Dependence modeling with copulas
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.
Summary of the master project:
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.
Contact information:
Kees
Vuik
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