Volatility and Risk Institute

NYU's Volatility Institute is expanding into the Volatility and Risk Institute (VRI). The four key initiatives are Climate Risk (run by Johannes Stroebel), Cyber Risk (run by Randal Milch), Financial Risk (run by Viral Acharya), and Geopolitical Risk (run by Thomas Philippon). Details here. This is a big deal. Great to see climate given such obvious and appropriate prominence. And notice how interconnected are climate, financial, and geopolitical risks.

The following is adapted from an email from Rob Engle and Dick Berner:

The Volatility Institute and its V-lab have, for the past decade, assessed risk through the lens of financial volatility, providing real-time measurement and forecasts of volatility and correlations for a wide spectrum of financial assets, and SRISK, a powerful measure of the resilience of the global financial system. Adopting an interdisciplinary approach, the VRI will build on that foundation to better assess newly emerging nonfinancial and financial risks facing today’s business leaders and policymakers, including climate-related, cyber/operational and geopolitical risks, as well as the interplay among them. 

The VRI will be co-directed by two NYU Stern faculty: Nobel Laureate Robert Engle, Michael Armellino Professor of Management and Financial Services and creator of the V-lab; and Richard Berner, Professor of Management Practice and former Director of the Office of Financial Research, established by the Dodd–Frank Wall Street Reform and Consumer Protection Act to help promote financial stability by delivering high-quality financial data, standards and analysis to policymakers and the public. 

The VRI will serve as the designated hub to facilitate, support and promote risk-related research, and external and internal engagement among scholars, practitioners and policymakers. To realize its interdisciplinary potential, the VRI will engage the expertise of faculty across New York University, including at the Courant Institute of Mathematical Sciences, Law School, Tandon School of Engineering, Wagner School of Public Policy and Wilf Family Department of Politics in the Faculty of Arts & Science. 

Hedging Realized vs. Expected Volatility

Not all conferences can be above average, let alone in the extreme right tail of the distribution, so it's wonderful when it happens, as with last week's AP conference. Fine papers all -- timely, thought provoking, and empirically sophisticated.  Thanks to Jan Eberly and Konstantin Milbradt for assembling the program, here (including links to papers). 

I keep thinking about the Dew-Becker-Giglio-Kelly paper. For returns r, they produce evidence that (1) investors are willing to pay a lot to insure against movements in realized volatility, r^2_{t}, but (2) investors are not willing to pay to insure against movements in expected future realized volatility (conditional variance), E_t(r^2_{t+1} | I_t). On the one hand, as a realized volatility guy I'm really intrigued by (1). On the other hand, it seems hard to reconcile (1) and (2), a concern that was raised at the meeting. On the third hand, maybe it's not so hard.  Hmmm...

The Housing Risk Premium is Huge

Earlier I blogged on Jorda et al.'s fascinating paper, "The Rate of Return on Everything".  Now they're putting their rich dataset to good use.  Check out the new paper, NBER w.p. 25653.

The Total Risk Premium Puzzle
Òscar Jordà, Moritz Schularick, and Alan M. Taylor

Abstract:
The risk premium puzzle is worse than you think. Using a new database for the U.S. and 15 other advanced economies from 1870 to the present that includes housing as well as equity returns (to capture the full risky capital portfolio of the representative agent), standard calculations using returns to total wealth and consumption show that: housing returns in the long run are comparable to those of equities, and yet housing returns have lower volatility and lower covariance with consumption growth than equities. The same applies to a weighted total-wealth portfolio, and over a range of horizons. As a result, the implied risk aversion parameters for housing wealth and total wealth are even larger than those for equities, often by a factor of 2 or more. We find that more exotic models cannot resolve these even bigger puzzles, and we see little role for limited participation, idiosyncratic housing risk, transaction costs, or liquidity premiums. 

