Jan de Kort

Site of the project:

ABN AMRO

start of the project: January 2007

In March 2007 the Interim Thesis has been appeared.

The Master project has been finished in September 2007 by the completion of the Masters Thesis and a final presentation has been given. For working address etc. we refer to our alumnipage.

Within the hybrids group of PA a model is in place which can price derivatives on multiple assets. Across assets the dependence is currently modeled by a Gaussian copula (i.e. correlation matrix). However empirical data on the relevant underlyings (especially commodities) show evidence of (asymmetric) tail dependence, something which the Gaussian copula is not able to capture.

The concept of a copula is simple but elegant. A bivariate copula is a function C(u,v): [0,1] x [0,1] -> [0,1] such that,

C(u,0) = C(0,v) = 0 (grounded)

C(u,1) = u and C(1,v) = v (marginals recovered)

C(u,v) - C(a,v) - C(u,b) + C(a,b) >= 0 (increasing)

Let F and G be marginal distribution functions and let C be a copula. Then C[F(x), G(y)] is a joint distribution with marginals F and G. Conversely any joint distribution with marginals F and G can be expressed as C[F(x), G(y)] where C is a copula.

It connects multiple univariate probability distributions such that the marginal distributions are preserved. Hence a copula is a measure of dependence which is independent of the marginal distributions. An important concept in dependence modeling is tail dependence (the probability of joint extreme events). Different parametric copulas have different degrees of upper and lower tail dependence, Gaussian (no tail dependence), Student-t (symmetric both upper and lower tail dependence), Clayton (only upper tail dependence) and Gumbel (only lower tail dependence).

The goal of the project is to combine different well known parametric forms (at most three: upper tail, lower tail, general) into a flexible parametric copula. This copula then has to be estimated efficiently from actual market data. Model selection (criteria), as well as hypothesis tests have to be implemented and in the end automated.

Approach

The copula has to have a parametric form! This is needed for interpretation and implementation purposes. Semi or non parametric approaches will lead to over fitting.

The best way to start is to use the fact that a linear combination of copulas is still a copula. In this way we can use a combination of Gaussian, Clayton and Gumbel copula to obtain a copula with the desired degree of upper and lower tail dependence. This is relatively easy to do in a bivariate setting. The approach is outlined in Hu (2004) with the EM algorithm used for estimation explained in Bilmes (1998) among others.

Model selection is important since the inclusion of tail dependence should only take place if estimated to be significant. The copula should be as parsimonious as possible.

Within the practical phase of the project the following points have to be taken into account: forward vs. spot dependence: since the payoff is priced under the forward measure the stochastic processes are forward prices, hence the dependence modeling should focus on forward prices as well. This complicates matters slightly as the forward maturity changes as time progresses (e.g. a 5 year forward at inception of the deal becomes a 4 year forward after 1 year has passed). The convenient representation of the combined space and time copula as a (correlation-)matrix ceases to exist. The frequency of the data which is used for estimation: I?ll explain this by an example. For a 5 year maturity deal it doesn?t make sense to estimate the copula using three months of daily data. Rather at least 5 years of biweekly or monthly data is required.

When the above procedure is implemented the instantaneous dependence between forward prices can be estimated. The terminal dependence between forward prices is then the integrated instantaneous dependence. It is this latter dependence which determines derivatives prices. The relation between these types of dependencies can be investigated. Furthermore it is interesting to see how dependency (e.g. tail dependence) varies with forward maturity.