# cumulativeZ (3) - Linux Manuals

## cumulativeZ: Abstract base class for one-factor copula models.

## 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 ]

or

[ 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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