1. Couple of days ago, a paper from the great Bouchaud was brought to attention by Twitter-sphere’s Vol-Potter. One of the sections “My Kingdom for a Copula” introduced copula methods.
A thread; Portfolio Risk-Management, and What The Hell is a Copula?
A thread; Portfolio Risk-Management, and What The Hell is a Copula?
https://twitter.com/volquant/status/1443434292630147072
2. The invariant of trading rule is to never go “bust”. Unless you are Hwang, chances are, getting re-financed is not an option; once you’re out, you are OUT. Unfortunately, correlation dynamics are not stable. To make matters worse, correlations increase in extreme conditions.
3. In Bouchaud’s words: tail correlations in equity markets are notoriously higher than bulk correlations. In order to deal with non-linear correlations, mathematics has afforded us with a seemingly powerful tool – “copulas”.
So, what is a copula?
So, what is a copula?
4. The measure of tail dependence can be defined as the conditional probability that some random variable y2 being lower than tail1, given another random variable y1 being lower than tail2. In other words, chances that y2 is getting pummelled, if y1 is.
5. It is a well known fact of the asset price process that prices have log-normal distribution. Unfortunately, with multivariate normal distributions, we see that they are underestimate the tail dependence/joint distribution under market stress. Enter copulas.
6. Copulas are a class of CDF functions, whose marginal distributions are univariate uniform. (Abe) Sklar's theorem cleverly shows that any multivariate joint distribution can be written in terms of univariate marginal distribution functions and a copula.
7. If this copula were to describe the dependence structure between the variables, then voila! The advantage behind this model was that we can now estimate marginal distributions of random variables and the joint distribution separately.
8. Stocks displaying different return distributions and all kinds of leptokurtic idiosyncrasies could be coherently approximated by their marginals, independent of the copulae. There is a whole class of em, so let’s see an example.
9. One of the most useful copula is t-copula. Suppose we have the return distributions of AAPL, and GME, which obviously display different characteristics. Using 2 marginal distributions, the difference in tails are captured accordingly by their degrees of freedom.
10. The joint distribution is then modelled by the copula choice, in this case the t-copula, parameterised by their correlation matrix. Mathematicians found that this approach explained empirical data more closely.
11. Even better, it turns out that you can mix and match copulas of varying tail dependence, meaning now you can vary the kind of left/right tail fit according to your copula buffet and fitting them using (Pseudo) Maximum Likelihood techniques.
12. Cue Bouchaud’s critique: ...unfortunate, but typical pattern of mathematical finance, the introduction of copulas…followed by a calibration spree, with academics and financial engineers alike frantically looking for copulas to best represent their pet multivariate problem.
13. The all but common “fitting” rave is but one with the copula crowd, where statisticians and engineers alike parade around models without critical appraisal by which their dependence lies. Alas, a genius with a model can be more dangerous than a trading hamster.
14. In 2000, David Li developed the Gaussian Copula model of credit risk. Heralded as the “world’s most influential actuary”, his heydays came to an end as “recipe for disaster” in the post crisis of 2009.
15. Since that debacle, improvements such as the Marshall-Olkin family of copula functions have been used to model survival probabilities and credit correlations.
16. With that being said, copula methods are still fantastic techniques. Bouchaud’s paper then goes about the “right way to do copulas”.
17. Importance of doing this well cannot be understated. Modelling tails in a robust fashion allows avoidance of the bust conundrum. Using Monte Carlo, it is also not difficult to obtain other risk metrics such as VaR/Expected Shortfall, which for some reason are still popular.
18. FIN. Please retweet/give me a follow if you liked my content! More papers on copula methods:
sciencedirect.com/science/articl…
sciencedirect.com/science/articl…
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