Quants and the financial crisis

A time long friend, Econtalk interviews Riccardo Rebonato of the Royal Bank of Scotland. Rebonato authored Plight of the Fortune Tellers: Why We Need to Manage Financial Risk Differently and talks with EconTalk host Russ Roberts about the challenges of measuring risk and making decisions and creating regulation in the face of risk and uncertainty.

Rebonato in the interview discusses the role of the quant, the role of model based risk management, the data used to calibrate quant models, data relevance after systemic changes, recovery after the crises and incentive structure for players in the market.

Well worth a listen.

Gumbel here, clayton there

Copulae can at the same time be both very useful and somewhat impractical.

Useful, because they allow for a more precise modelling of co-variation between variables, empowering our models with the non-symmetry, non-normality found ever so often in the real world.

Impractical, becuse multivariate co-variation can be very hard to model, even worse to parameterize. A good description of A ∩ B can be done, but when you model A ∩ B ∩ C ∩ D ∩ E ..., the fitting is so complex that the previously so well described relation between A ∩ B is no longer precise.

Solution
The solution is closer than it might seem. Keep modelling the pairs, and model the mulitvariate whole as a cascade of two-by-two pairs. A refreshing paper by Aas, Czado, Frigessi, and Bakken was publicized in Insurance: Mathematics and Economics, 2007: Pair-copula constructions of multiple dependence.

Introducing call tilts and put tilts on FStat

If all call-options are expensive and all put-options are cheap, does the market expect the stock to rise?

Prediction markets are all the buzz (Cass Sunstein or Robin Hanson), ranging from sports to macro indicators. Heck, even Christopher Cox, chairman of the U.S. Securities and Exchange Commission is discussing wheither to police them.

Can we learn something from the derivatives market?

On FStat, we compare the simulated price of an option to it's market price. The ratio of the two gives a tilt. If the market price is higher than the simulated price, the tilt is above 100%. If the market price is lower, the tilt is less than 100%.

In the above picture, we can see the calls are expensive (i.e. market price > simulated price), but the put's are fairly valued.

Wheither or not this might be a decent predictor remains to be seen, but it'll be fun to find out.

t-copula and various measures of dependance

The t-copula can use Spearman's rho to describe the dependance between the variables. To build the all important intuition a graph helps clear things up.

One really sees how the copula takes into account the relationship from a very negative dependance (when one rises the other falls, rho < 0), via a neutral relationship (one rises, the other is not incluenced, rho=0), to a strong positive relationship (one rises, the other rises as well, rho > 0).








Try it out in R:

  library("copula")

  rho_list <- c(-0.9, -0.7, -0.5, -0.3, 0, 0.3, 0.5, 0.7, 0.9)

  par(mfrow=c(3,3))
  for(i in 1:length(rho_list)){
    rho <- rho_list[i]
    t.cop    <- tCopula(c(rho), dim=2, dispstr="ex", df=3)
    persp(t.cop, dcopula, col='lightblue', main=paste("rho", rho), 
          phi=20, xlab='U', ylab='V', zlab='tCopula', n=30)
  }

Some copulas

The word 'copula' originates from the Latin noun for a "link or tie" that connects two different things. Words such as 'couple' or even 'copulate' stem from the same source.

In Mathematics, copulas are used to couple two or more univariate distributions. The univariate distributions can be specified in isolation, and then coupled afterwards. For example can a copula join together a Pareto distribution and a Gaussian distribution, or two Student-t distributions with various degrees of freedom. This kind of flexibility is not found in traditional multivariate statistics. For example, in a mulitvariate Student-t distribution, all marginal distributions must share the same tail heavyness.

More formally, Wolfram describes copulas as follows:

Let H be a two-dimensional distribution function with marginal distribution functions F and G. Then there exists a copula C such that H(x,y)=C(F(x),G(y)).

Conversely, for any univariate distribution functions F and G and any copula C, the function H is a two-dimensional distribution function with marginals F and G. Furthermore, if F and G are continuous, then C is unique.

Weisstein, Eric W. "Sklar's Theorem." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/SklarsTheorem.html

There exists a lot of Copulas, among which the Gaussian and Student-t are the most popular. These two, so called elliptical copulas, are symmetrical in that they deliver equal probabilities for both high and low joint movements.

One significant difference exists though. A Gaussian copula has zero probability for joint extreme movements, while a Student-t distribution has non-zero probability for joint extreme movements. In for example a financial application, if the value of one stock hits rock bottom, only the Student-t copula allows the other stock to plunge as well. Therefore, in risk management, where everything that can go wrong, will go wrong at the same time, the student-t copula should be prefered.

Risk and Asset Allocation - a gem of a book

Attilio Meucci's book Risk and Asset Allocation is a true gem. Clear and practical without sacrificing quality or detail. Meucci really sets the bar for scientific writing.

Note also the extensive amount of extra material at his homepage. Extra appendices, presentation slides covering the whole book, Matlab code.

Last but not least, Meucci donates heavily to charities from his book project and teachings. Just go buy the book. Now.

fstat.no - even cooler

fstat.no - the place to be for fat-tailed finance. Complete coverage of the Norwegian derivatives market, daily updates of market prices and monte carlo simulations, Normal Inverse Gauss (NIG) distributions, volatility plots, heavyness plots in NIG-triangles. It's all there, and it's getting better by the day.

Enjoy