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density: Abstract base class for one-factor copula models.

Command to display density manual in Linux: $ man 3 density

NAME

QuantLib::OneFactorCopula - Abstract base class for one-factor copula models.

SYNOPSIS

#include <ql/experimental/credit/onefactorcopula.hpp>

Inherits QuantLib::LazyObject.

Inherited by OneFactorGaussianCopula, OneFactorGaussianStudentCopula, OneFactorStudentCopula, and OneFactorStudentGaussianCopula.

Public Member Functions

OneFactorCopula (const Handle< Quote > &correlation, Real maximum=5, Size integrationSteps=50)

virtual Real density (Real m) const =0
Density function of M.
virtual Real cumulativeZ (Real z) const =0
Cumulative distribution of Z.
virtual Real cumulativeY (Real y) const
Cumulative distribution of Y.
virtual Real inverseCumulativeY (Real p) const
Inverse cumulative distribution of Y.
Real correlation () const
Single correlation parameter.
Real conditionalProbability (Real prob, Real m) const
Conditional probability.
std::vector< Real > conditionalProbability (const std::vector< Real > &prob, Real m) const
Vector of conditional probabilities.
Real integral (Real p) const

template<class F > Real integral (const F &f, std::vector< Real > &probabilities) const

template<class F > Distribution integral (const F &f, const std::vector< Real > &nominals, const std::vector< Real > &probabilities) const

int checkMoments (Real tolerance) const

Protected Attributes

Handle< Quote > correlation_

Real max_

Size steps_

std::vector< Real > y_

std::vector< Real > cumulativeY_

Detailed Description

Abstract base class for one-factor copula models.

Reference: John Hull and Alan White, The Perfect Copula, June 2006

Let $Q_i(t)$ be the cumulative probability of default of counterparty i before time t.

In a one-factor model, consider random variables [ Y_i = a_i,M+qrt{1-a_i^2}:Z_i ] where $M$ and $Z_i$ have independent zero-mean unit-variance distributions and $-1

lation between $Y_i$ and $Y_j$ is then $a_i a_j$.

Let $F_Y(y)$ be the cumulative distribution function of $Y_i$. $y$ is mapped to $t$ such that percentiles match, i.e. $F_Y(y)=Q_i(t)$ or $y=F_Y^{-1}(Q_i(t))$.

Now let $F_Z(z)$ be the cumulated distribution function of $Z_i$. For given realization of $M$, this determines the distribution of $y$: [ Prob ,(Y_i < y|M) = F_Z

specified in derived classes. The distribution function of $ Y $ is then given by the convolution [ F_Y(y) = Prob,(Y<y) = int_{-infty}^infty,int_{-infty}^{infty}: D_Z(z),D_M(m) quad Theta

ta (x) =

re $ D_Z(z) $ and $ D_M(m) $ are the probability densities of $ Z$ and $ M, $ respectively.

This convolution can also be written [ F(y) = Prob ,(Y < y) = int_{-infty}^infty D_M(m),dm: int_{-infty}^{g(y,a,m)} D_Z(z),dz, qquad g(y,a,m) = ac{y - adot m}{qrt{1-a^2}}, qquad a < 1 ]

[ F(y) = Prob ,(Y < y) = int_{-infty}^infty D_Z(z),dz: int_{-infty}^{h(y,a,z)} D_M(m),dm, qquad h(y,a,z) = ac{y - qrt{1 - a^2}dot z}{a}, qquad a > 0. ].PPIn general, $ F_Y(y) $ needs to be computed numerically.

Possible enhancements

: Improve on simple Euler integration

Member Function Documentation

virtual Real density (Real m) const [pure virtual]

Density function of M.

Derived classes must override this method and ensure zero mean and unit variance.

Implemented in OneFactorGaussianCopula, OneFactorStudentCopula, OneFactorGaussianStudentCopula, and OneFactorStudentGaussianCopula.

virtual Real cumulativeZ (Real z) const [pure virtual]

Cumulative distribution of Z.

Derived classes must override this method and ensure zero mean and unit variance.

Implemented in OneFactorGaussianCopula, OneFactorStudentCopula, OneFactorGaussianStudentCopula, and OneFactorStudentGaussianCopula.

virtual Real cumulativeY (Real y) const [virtual]

Cumulative distribution of Y.

This is the default implementation based on tabulated data. The table needs to be filled by derived classes. If analytic calculation is feasible, this method can also be overridden.

Reimplemented in OneFactorGaussianCopula.

virtual Real inverseCumulativeY (Real p) const [virtual]

Inverse cumulative distribution of Y.

This is the default implementation based on tabulated data. The table needs to be filled by derived classes. If analytic calculation is feasible, this method can also be overridden.

Reimplemented in OneFactorGaussianCopula.

Real conditionalProbability (Real prob, Real m) const

Conditional probability.

[ l> conditionalProbability (const std::vector< Real > & prob, Real m) const"

Vector of conditional probabilities.

[ l integral (Real p) const"

Integral over the density $ ho(m) $ of M and the conditional probability related to p:

[ int_{-infty}^infty,dm,ho(m), F_Z

al integral (const F & f, std::vector< Real > & probabilities) const"

Integral over the density $ ho(m) $ of M and a one-dimensional function $ f $ of conditional probabilities related to the input vector of probabilities p:

[ int_{-infty}^infty,dm,ho(m), f (ls, const std::vector< Real > & probabilities) const"

Integral over the density $ ho(m) $ of M and a multi-dimensional function $ f $ of conditional probabilities related to the input vector of probabilities p:

[ int_{-infty}^infty,dm,ho(m), f (n and unit variance) of the distributions of M, Z, and Y by numerically integrating the respective density. Parameter tolerance is the maximum tolerable absolute error.

Author

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