> http://www.r-project.org/conferences/useR-2008/slides/Nakatani.pdf
>
> Cheers -- Paul
>
> -----Original Message-----
> From: [hidden email]
[mailto:[hidden email]] On Behalf Of Megh Dal
> Sent: Wednesday, October 05, 2011 12:15 PM
> To: [hidden email]
> Subject: [R-SIG-Finance] A question on volatility
>
> Dear all, I was trying to understand the correlation among the
volatilities in different financial market, however am in dilemma what could
be the rightful and acceptable-to-everyone approach. I thought to estimate
the volatilities of individual markets using some GARCH modeling, then just
calculate the correlation coefficient on the estimated time series of
estimated daily volatilities.
>
> Is it correct approach to understand the correlation? Can somebody point
me any online paper or any idea on the same?
>
> Thanks for your time.
>
> _______________________________________________
> [hidden email] mailing list
https://stat.ethz.ch/mailman/listinfo/r-sig-finance
> -- Subscriber-posting only. If you want to post, subscribe first.
> -- Also note that this is not the r-help list where general R questions
should go.
>
> _______________________________________________
> [hidden email] mailing list
> https://stat.ethz.ch/mailman/listinfo/r-sig-finance
> -- Subscriber-posting only. If you want to post, subscribe first.
> -- Also note that this is not the r-help list where general R questions
should go.
>

--
Patrick Burns
[hidden email]
http://www.burns-stat.com
http://www.portfolioprobe.com/blog
twitter: @portfolioprobe

_______________________________________________
[hidden email] mailing list
https://stat.ethz.ch/mailman/listinfo/r-sig-finance
-- Subscriber-posting only. If you want to post, subscribe first.
-- Also note that this is not the r-help list where general R questions
should go.

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Re: A question on volatility

Paul Ringseth
Yeah, you're all probably right.  But corr of vols are 4-th moment components.  I wouldn't think you could get anything meaningful from just returns time series.  If you could compare markets that trade variance swaps, then it's reasonable -- though you'd want to use on-the-run prices to prevent too much implicit vol bias.  But I think you still get good calibration by incorporating derivative prices -- work by Carr and Madan et al  shows that the usual time-conditional expectation of most reasonable functions of an underlying security can be represented as a (continuous) sum of European options on the same underlying.  I'd assume a Bayesian model here and calibrate using an adaptive MCMC, like Holenstein & Doucet's particle MCMC (viz.,  using a particle filter/learning pass to estimate the proposal distribution for Metropolis-Hastings.)   Perhaps, instead of assuming a multivariate stochastic vol model to be calibrated, I think I might want to represent vector variance as a!
  dynamic latent factor model -- observables could be traded variance swaps, Carr and Madan estimates from traded options and as Eric mentioned realized variance along the lines of Anderson et al.   Maybe try using a Gaussian process variant -- treed GPs for a clean non-parametric, non-stationary regression?  Or I don't know, perhaps a mitigation for non-stationarity could be a decomposition into cointegrated factors along the lines of http://ba.stat.cmu.edu/journal/2010/vol05/issue03/peters.pdf .    Damn -- I'm having too much fun here, I want to do it myself now.   And I'm getting way too far away from reality and I'm fairly new to this area.  ;-)  

But I have a question.  Assuming a straight returns time series approach using intraday etc., wouldn't you still need to do a multivariate covariance model?  The univariate time series will be correlated and therefore wouldn't independent calibration introduce significant error into any subsequent 4-moment calculation?  

