the Jon Egil Strand blog

Introducing call tilts and put tilts on FStat

2008-08-07T16:34:00.007+02:00

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

2008-07-11T13:34:00.012+02:00

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

2008-07-08T15:47:00.005+02:00

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

2008-06-07T21:11:00.000+02:00

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

2008-05-12T21:24:00.000+02:00

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

10 x speed increase in gaussian random numbers in excel

2008-04-17T20:52:00.003+02:00

For generating Gaussian random numbers in Excel, one could simply do

=NORMSINV(RAND())

but the performance is horrible. Instead, try this VBA function from www.vbnumericalmethods.com.
It will easily give you a 10x speed increase:


'From Jack at www.vbnumericalmethods.com
'Simulate a Gaussian variable N(0,1)
Public Function Gauss()
          Dim fac As Double, r As Double, V1 As Double, V2 As Double
      10  V1 = 2 * Rnd - 1
          V2 = 2 * Rnd - 1
          r = V1 ^ 2 + V2 ^ 2
          If (r >= 1) Then GoTo 10
          fac = Sqr(-2 * Log(r) / r)
          Gauss = V2 * fac
End Function

Young entrepreneur

2008-04-03T21:47:00.004+02:00

Do you ever google yourself? I did so today, and found this article in Universitas from 2003. Looking back five years feels good, I recognize both who I was and who I am.

The article describes a venture we did in 2003-2005. Building your own business and making it cash-positive is a great, although hard, experience. The mindset of business rally can't be learned any other way, and the start-up experience really helped shape who I am as a professional.