Long Memory / Scaling Laws in Return Volatility

The 25-year accumulation of evidence for long memory / fractional integration / self-similarity / scaling laws in financial asset return volatility continues unabated.  For the latest see this nice new paper from Bank of Portugal, in particular its key Table 6. Of course the interval estimates of the fractional integration parameter "d" are massively far from both 0 and 1 -- that's the well-known long memory. But what's new and interesting is the systematic difference in the intervals depending on whether one uses absolute or range-based volatility. The absolute d intervals tend to be completely below 1/2 (0<d<1/2 corresponds to covariance-stationary dynamics), whereas the range-based d intervals tend to include 1/2 (1/2<d<1 corresponds to mean-reverting but not covariance- stationary dynamics, due to infinite unconditional variance). 

Realized vol based on the range is less noisy than realized vol based on absolute returns. But least noisy of all, and not considered in the paper above, is realized vol calculated directly from high-frequency return data (HFD-vol), as done by numerous authors in recent decades. Interestingly, recent work for HFD-vol also reports d intervals that tend to poke above 1/2. See this earlier post.

Factor Model w Time-Varying Loadings

Markus Pelger has a nice paper on factor modeling with time-varying loadings in high dimensions. There are many possible applications. He applies it to level-slope-curvature yield-curve models. 

For me another really interesting application would be measuring connectedness in financial markets, as a way of tracking systemic risk. The Diebold-Yilmaz (DY) connectedness framework is based on a high-dimensional VAR with time-varying coefficients, but not factor structure. An obvious alternative in financial markets, which we used to discuss a lot but never pursued, is factor structure with time-varying loadings, exactly in Pelger! 

It would seem, however, that any reasonable connectedness measure in a factor environment would need to be based not only time-varying loadings but also time-varying idiosynchratic shock variances, or more precisely a time-varying noise/signal ratio (e.g., in a 1-factor model, the ratio of the idiosyncratic shock variance to the factor innovation variance). That is, connectedness in factor environments is driven by BOTH the size of the loadings on the factor(s) AND the amount of variation in the data explained by the factor(s). Time-varying loadings don't really change anything if the factors are swamped by massive noise. 

Typically one might fix the factor innovation variance for identification, but allow for time-varying idiosyncratic shock variance in addition to time-varying factor loadings. It seems that Pelger's framework does allow for that. Crudely, and continuing the 1-factor example, consider y_t  =  lambda_t  f_t  +  e_t. His methods deliver estimates of the time series of loadings lambda_t and factor f_t, robust to heteroskedasticity in the idiosyncratic shock e_t. Then in a second step one could back out an estimate of the time series of e_t and fit a volatility model to it. 
Then the entire system would be estimated and one could calculate connectedness measures based, for example, on variance decompositions as in the DY framework. 

Machine Learning, Volatility, and the Interface

Just got back from the NBER Summer Institute. Lots of good stuff happening in the Forecasting and Empirical Methods group. The program, with links to papers, is here.

Lots of room for extensions too. Here's a great example. Consider the interface of the Gu-Kelly-Xiu and Bollerslev-Patton-Quagvleg papers. At first you might think that there is no interface. 

Kelly-Xiu is about using off-the-shelf machine-learning methods to model risk premia in financial markets; that is, to construct portfolios that deliver superior performance. (I had guessed they'd get nothing, but I was massively wrong.) Bollerslev et al. is about predicting realized covariance by exploiting info on past signs (e.g., was yesterday's covariance cross-product pos-pos, neg-neg, pos-neg, or neg-pos?). (They also get tremendous results.)

But there's actually a big interface.

Note that Kelly-Xiu is about conditional mean dynamics -- uncovering the determinants of expected excess returns. You might expect even better results for derivative assets, as the volatility dynamics that drive options prices may be nonlinear in ways missed by standard volatility models. And that's exactly the flavor of the Bollerslev et al. results -- they find that a tree structure conditioning on sign is massively successful.

But Bollerslev et al. don't do any machine learning. Instead they basically stumble upon their result, guided by their fine intuition. So here's a fascinating issue to explore: Hit the Bollerslev et al. realized covariance data with machine learning (in particular, tree methods like random forests) and see what happens. Does it "discover" the Bollerslev et al. result? If not, why not, and what does it discover? Does it improve upon Bollerslev et al.?

