This post is about copulas and heavy tails. In a previous post we discussed the concept of correlation structure. The aim is to characterize the correlation across the distribution. Prior to the global financial crisis many investors were under the impression that they were diversified, and they were, for how things looked there and then. Alas, when things went south, correlation in the new southern regions turned out to be different\stronger than that in normal times. The hard-won diversification benefits evaporated exactly when you needed them the most. This adversity has to do with fat-tail in the joint distribution, leading to great conceptual and practical difficulties. Investors and bankers chose to swallow the blue pill, and believe they are in the nice Gaussian world, where the math is magical and elegant. Investors now take the red pill, where the math is ugly and problems abound.
Last time we showed how to estimate a CCC and DCC volatility model. Here I describe an advancement labored by Engle and Kelly (2012) bearing the name: Dynamic equicorrelation. The idea is nice and the paper is well written.
Departing where the previous post ended, once we have (say) the DCC estimates, instead of letting the variance-covariance matrix be, we force some structure by way of averaging correlation across assets. Generally speaking, correlation estimates are greasy even without any breaks in dynamics, so I think forcing some structure is for the better.
Given a constant speed, time and distance are fully correlated. Provide me with the one, and I’ll give you the other. When two variables have nothing to do with each other, we say that they are not correlated.
You wish that would be the end of it. But it is not so. As it is, things are perilously more complicated. By far the most familiar correlation concept is the Pearson’s correlation. Pearson’s correlation coefficient checks for linear dependence. Because of it, we say it is a parametric measure. It can return an actual zero even when the two variables are fully dependent on each other (link to cool chart).
When hopping from univariate volatility forecasts to multivariate volatility forecast, we need to understand that now we have to forecast not only the univariate volatility element, which we already know how to do, but also the covariance elements, which we do not know how to do, yet. Say you have two series, then this covariance element is the off-diagonal of the 2 by 2 variance-covariance matrix. The precise term we should use is “variance-covariance matrix”, since the matrix consists of the variance elements on the diagonal and the covariance elements on the off-diagonal. But since it is very tiring to read\write “variance-covariance matrix”, it is commonly referred to as the covariance matrix, or sometimes less formally as var-covar matrix.
In this day and age, paralleling and mining big data, I like to think about the new complications that follow this abundance. By way of analogy, Alzheimer’s dementia is an awful condition, but we are only familiar with it since medical advances allow for higher life expectancy. Better abilities allow for new predicaments. One of those new predicament is what I call out-of-sample data snooping.
Recently, volatility has been on the up. Generally, we associate rising volatility with a bear regime, but we also know there is a percolating oil shock. Is the volatility we see in the stock market broad-based, or is it the effect brought about by sharp the drop in oil prices (so related to the energy sector)? I propose here a practical way to take a closer look at it.
5 weeks ago we took a look at the rising volatility in the (US) equity markets via a time-series threshold model for the VIX. The estimate suggested we are crossing (or crossed) to the more volatile regime. Here, taking somewhat different Hidden Markov Model (HMM) approach we gather more corroboration (few online references at the bottom if you are not familiar with HMM models. The word hidden since the state is ‘invisible’).
If you google-finance AMZN you can see the beta is 0.93. I already wrote in the past about this illusive concept. Beta is suppose to reflect the risk of an instrument with respect for example to the market. However, you can estimate this measure in all kind of ways.
In the past, I wrote about robust regression. This is an important tool which handles outliers in the data. Roger Koenker is a substantial contributor in this area. His website is full of useful information and code so visit when you have time for it. The paper which drew my attention is “Quantile Autoregression” found under his research tab, it is a significant extension to the time series domain. Here you will find short demonstration for stuff you can do with quantile autoregression in R.
In portfolio management, risk management and derivative pricing, volatility plays an important role. So important in fact that you can find more volatility models than you can handle (Wikipedia link). What follows is to check how well each model performs, in and out of sample. Here are three simple things you can do:
In the last few decades there has been tremendous progress in the realm of volatility estimation. A major step is the additional use of intraday price path. It has been shown that estimates which consider intraday information are more accurate. Which is to say they converge faster to the real unobserved value of the true volatility.
In the post pairs trading issues one of the problems raised was the unstable estimates of the stock’s beta with respect to the market. Here is a suggestion for a possible solution, which is not really a solution but more stuff to do to make you feel less stupid when trading based on your fragile estimates.
In the last decade we have observed an increase in computational power, information availability, speed of execution and stock market competition in general. One might think that, as a result, we are prone to larger shocks that occur faster than what was common in the past. I crunched some numbers and was surprised to see that this is not the case.
Having stock market in mind, in the previous post: “Price is right, part one.”, I stated that we should not think in terms of “the price went up/down too much” but that “the current price level is wrong since…. and the market is not getting it because…”, bearing in mind that Mr. Market is not a weak player to say the least.
In this post I back this claim with the examination of a trading strategy that ignores economical arguments, thus is only based on relative price moves. Say you believe my previous post is horseshit, wouldn’t it be nice to short the market if it’s “too high” and to long it when it “went down too much”? Fine!, let’s have a look at the performance of such a strategy.