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/* -*- mode: c++; tab-width: 4; indent-tabs-mode: nil; c-basic-offset: 4 -*- */
/*
Copyright (C) 2008 Roland Lichters
This file is part of QuantLib, a free-software/open-source library
for financial quantitative analysts and developers - http://quantlib.org/
QuantLib is free software: you can redistribute it and/or modify it
under the terms of the QuantLib license. You should have received a
copy of the license along with this program; if not, please email
<quantlib-dev@lists.sf.net>. The license is also available online at
<https://www.quantlib.org/license.shtml>.
This program is distributed in the hope that it will be useful, but WITHOUT
ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or FITNESS
FOR A PARTICULAR PURPOSE. See the license for more details.
*/
/*! \file onefactorstudentcopula.hpp
\brief One-factor Student-t copula
*/
#ifndef quantlib_one_factor_student_copula_hpp
#define quantlib_one_factor_student_copula_hpp
#include <ql/experimental/credit/onefactorcopula.hpp>
#include <ql/math/distributions/studenttdistribution.hpp>
#include <ql/math/distributions/normaldistribution.hpp>
namespace QuantLib {
//! One-factor Double Student t-Copula
/*! The copula model
\f[ Y_i = a_i\,M+\sqrt{1-a_i^2}\:Z_i \f]
is specified here by setting the probability density functions
for \f$ Z_i \f$ (\f$ D_Z \f$) and \f$ M \f$ (\f$ D_M \f$) to
Student t-distributions with \f$ N_z \f$ and \f$ N_m \f$
degrees of freedom, respectively.
The variance of the Student t-distribution with \f$ \nu \f$
degrees of freedom is \f$ \nu / (\nu - 2) \f$. Since the
copula approach requires zero mean and unit variance
distributions, variables \f$ Z \f$ and \f$ M \f$ are scaled by
\f$ \sqrt{(N_z - 2) / N_z} \f$ and \f$ \sqrt{(N_m - 2) / N_m}, \f$
respectively.
\todo Improve performance/accuracy of the calculation of
inverse cumulative Y. Tabulate and store it for selected
correlations?
*/
class OneFactorStudentCopula : public OneFactorCopula {
public:
OneFactorStudentCopula (const Handle<Quote>& correlation,
int nz, int nm,
Real maximum = 10, Size integrationSteps = 200);
Real density(Real m) const override;
Real cumulativeZ(Real z) const override;
private:
//! Observer interface
void performCalculations() const override;
StudentDistribution density_; // density of M
CumulativeStudentDistribution cumulative_; // cumulated density of Z
int nz_; // degrees of freedom of Z
int nm_; // degrees of freedom of M
Real scaleM_; // scaling for m to ensure unit variance
Real scaleZ_; // scaling for z to ensure unit variance
// This function is used to update the table of the cumulative
// distribution of Y. It is invoked by performCalculations() when the
// correlation handle is amended.
Real cumulativeYintegral (Real y) const;
};
inline Real OneFactorStudentCopula::density (Real m) const {
return density_(m / scaleM_) / scaleM_;
}
inline Real OneFactorStudentCopula::cumulativeZ (Real z) const {
return cumulative_(z / scaleZ_);
}
//! One-factor Gaussian-Student t-Copula
/*! The copula model
\f[ Y_i = a_i\,M+\sqrt{1-a_i^2}\:Z_i \f]
is specified here by setting the probability density functions
for \f$ Z_i \f$ (\f$ D_Z \f$) to a Student t-distributions
with \f$ N_z \f$ degrees of freedom, and for \f$ M \f$
(\f$ D_M \f$) to a Gaussian.
The variance of the Student t-distribution with \f$ \nu \f$
degrees of freedom is \f$ \nu / (\nu - 2) \f$. Since the
copula approach requires zero mean and unit variance
distributions, \f$ Z \f$ is scaled by \f$ \sqrt{(N_z - 2) /
N_z}.\f$
\todo Improve performance/accuracy of the calculation of
inverse cumulative Y. Tabulate and store it for selected
correlations?
*/
class OneFactorGaussianStudentCopula : public OneFactorCopula {
public:
OneFactorGaussianStudentCopula (const Handle<Quote>& correlation,
int nz,
Real maximum = 10,
Size integrationSteps = 200);
Real density(Real m) const override;
Real cumulativeZ(Real z) const override;
private:
//! Observer interface
void performCalculations() const override;
NormalDistribution density_; // density of M
CumulativeStudentDistribution cumulative_; // cumulated density of Z
int nz_; // degrees of freedom of Z
Real scaleZ_; // scaling for z to ensure unit variance
// This function is used to update the table of the cumulative
// distribution of Y. It is invoked by performCalculations() when the
// correlation handle is amended.
Real cumulativeYintegral (Real y) const;
};
inline Real OneFactorGaussianStudentCopula::density (Real m) const {
return density_(m);
}
inline Real OneFactorGaussianStudentCopula::cumulativeZ (Real z) const {
return cumulative_(z / scaleZ_);
}
//! One-factor Student t - Gaussian Copula
/*! The copula model
\f[ Y_i = a_i\,M+\sqrt{1-a_i^2}\:Z_i \f]
is specified here by setting the probability density functions
for \f$ Z_i \f$ (\f$ D_Z \f$) to a Gaussian and for \f$ M \f$
(\f$ D_M \f$) to a Student t-distribution with \f$ N_m \f$
degrees of freedom.
The variance of the Student t-distribution with \f$ \nu \f$
degrees of freedom is \f$ \nu / (\nu - 2) \f$. Since the
copula approach requires zero mean and unit variance
distributions, \f$ M \f$ is scaled by \f$ \sqrt{(N_m - 2) /
N_m}. \f$
\todo Improve performance/accuracy of the calculation of
inverse cumulative Y. Tabulate and store it for selected
correlations?
*/
class OneFactorStudentGaussianCopula : public OneFactorCopula {
public:
OneFactorStudentGaussianCopula (const Handle<Quote>& correlation,
int nm,
Real maximum = 10,
Size integrationSteps = 200);
Real density(Real m) const override;
Real cumulativeZ(Real z) const override;
private:
//! Observer interface
void performCalculations() const override;
StudentDistribution density_; // density of M
CumulativeNormalDistribution cumulative_; // cumulated density of Z
int nm_; // degrees of freedom of M
Real scaleM_; // scaling for m to ensure unit variance
// This function is used to update the table of the cumulative
// distribution of Y. It is invoked by performCalculations() when the
// correlation handle is amended.
Real cumulativeYintegral (Real y) const;
};
inline Real OneFactorStudentGaussianCopula::density (Real m) const {
return density_(m / scaleM_) / scaleM_;
}
inline Real OneFactorStudentGaussianCopula::cumulativeZ (Real z) const {
return cumulative_(z);
}
}
#endif
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