Deep Neural Nets for Volatility Dynamics

There doesn't seem to be much need for nonparametric nonlinear modeling in empirical macro and finance. Not that lots of smart people haven't tried. The two key nonlinearities (volatility dynamics and regime switching) just seem to be remarkably well handled by tightly-parametric customized models (GARCH/SV and Markov-switching, respectively). 

But the popular volatility models are effectively linear (ARMA) in squares. Maybe that's too rigidly constrained. Volatility dynamics seem like something that could be nonlinear in ways much richer than just ARMA in squares. 

Here's an attempt using deep neural nets. I'm not convinced by the paper -- much more thorough analysis and results are required than the 22 numbers reported in the "GARCH" and "stocvol" columns of its Table 1 -- but I'm intrigued.

It's quite striking that neural nets, which have been absolutely transformative in other areas of predictive modeling, have thus far contributed so little in economic / financial contexts. Maybe the "deep" versions will change that, at least for volatility modeling. Or maybe not. 

Classic Jacod (1994) Paper

J. Financial Econometrics will soon publish Jean Jacod's brilliant and beautiful 1994 paper, "Limit of Random Measures Associated with the Increments of a Brownian Semimartingale", which I just had the pleasure of reading for the first time. (Ungated version here.) Along with several others, I was asked to supply some comments for the issue's introduction. What follows is adapted from those comments, providing some historical background. (Except that it's not really historical background -- keep reading...)

Jacod's paper effectively lays the foundation for the vast subsequent econometric "realized volatility" (empirical quadratic variation) literature of the past twenty years.  Reading it leads me to recall my early realized volatility work with Torben Andersen and Tim Bollerslev in the late 1990's and early 2000's. It started in the mid-1990's at a meeting of the NBER Asset Pricing Program, where I was the discussant for a paper of theirs, eventually published as Andersen and Bollerslev (1998). They were using realized volatility as the "realization" in a study of GARCH volatility forecast accuracy, and my discussion was along the lines of, "That's interesting, but I think you've struck gold without realizing it -- why not skip the GARCH and instead simply characterize, model, and forecast realized volatility directly?".

So we decided to explore realized volatility directly. Things really took off with Andersen et al. (2001) and Andersen et al. (2003). The research program was primarily empirical, but of course we also wanted to advance the theoretical foundations. We knew some relevant stochastic integration theory, and we made progress culminating in Theorem 2 of Andersen et al. (2003). Around the same time, Ole Bardorff-Nielsen and Neil Shephard were also producing penetrating and closely-related results (most notably Barndorff-Nielsen and Shephard, 2002). Very exciting early times.

Now let's return to Jacod's 1994 paper, and consider it against the above historical background of early econometric realized volatility papers. Doing so reveals not only its elegance and generality, but also its prescience: It was written well before the "historical background"!! One wonders how it went unknown and unpublished for so long.

References

Andersen, T. G. and T. Bollerslev (1998), "Answering the Skeptics: Yes, Standard Volatility Models do Provide Accurate Forecasts," International Economic Review, 39, 885-905.

Andersen, T.G., T. Bollerslev, F.X. Diebold, and P. Labys (2001), "The Distribution of Realized Exchange Rate Volatility," Journal of the American Statistical Association, 96, 42-55.

Andersen, T.G., T. Bollerslev, F.X. Diebold, and P. Labys (2003), "Modeling and Forecasting Realized Volatility," Econometrica, 71, 579-625.

Barndorff-Nielsen, O. and N. Shephard (2002), "Econometric Analysis of Realized Volatility and its Use in Estimating Stochastic Volatility Models," Journal of the Royal Statistical Society, 64,
253-280.

Jacod, J. (1994), "Limit of Random Measures Associated with the Increments of a Brownian Semimartingale," Manuscript, Institute de Mathematiques de Jussieu, Universite Pierre et Marie Curie, Paris.