Cheers -- Paul

-----Original Message-----
From: [hidden email] [mailto:[hidden email]] On Behalf Of Eric Zivot
Sent: Wednesday, October 05, 2011 1:53 PM
To: 'Patrick Burns'; [hidden email]
Subject: Re: [R-SIG-Finance] A question on volatility

I agree with Pat. Time varying correlations in a multivariate GARCH model are different from the correlations between volatility series. Because volatility is "unobservable" (i.e, except for special cases like the VIX) and derived measures like implied volatility are model based (e.g. derived from Black-Scholes) it is not straightforward to define and measure correlations between volatilities. One model-based approach in which volatility is a random variable is the stochastic volatility model. One can build multivariate models in which the correlation to volatility shocks is parameterized (but this is not the correlation between volatilities). GARCH models produce very noisy estimate of volatility and so the correlations computed from GARCH volatilities are likely to be very  noisy as well. A better approach would be to compute volatilities using intra-day high frequency data (e.g. realized volatility) - see the realized package. This would give you much more precise estimat!
 es of volatility. Then the problem would be to model the correlation between the observed volatilities. For example, simple EWMAs. One could even consider a simple vector autoregressive model for a multi-variate time series of volatilities. This is what Andersen, Bollerslev, Diebold and Labys did in their Econometrica paper. One potential problem is that the realized volatility series tend to be non-stationary. Just some thoughts.

-----Original Message-----
From: [hidden email]
[mailto:[hidden email]] On Behalf Of Patrick Burns
Sent: Wednesday, October 05, 2011 1:39 PM
To: [hidden email]
Subject: Re: [R-SIG-Finance] A question on volatility

Paul,

If my understanding of Megh's question is correct, then you've misinterpreted it.  I think the correlations that are being sought are the correlations between the volatilities of the assets, not the correlations of the asset returns.

In any case, I'll attempt to give a bit of an answer to the question as I understand it.

I'm uneasy about correlation of volatilities because they are quite skewed.  Certainly favor rank correlations over Pearson correlation.

Somewhere in Engle's body of work is a paper (or more) on the transmission of volatility.  I don't recall at all what the technique was, and vaguely remember it being a mildly satisfying answer.

On 05/10/2011 21:10, Paul Ringseth wrote:
> Hi:
>
> You really need to jointly estimate the correlations with the variances.
The easiest technique (but not the best) is Orthogonal GARCH from Carl Alexander's papers (http://www.carolalexander.org/publish/download/DiscussionPapers/OrthogonalG
ARCH_Primer.pdf ).  Recently Engle has recommended a factor DCC-GARCH variant using a heuristic, he calls the MacGyver technique, for large covariance matrices
(http://papers.ssrn.com/sol3/papers.cfm?abstract_id=1293628 ).    Then
Engle, Shephard and Sheppard came up with an exceptionally interesting technique for fitting all parameters in any large covariance matrix http://www.economics.ox.ac.uk/Research/wp/pdf/paper403.pdf -- the estimator is essentially the sum of the quasi-MLE's of all pairs.  Also you should check out Engle's new book -- Anticipating Correlations ( http://press.princeton.edu/titles/8768.html ).
>
> Whatever you end up doing, you should backtest and compare to
> published
results, for example at Engle's volatility lab -- http://vlab.stern.nyu.edu/analysis .
>
> But as long as the dimensionality of the desired correlation /
> covariance
matrix is not too large (<= 16 should be ok ), a straightforward DCC-GARCH fit should work.  Here's some R code:
>
> http://www.r-project.org/conferences/useR-2008/slides/Nakatani.pdf
>
> Cheers -- Paul
>
> -----Original Message-----
> From: [hidden email]
[mailto:[hidden email]] On Behalf Of Megh Dal
> Sent: Wednesday, October 05, 2011 12:15 PM
> To: [hidden email]
> Subject: [R-SIG-Finance] A question on volatility
>
> Dear all, I was trying to understand the correlation among the
volatilities in different financial market, however am in dilemma what could be the rightful and acceptable-to-everyone approach. I thought to estimate the volatilities of individual markets using some GARCH modeling, then just calculate the correlation coefficient on the estimated time series of estimated daily volatilities.
>
> Is it correct approach to understand the correlation? Can somebody
> point
me any online paper or any idea on the same?
>
> Thanks for your time.
>
> _______________________________________________
> [hidden email] mailing list
https://stat.ethz.ch/mailman/listinfo/r-sig-finance
> -- Subscriber-posting only. If you want to post, subscribe first.
> -- Also note that this is not the r-help list where general R
> questions
should go.
>
> _______________________________________________
> [hidden email] mailing list
> https://stat.ethz.ch/mailman/listinfo/r-sig-finance
> -- Subscriber-posting only. If you want to post, subscribe first.
> -- Also note that this is not the r-help list where general R
> questions
should go.
>

--
Patrick Burns
[hidden email]
http://www.burns-stat.com
http://www.portfolioprobe.com/blog
twitter: @portfolioprobe

_______________________________________________
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-- Subscriber-posting only. If you want to post, subscribe first.
-- Also note that this is not the r-help list where general R questions should go.

