Abstract
Copula functions can be utilized in financial applications to determine the dependence structure of the financial asset returns in the portfolio. Empirical evidence has proved the inadequacy of the multi-normal distribution, traditionally adopted to model the financial asset returns distribution. Copula functions can be employed in a flexible way for building efficient algorithms and to simulate a more adequate distribution of the financial assets. This paper aims to describe some simple statistical procedures currently employed to calibrate the copula functions to the financial market data. Furthermore, we present some useful methods for choosing which copula function better fits the real financial data. Also, some algorithms to simulate random variates from certain types of copula functions are illustrated. Finally, for illustration purposes, the previous procedures described are applied to two Italian equities. In particular, we show how to generate efficient Monte Carlo scenarios of equity log-returns in the bivariate case using different types of copula functions.
Highlights
The study of the copula functions is very relevant because of their implementation in the field of financial portfolio risk management
The copula functions are used in financial applications since 2000, following the seminal researches of [1, 2]
Empirical evidence has widely proved that the multinormal distribution is inadequate to model portfolio’s financial asset returns distribution at least from two points of view: (1) The empirical marginal distributions are skewed and fat-tailed
Summary
The study of the copula functions is very relevant because of their implementation in the field of financial portfolio risk management. In this paper, the described statistical methods for calibrating, selecting, and simulating copula functions are implemented to an empirical financial data set concerning the log-returns of two Italian equities: Olivetti and TIM. When it is possible, we show as the copula approach performs better than the bivariate normal distribution in modeling the real financial data. The following recursive procedure is used to estimate the parameter R of the tν-Student copula (Eq 5): The Method of Inference Functions for Margins (IFM). One can generate random variates from the n-dimensional Gaussian copula running the following algorithm: Find the Cholesky decomposition A of the matrix R; Simulate n independent standard normal random variates z The vector (u1, u2) is generated from the Farlie–Gumbel –Morgenstern copula
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