Long Memory in Realized Volatility

A noteworthy aspect of long memory in realized asset return volatility is that in many leading cases it's basically undeniable on the basis of a variety of evidence -- the question isn't existence but rather strength.  Hence it's useful to have a broad and comparable set of state-of-the-art (local Whittle) estimates together in one place, as in the interesting paper below.  For the most part it gets d in [.4, .6], consistent with my personal experience of d usually around .45, in the covariance stationary (finite variance) region d<.5, but close to the boundary.

http://d.repec.org/n?u=RePEc:han:dpaper:dp-601&r=ecm

The Memory of Volatility

Date:	2017-07
By:	Wenger, Kai ; Leschinski, Christian ; Sibbertsen, Philipp
The focus of the volatility literature on forecasting and the predominance of the conceptually simpler HAR model over long memory stochastic volatility models has led to the fact that the actual degree of memory estimates has rarely been considered. Estimates in the literature range roughly between 0.4 and 0.6 - that is from the higher stationary to the lower non-stationary region. This difference, however, has important practical implications - such as the existence or non-existence of the fourth moment of the return distribution. Inference on the memory order is complicated by the presence of measurement error in realized volatility and the potential of spurious long memory. In this paper we provide a comprehensive analysis of the memory in variances of international stock indices and exchange rates. On the one hand, we find that the variance of exchange rates is subject to spurious long memory and the true memory parameter is in the higher stationary range. Stock index variances, on the other hand, are free of low frequency contaminations and the memory is in the lower non-stationary range. These results are obtained using state of the art local Whittle methods that allow consistent estimation in presence of perturbations or low frequency contaminations.
Keywords:	Realized Volatility; Long Memory; Perturbation; Spurious Long Memory
JEL:	C12 C22 C58 G15
URL:	http://d.repec.org/n?u=RePEc:han:dpaper:dp-601&r=ecm

Exogenous vs. Endogenous Volatility Dynamics

I always thought putting exogenous volatility dynamics in macro-model shocks was a cop-out.  Somehow it seemed more satisfying for volatility to be determined endogenously, in equilibrium.  Then I came around:  We allow for shocks with exogenous conditional-mean dynamics (e.g., AR(1)), so why shouldn't we allow for shocks with exogenous conditional-volatility dynamics?  Now I might shift back, at least in part, thanks to new work by Sydney Ludvigson, Sai Ma, and Serena Ng, "Uncertainty and Business Cycles: Exogenous Impulse or Endogenous Response?", which attempts to sort things out. The October 2016 version is here.  It turns out that real (macro) volatility appears largely endogenous, whereas nominal (financial market) volatility appears largely exogenous. 

Dense Data for Long Memory

From the last post, you might think that efficient learning about low-frequency phenomena requires tall data. Certainly efficient estimation of trend, as stressed in the last post, does require tall data. But it turns out that efficient estimation of other aspects of low-frequency dynamics sometimes requires only dense data. In particular, consider a pure long memory, or "fractionally integrated", process, \( (1-L)^d x_t = \epsilon_t \), 0 < \( d \) < 1/2. (See, for example, this or this.) In a general \( I(d) \) process, \(d\) governs only low-frequency behavior (the rate of decay of long-lag autocorrelations toward zero, or equivalently, the rate of explosion of low-frequency spectra toward infinity), so tall data are needed for efficient estimation of \(d\). But in a pure long-memory process, one parameter (\(d\)) governs behavior at all frequencies, including arbitrarily low frequencies, due to the self-similarity ("scaling law") of pure long memory. Hence for pure long memory a short but dense sample can be as informative about \(d\) as a tall sample. (And pure long memory often appears to be a highly-accurate approximation to financial asset return volatilities, as for example in ABDL.)

Big Data for Volatility vs.Trend

Although largely uninformative for some purposes, dense data (high-frequency sampling) are highly informative for others.  The massive example of recent decades is volatility estimation.  The basic insight traces at least to Robert Merton's early work. Roughly put, as we sample returns arbitrarily finely, we can infer underlying volatility (quadratic variation) arbitrarily well.

So, what is it for which dense data are "largely uninformative"?  The massive example of recent decades is long-term trend.  Again roughly put and assuming linearity, long-term trend is effectively a line segment drawn between a sample's first and last observations, so for efficient estimation we need tall data (long calendar span), not dense data.

