## Tuesday, May 01, 2012

### mildly mathematical: it takes more than one formula to kill wall st ..

i'm running a kind of gauntlet again: over a span of 7 days, i'll be giving two talks.

the good thing is that the subjects aren't too different from talks i've given before, so the preparations won't be a problem.  (i even have slides for one of them, depending on how lazy i feel ..)
maybe it's some kind of spring fever or something [0],
maybe it's the jolly feeling of vappu, a national holiday in finland ..

.. but i want to write today.
on the other hand, i just can't get myself to do it.
*sighs*
so yes, procrastination ensues.

having seen a few headlines about it already, i expect a few more math blogs will link to the bbc news article about the black-schοles equation, or more notoriously, "the fοrmula that kιlled wall street."  so rather than discussing that in detail, i figured that i'd re-post an article from wired that i read and enjoyed, a few years ago.

it's about gaussιan cοpula functions, which gave rise to yet another "fοrmula that kιlled wall street" [1].  at the heart of the article is probability and correlation, which can be explained quite intuitively.

here's an excerpt:
-- ✂ -- -- -- -- -- -- -- --
The reason that ratings agencies and investors felt so safe with the triple-A tranches was that they believed there was no way hundreds of homeowners would all default on their loans at the same time. One person might lose his job, another might fall ill. But those are individual calamities that don't affect the mortgage pool much as a whole: Everybody else is still making their payments on time.

But not all calamities are individual, and tranching still hadn't solved all the problems of mortgage-pool risk. Some things, like falling house prices, affect a large number of people at once. If home values in your neighborhood decline and you lose some of your equity, there's a good chance your neighbors will lose theirs as well. If, as a result, you default on your mortgage, there's a higher probability they will default, too. That's called correlation—the degree to which one variable moves in line with another—and measuring it is an important part of determining how risky mortgage bonds are.

Investors like risk, as long as they can price it. What they hate is uncertainty—not knowing how big the risk is.
As a result, bond investors and mortgage lenders desperately want to be able to measure, model, and price correlation.
-- ✂ -- -- -- -- -- -- -- --
as for the gory details, here's the diagram that is completely lifted from the article from wired [2]:

Here's what killed your 401(k)   David X. Lι's Gaussian copula function as first published in 2000. Investors exploited it as a quick—and fatally flawed—way to assess risk. A shorter version appears on this month's cover of Wired.

#### Probability

Specifically, this is a joint default probability—the likelihood that any two members of the pool (A and B) will both default. It's what investors are looking for, and the rest of the formula provides the answer.

#### Survival times

The amount of time between now and when A and B can be expected to default. Lι took the idea from a concept in actuarial science that charts what happens to someone's life expectancy when their spouse dies.

#### Equality

A dangerously precise concept, since it leaves no room for error. Clean equations help both quants and their managers forget that the real world contains a surprising amount of uncertainty, fuzziness, and precariousness.

#### Copula

This couples (hence the Latinate term copula) the individual probabilities associated with A and B to come up with a single number. Errors here massively increase the risk of the whole equation blowing up.

#### Distribution functions

The probabilities of how long A and B are likely to survive. Since these are not certainties, they can be dangerous: Small miscalculations may leave you facing much more risk than the formula indicates.

#### Gamma

The all-powerful correlation parameter, which reduces correlation to a single constant—something that should be highly improbable, if not impossible. This is the magic number that made Lι's copula function irresistible.

[0] in my previous postdoc, the semester would typically end around this time of year; i'd even have grades submitted by now.  coincidentally enough, the same calendar also fits the universities of my ph.d and my bachelor's degree.

so after 12 years of conditioning, i think my body expects to be tired by 1st of May .. \-:

[1] it's funny how the media will always refer to a "wall street killer" to sell a story.  meanwhile, not too recently ago protesters were trying to occupy wall street .. which suggests that wall st has been alive and well for most of this time.

it just goes to show you, as a general rule, that people want to make money. any mechanism for doing so -- whether technical or social -- will survive as long as that desire survives.  i'd even go as far as to say that if some version of the apocalypse comes -- say skynet does launch those nukes and we have to fight those damned T-800s -- and if we somehow pull through, one of the first things we'll re-institute would probably be some sort of stock market.

[2] that said, owners of wired.com: please don't sue.

i think it's clear to everyone that it's your succinct explanation and not mine.  besides, i would have rendered the actual formula with LaTeX and mathjax.

Leonid said...

Your undergraduate institution was on the same schedule as the place of your 1st postdoc? A curious coincidence.

janus said...

these things occasionally happen, L. (-:

to be fair, it's not impossible for scheduling to change, even in the same institution. i think cinci is going to switch from quarters to semesters soon(?) ..

Anonymous said...

Leo- :) :)

Janus: Yes, Cincy is going to semesters. So, summer is only about 6 weeks long this year.