## From broken hearts to Markov chains to Gaussian copulas to quant finance

Posted by alifinmath on May 1, 2009

From the Emerald Tablet:

As above so below;

as below so above;

as within so without;

as without so within

This probably sums part of the faith of most theoretical physicists and mathematicians: the belief in correspondence in the cosmos — that diverse phenomena are subtly interrelated. And for this, mathematics is ideal as it abstracts certain features from concrete phenomena that allow comparisons and correspondences to be drawn. Thus, to give one of the earliest examples, both the falling of an apple and the trajectory of the moon can be shown to be equivalent (with a bit of vector analysis) and thus it can be shown that one force is responsible for both — gravity. This was quite something in Newton’s time as it was thought that terrestrial phenomena were governed by laws differing from heavenly phenomena.

More recently, it’s been noticed that people often die soon after their spouse’s death — a phenomenon known as “broken-heart syndrome.” There’s an interesting piece in the FT on this:

In a March 2008 study, Jaap Spreeuw and Xu Wang of the Cass Business School observed that in the year following a loved one’s death, women were more than twice as likely to die than normal, and men more than six times as likely. “

The article goes on to explain how ideas from Markov chains (a part of a more general framework of ideas known as “stochastic processes”) can be used to model the broken-heart syndrome. Markov chains are fun, and a fair amount of the theory can be explained even to a well-prepared high school senior who knows how to play with matrices (indeed there are college texts on linear algebra that incorporate some elementary Markov chain theory as a supplementary topic). There’s a simple example of a Markov chain used to find the “steady state vector” with respect to a very simple weather model here. Markov chains allow all sorts of diverse phenomena to be modeled — economic, physical, meteorological. So the fact that Markov chain theory can be used to model broken hearts should not be cause for amazement. Though perhaps we would like to think that affairs of the heart are not really amenable to quantitative treatment. But perhaps they are *statistically*, “in the large.”

The FT piece goes on to discuss the math being used to model default probabilities:

And if he could apply the broken hearts maths to broken companies, he’d have a way of mathematically modelling the effect that one company’s default would have on the chance of default for others.

The problem with relying on Markov chains was that they painted a far too mechanical, physical – atomic, even – picture of the human lifespan. Li reasoned that with a copula that showed a probable distribution of outcomes, a more accurate, encompassing picture of the broken heart or, for that matter, the broken company, could be devised.

But of course, with 20/20 hindsight, we know that this modeling didn’t work (in fact Nassim Taleb had been telling us this all along). Economic and financial phenomena are not amenable to modeling — or at least not with the math toolkit we presently have. Attempts to do so (other than the Gaussian copula, the most famous example is the Black-Scholes PDE) have exploded in our faces, discrediting (rightfully so) the whole fledgling field of quant finance. We have to be careful in what correspondences we draw in the cosmos. The heat conduction PDE cannot be used — as the Black-Scholes equation — to successfully model option prices. As Mackenzie points out in his book, An Engine, Not a Camera, the primary purpose of the equations, of the quant approach, was not to model, but rather to create and shape the markets the quant approach was purportedly modeling.

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