Assembling everything, for estimating yesterday's stock-market volatility you'd love to have yesterday's 1-minute intra-day returns, but for estimating the expected return on the stock market (the slope of a linear log-price trend) you'd much rather have 100 years of annual returns, despite the fact that a naive count would say that 1 day of 1-minute returns is a much "bigger" sample.

So different aspects of Big Data -- in this case dense vs. tall -- are of different value for different things.  Dense data promote accurate volatility estimation, and tall data promote accurate trend estimation.

On Forecasting Variation and Covariation

One hallmark of a great idea is that it's "obvious" (ex post). Fantastic recent work by Bollerslev, Patton, and Quaedvlieg (BPQ) certainly passes that test.

BPQ build on the classic Barndorff-Nielsen and Shephard result that the precision with which realized variation and covariation are estimated is time-varying but can be estimated (let's just speak of "variation" for short, whether univariate or multivariate). Put differently, the measurement error in realized variation is heteroskedastic but can be estimated. Hence, for optimal variation prediction, one should presumably weight the recent past differently depending on the estimated size of the measurement error. BPQ do it and get large predictive gains. Check it out here. (This is the new and multivariate (covariance) paper, which cites the earlier univariate (variance) paper.)

Why didn't I think of that? I mean, really, the Barndorff-Nielsen and Shephard result is more than a decade old, and I know it well. Can I not put two and two together? Damn.

But seriously, congratulations to BPQ.

Long Memory Stochastic Volatility

Check out Mark Jensen's new paper.  Long memory is a key feature of realized high-frequency asset-return volatility, yet it remains poorly understood. Jensen's approach may help change that. Of particular interest are: (1) its ability to handle seamlessly d in [0, 1[, despite the fact that the unconditional variance is infinite for d in ].5, 1[, and (2) closely related, the important role played by wavelets. 

Details:

Robust estimation of nonstationary, fractionally integrated, autoregressive, stochastic volatility

Date:	2015-11-01
By:	Jensen, Mark J. (Federal Reserve Bank of Atlanta)
Empirical volatility studies have discovered nonstationary, long-memory dynamics in the volatility of the stock market and foreign exchange rates. This highly persistent, infinite variance—but still mean reverting—behavior is commonly found with nonparametric estimates of the fractional differencing parameter d, for financial volatility. In this paper, a fully parametric Bayesian estimator, robust to nonstationarity, is designed for the fractionally integrated, autoregressive, stochastic volatility (SV-FIAR) model. Joint estimates of the autoregressive and fractional differencing parameters of volatility are found via a Bayesian, Markov chain Monte Carlo (MCMC) sampler. Like Jensen (2004), this MCMC algorithm relies on the wavelet representation of the log-squared return series. Unlike the Fourier transform, where a time series must be a stationary process to have a spectral density function, wavelets can represent both stationary and nonstationary pr! ocesses. As long as the wavelet has a sufficient number of vanishing moments, this paper's MCMC sampler will be robust to nonstationary volatility and capable of generating the posterior distribution of the autoregressive and long-memory parameters of the SV-FIAR model regardless of the value of d. Using simulated and empirical stock market return data, we find our Bayesian estimator producing reliable point estimates of the autoregressive and fractional differencing parameters with reasonable Bayesian confidence intervals for either stationary or nonstationary SV-FIAR models.
Keywords:	Bayes; infinite variance; long-memory; Markov chain Monte Carlo; mean-reverting; wavelets
JEL:	C11 C14 C22
URL:	http://d.repec.org/n?u=RePEc:fip:fedawp:2015-12&r=ecm

Measuring and Monitoring Connectedness

I'm at the IMF soon for a couple days of lecturing on Diebold-Yilmaz's Connectedness. It was published earlier this year, and preparing for the IMF jogged my memory: I brilliantly forgot to announce it in a No Hesitations post. Anyway, it's available at the usual online shops (where you can also read the T.O.C. and first chapter), or directly from Oxford University Press. There's also a web site. Special thanks to Eric Ghysels, who put us in touch with our fine editor, Scott Parris. 

