OneFactorCopula Class Referenceabstract

Abstract base class for one-factor copula models. More...

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

Inheritance diagram for OneFactorCopula:
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## Public Member Functions

| OneFactorCopula (const Handle< Quote > &correlation, Real maximum=5.0, Size integrationSteps=50, Real minimum=-5.0)
virtual Real | density (Real m) const =0
| Density function of M. More...
virtual Real | cumulativeZ (Real z) const =0
| Cumulative distribution of Z. More...
virtual Real | cumulativeY (Real y) const
| Cumulative distribution of Y. More...
virtual Real | inverseCumulativeY (Real p) const
| Inverse cumulative distribution of Y. More...
Real | correlation () const
| Single correlation parameter.
Real | conditionalProbability (Real prob, Real m) const
| Conditional probability. More...
std::vector< Real > | conditionalProbability (const std::vector< Real > &prob, Real m) const
| Vector of conditional probabilities. More...
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
Public Member Functions inherited from LazyObject
void | update ()
void | recalculate ()
void | freeze ()
void | unfreeze ()
void | alwaysForwardNotifications ()
Public Member Functions inherited from Observable
| Observable (const Observable &)
Observable & | operator= (const Observable &)
void | notifyObservers ()
Public Member Functions inherited from Observer
| Observer (const Observer &)
Observer & | operator= (const Observer &)
std::pair< iterator, bool > | registerWith (const ext::shared_ptr< Observable > &)
void | registerWithObservables (const ext::shared_ptr< Observer > &)
Size | unregisterWith (const ext::shared_ptr< Observable > &)
void | unregisterWithAll ()
virtual void | deepUpdate ()

## Protected Member Functions

Size | steps () const
Real | dm (Size i) const
Real | m (Size i) const
Real | densitydm (Size i) const
Protected Member Functions inherited from LazyObject
virtual void | calculate () const
virtual void | performCalculations () const =0

## Protected Attributes

Handle< Quote > | correlation_
Real | max_
Size | steps_
Real | min_
std::vector< Real > | y_
std::vector< Real > | cumulativeY_
Protected Attributes inherited from LazyObject
bool | calculated_
bool | frozen_
bool | alwaysForward_

## Additional Inherited Members

Public Types inherited from Observer
typedef boost::unordered_set< ext::shared_ptr< Observable > > | set_type
typedef set_type::iterator | iterator

## 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

\[ Y_i = a_i\,M+\sqrt{1-a_i^2}\:Z_i \]

where \(M\) and \(Z_i\) have independent zero-mean unit-variance distributions and \(-1\leq a_i \leq 1\). The correlation 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 \(y\):

\[ Prob \,(Y_i < y|M) = F_Z \left( \frac{y-a_i\,M}{\sqrt{1-a_i^2}}\right) \qquad \mbox{or} \qquad Prob \,(t_i < t|M) = F_Z \left( \frac{F_Y^{-1}(Q_i(t))-a_i\,M} {\sqrt{1-a_i^2}} \right) \]

The distribution functions of \( M, Z_i \) are specified in derived classes. The distribution function of \( Y \) is then given by the convolution

\[ F_Y(y) = Prob\,(Y<y) = \int_{-\infty}^\infty\,\int_{-\infty}^{\infty}\: D_Z(z)\,D_M(m) \quad \Theta \left(y - a\,m - \sqrt{1-a^2}\,z\right)\,dm\,dz, \qquad \Theta (x) = \left\{ \begin{array}{ll} 1 & x \geq 0 \\ 0 & x < 0 \end{array}\right. \]

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

This convolution can also be written

\[ F(y) = Prob \,(Y < y) = \int_{-\infty}^\infty D_M(m)\,dm\: \int_{-\infty}^{g(y,a,m)} D_Z(z)\,dz, \qquad g(y,a,m) = \frac{y - a\cdot m}{\sqrt{1-a^2}}, \qquad a < 1 \]

or

\[ F(y) = Prob \,(Y < y) = \int_{-\infty}^\infty D_Z(z)\,dz\: \int_{-\infty}^{h(y,a,z)} D_M(m)\,dm, \qquad h(y,a,z) = \frac{y - \sqrt{1 - a^2}\cdot z}{a}, \qquad a > 0. \]

In general, \( F_Y(y) \) needs to be computed numerically.

## Member Function Documentation

## density()

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 OneFactorStudentGaussianCopula, OneFactorGaussianStudentCopula, OneFactorStudentCopula, and OneFactorGaussianCopula.

## cumulativeZ()

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 OneFactorStudentGaussianCopula, OneFactorGaussianStudentCopula, OneFactorStudentCopula, and OneFactorGaussianCopula.

## cumulativeY()

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.

## inverseCumulativeY()

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.

## conditionalProbability() [1/2]

Real conditionalProbability | ( | Real | prob,
|  | Real | m
| ) |  | const

Conditional probability.

\[ \hat p(m) = F_Z \left( \frac{F_Y^{-1}(p)-a\,m}{\sqrt{1-a^2}}\right) \]

## conditionalProbability() [2/2]

std::vector<Real> conditionalProbability | ( | const std::vector< Real > & | prob,
|  | Real | m
| ) |  | const

Vector of conditional probabilities.

\[ \hat p_i(m) = F_Z \left( \frac{F_Y^{-1}(p_i)-a\,m}{\sqrt{1-a^2}} \right) \]

## integral() [1/3]

Real integral | ( | Real | p | ) | const

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

\[ \int_{-\infty}^\infty\,dm\,\rho(m)\, F_Z \left( \frac{F_Y^{-1}(p)-a\,m}{\sqrt{1-a^2}}\right) \]

## integral() [2/3]

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

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

\[ \int_{-\infty}^\infty\,dm\,\rho(m)\, f (\hat p_1, \hat p_2, \dots, \hat p_N), \qquad \hat p_i (m) = F_Z \left( \frac{F_Y^{-1}(p_i)-a\,m}{\sqrt{1-a^2}} \right) \]

## integral() [3/3]

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

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

\[ \int_{-\infty}^\infty\,dm\,\rho(m)\, f (\hat p_1, \hat p_2, \dots, \hat p_N), \qquad \hat p_i = F_Z \left( \frac{F_Y^{-1}(p_i)-a\,m}{\sqrt{1-a^2}}\right) \]

## checkMoments()

int checkMoments | ( | Real | tolerance | ) | const

Check moments (unit norm, zero mean 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.