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Re: A question on volatility

Adrian Trapletti-2
In reply to this post by Megh
Hi Megh,

As a practitioner I would use something like

x1 <- get.hist.quote(instrument = "^gspc", start = "1990-01-01")
x2 <- get.hist.quote(instrument = "^dji", start = "1990-01-01")  ## both
need to be synchronized in time

r1 <- log(x1[, 2])-log(x1[, 3]) ## range as proxy for vola
r2 <- log(x2[, 2])-log(x2[, 3]) ## not ()^2 to avoid possibly non-finite
fourth moment

r <- merge(r1, r2)

plot(r)

rcor <- rollapply(r, width = 250, FUN = function(z) cor(z[, 1], z[, 2],
method = "pearson"),
                  by.column = FALSE, align = "left") ## method !=
"pearson" for rank correlations

plot(rcor)

as a starting point. As a next step I would use a better proxy for vola
from the zoo of realized vola based estimators.

Best regards
Adrian

> Dear all, I was trying to understand the correlation among the?volatilities?in different financial market, however am in dilemma what could be the rightful and acceptable-to-everyone approach. I thought to estimate the volatilities of?individual?markets using some GARCH modeling, then just calculate the correlation coefficient on the estimated time series of estimated daily volatilities.?
>
> Is it correct approach to understand the correlation? Can somebody point me any online paper or any idea on the same?
>
> Thanks for your time.

--
Dr. Adrian Trapletti
Steinstrasse 9b
8610 Uster
Switzerland

Phone : +41 (0) 44 9945630
Mobile : +41 (0) 79 1037131

Email : [hidden email]

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Re: A question on volatility

Adrian Trapletti-2
I forgot to mention: Usually it is better to work with the logarithmic
vola proxy, i.e., use

r<-log(r)

after the merge.

Best regards
Adrian

Adrian Trapletti wrote:

> Hi Megh,
>
> As a practitioner I would use something like
>
> x1 <- get.hist.quote(instrument = "^gspc", start = "1990-01-01")
> x2 <- get.hist.quote(instrument = "^dji", start = "1990-01-01")  ##
> both need to be synchronized in time
>
> r1 <- log(x1[, 2])-log(x1[, 3]) ## range as proxy for vola
> r2 <- log(x2[, 2])-log(x2[, 3]) ## not ()^2 to avoid possibly
> non-finite fourth moment
>
> r <- merge(r1, r2)
>
> plot(r)
>
> rcor <- rollapply(r, width = 250, FUN = function(z) cor(z[, 1], z[,
> 2], method = "pearson"),
>                  by.column = FALSE, align = "left") ## method !=
> "pearson" for rank correlations
>
> plot(rcor)
>
> as a starting point. As a next step I would use a better proxy for
> vola from the zoo of realized vola based estimators.
>
> Best regards
> Adrian
>
>> Dear all, I was trying to understand the correlation among
>> the?volatilities?in different financial market, however am in dilemma
>> what could be the rightful and acceptable-to-everyone approach. I
>> thought to estimate the volatilities of?individual?markets using some
>> GARCH modeling, then just calculate the correlation coefficient on
>> the estimated time series of estimated daily volatilities.?
>>
>> Is it correct approach to understand the correlation? Can somebody
>> point me any online paper or any idea on the same?
>>
>> Thanks for your time.
>

--
Dr. Adrian Trapletti
Steinstrasse 9b
8610 Uster
Switzerland

Phone : +41 (0) 44 9945630
Mobile : +41 (0) 79 1037131

Email : [hidden email]

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