Abstract base class for one-factor copula models. More...
#include <ql/experimental/credit/onefactorcopula.hpp>
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 |
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void | update () |
void | recalculate () |
void | freeze () |
void | unfreeze () |
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Observable (const Observable &) | |
Observable & | operator= (const Observable &) |
void | notifyObservers () |
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Observer (const Observer &) | |
Observer & | operator= (const Observer &) |
std::pair< std::set < boost::shared_ptr < Observable > >::iterator, bool > | registerWith (const boost::shared_ptr< Observable > &) |
Size | unregisterWith (const boost::shared_ptr< Observable > &) |
void | unregisterWithAll () |
Protected Member Functions | |
Size | steps () const |
Real | dm (Size i) const |
Real | m (Size i) const |
Real | densitydm (Size i) const |
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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_ |
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bool | calculated_ |
bool | frozen_ |
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.
Density function of M.
Derived classes must override this method and ensure zero mean and unit variance.
Implemented in OneFactorStudentGaussianCopula, OneFactorGaussianStudentCopula, OneFactorStudentCopula, and OneFactorGaussianCopula.
Cumulative distribution of Z.
Derived classes must override this method and ensure zero mean and unit variance.
Implemented in OneFactorStudentGaussianCopula, OneFactorGaussianStudentCopula, OneFactorStudentCopula, and OneFactorGaussianCopula.
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.
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.
Conditional probability.
\[ \hat p(m) = F_Z \left( \frac{F_Y^{-1}(p)-a\,m}{\sqrt{1-a^2}}\right) \]
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 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 over the density \( \rho(m) \) of M and a one-dimensional function \( f \) of conditional probabilities related to the input vector of probabilities p:
\[ \int_{-\infty}^\infty\,dm\,\rho(m)\, f (\hat p_1, \hat p_2, \dots, \hat p_N), \qquad \hat p_i (m) = F_Z \left( \frac{F_Y^{-1}(p_i)-a\,m}{\sqrt{1-a^2}} \right) \]
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:
\[ \int_{-\infty}^\infty\,dm\,\rho(m)\, f (\hat p_1, \hat p_2, \dots, \hat p_N), \qquad \hat p_i = F_Z \left( \frac{F_Y^{-1}(p_i)-a\,m}{\sqrt{1-a^2}}\right) \]