Online Volatility Data and Labs

I am reminded that I had planned to post on data/analysis sites that focus on financial asset return volatility measurement and modeling.

To my mind, the key trio is implied vol, GARCH vol, and realized vol. For implied vol it's the VIX at CBOE. For GARCH vol it's Rob Engle's V-Lab at NYU. For realized vol it's Neil Shephard's Realized Library at Oxford.

Yes, conspicuously missing is stochastic volatility. It's an academic simulator's paradise, but largely missing from serious/practical industry application. It's no accident; the benefit/cost ratio is just too low to excite many real financial-market modelers. One could argue that ten years from now things will look different. Perhaps, but I'm not at all sure. 

Interesting New Work on Yield Curve Modeling

Loved last week's PIER lectures at Penn. Good people, good times, good spring weather.  (Please join us next year in May 2016! More information in due course.) On Thursday we did yield curves, which had me thinking about what's new that I like in that area. Not surprisingly, I'm a fan of dynamic Nelson-Siegel (DNS), arbitrage-Free Nelson-Siegel (AFNS), and the many variations.  (See the Diebold-Rudebusch 2013 book.) What's more surprising is that although Nelson-Siegel is almost thirty years old, and DNS/AFNS is almost a teenager, interesting and useful new variations keep coming along.

The most important new work concerns imposition of the zero lower bound (ZLB). Fischer Black's "shadow rate" approach has influenced me most. Recently it's been taken to new heights by Glenn Rudebusch and coauthors at the Federal Reserve Bank of San Francisco (e.g., Christensen and Rudebusch 2015 -- just published in Journal of Financial Econometrics), and Leo Krippner at the Reserve Bank of New Zealand (see his wonderful 2015 book). The amazing thing is that one can stay in the DNS/AFNS framework -- the key tractable subclass of Gaussian affine models -- and still respect the ZLB by appropriately truncating simple simulations. The figure below, assembled from some of Krippner's, says it all. Also see these slides.   

I'm also partial to shadow-rate ZLB work by Cynthia Wu and coauthors at Chicago and San Diego (e.g. Wu and Xia, 2014). (Thanks to Jim Hamilton, her Ph.D. advisor, for reminding me!) See the monthly Wu-Xia shadow short rate series, produced and published to the web by FRB Atlanta.

Last and not at all least is the recent "ARG0" work of Monfort et al., which imposes the ZLB in a very different and elegant way. Again see these slides.   

Another interesting strand of recent DNS/AFNS progress concerns modeling the interaction of bond yield factors, macro fundamentals, and central bank policy.  More on that sometime soon.

Volatility Institute 2015

I'm baaaaaack...

Speaking of being back, I'm just back from the Rob Engle / NYU Volatility Institute Annual Conference.  (Well, more or less just back.) Great people, great science, tightly-focused on a fascinating and timely area, the bond market and yield-curve modeling.  Program and links to papers here.  I think they'll post slides soon as well.  Mine are here.  Shortly I'll blog separately on what I see as the two key econometric approaches to arbitrage-free yield-curve modeling in zero-lower-bound environments:  The ARG0 approach of Monfort et al. (the new paper I discussed) and the shadow-rate approach of Krippner et al. (going way back to Fischer Black.)

Nassim Taleb Graphic

This arrived a couple weeks ago from Nassim Taleb. Regardless of where your view falls on the black swan spectrum, I hope you'll like the graphic. One hallmark of a good graphic is that it repays careful study, as with a good map (which is a good graphic). Nassim's Genealogy certainly passes that test. I found myself thinking about its contents and assertions for a long time. (You can blow it up in your browser by clicking on it. That should do the trick, but if it's still not big enough, start hitting ctrl+.)   

DDH Now in Chinese

For my Chinese readers:
A Chinese version of the Diebold-Doherty-Herring risk management book just appeared. Interesting surprise. I knew nothing about it until it arrived in the snail mail, just as with the earlier Chinese version of the Diebold-Rudebusch yield curve book. Ya gotta love Princeton University Press. They take care of business, with minimal hassle.