CALIBRATING AND SIMULATING COPULA FUNCTIONS: AN
APPLICATION TO THE ITALIAN STOCK MARKET
Claudio Romano
1
Abstract
Copula functions are always more used in financial applications to determine
the dependence structure of the asset returns in a portfolio. Empirical evidence
has proved the inadequacy of the multinormal distribution, commonly adopted
to model the asset return distribution. Copulas are flexible instruments used to
build efficient algorithms for a better simulation of this distribution.
The aim of this paper is describing the statistical procedures used to calibrate a
copula function to real market data. Then, some methods used to choose which
copula better fit data are presented. Finally a number of algorithms to simulate
random variate from certain types of copula are illustrated.
The procedures described are applied to a portfolio of Italian equities. We
show how to generate efficient Monte Carlo scenarios of equity logreturns in
the bivariate case using different copulas.
Keywords:
Copula Function, Dependence Structure, Multivariate Distribution
Function.
1
Corresponding author: Claudio Romano, Risk Management Function, Capitalia, Viale U.
Tupini, 180, 00144 – Rome, Italy. Email:
[email protected]
.
The author is grateful to Prof. G. Szegö for his valuable comments and suggestions that helped
improve the article substantially.
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CALIBRATING AND SIMULATING COPULA FUNCTIONS: AN
APPLICATION TO THE ITALIAN STOCK MARKET
Claudio Romano
Introduction
Copula functions are used in financial application since 1999
2
. Empirical
evidence has proved that the multinormal distribution is inadequate to model
portfolio asset return distribution under two points of view:
1) The empirical marginal distributions are skewed and fat tailed;
2) it does not consider the possibility of extreme joint comovement of
asset returns
3
. In other words, the dependence structure is different from
the Gaussian one.
Copula functions are a useful tool to implement efficient algorithms to simulate
asset return distributions in a more realistic way. In fact, they allow to model
the dependence structure indipendently from the marginal distributions. In this
way, we may construct a multivariate distribution with different margins and
the dependence structure given from the copula function.
Therefore, a crucial step is the selection and the calibration of the copula
function from real data. In this paper a collection of methods for calibrating,
selecting and simulating copula functions are presented. Our aim is to collect in
this article the principal contributions to the argument provided by the
international literature cited in the references.
Most of the method presented are applied to an empirical data set of the log
returns of two Italian equities. When it is possible, we show as the copula
approach performs better than the multinormal distribution in modelling real
data.
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 Spring '11
 Neveskyi
 Normal Distribution, Probability theory, Copula, Multivariate normal distribution, Frank copula

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