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/* -*- mode: c++; tab-width: 4; indent-tabs-mode: nil; c-basic-offset: 4 -*- */
/*
Copyright (C) 2013 Peter Caspers
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
<http://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.
*/
#include <ql/experimental/risk/creditriskplus.hpp>
#include <map>
using std::sqrt;
namespace QuantLib {
CreditRiskPlus::CreditRiskPlus(
const std::vector<Real> &exposure,
const std::vector<Real> &defaultProbability,
const std::vector<Size> §or,
const std::vector<Real> &relativeDefaultVariance,
const Matrix &correlation, const Real unit)
: exposure_(exposure), pd_(defaultProbability), sector_(sector),
relativeDefaultVariance_(relativeDefaultVariance),
correlation_(correlation), unit_(unit) {
m_ = exposure_.size();
QL_REQUIRE(m_ > 0, "no exposures given");
QL_REQUIRE(m_ == pd_.size(), "number of exposures ("
<< m_
<< ") must be equal to number of pds ("
<< pd_.size() << ")");
QL_REQUIRE(m_ == sector_.size(),
"number of exposures ("
<< m_
<< ") must be equal to number of exposure sectors ("
<< sector_.size() << ")");
n_ = correlation_.rows();
QL_REQUIRE(correlation_.columns() == n_,
"correlation matrix (" << n_ << "," << correlation_.columns()
<< ") must be a square matrix");
QL_REQUIRE(relativeDefaultVariance_.size() == n_,
"number of relative default variances ("
<< relativeDefaultVariance_.size() << ")"
<< " must be equal to number of sectors (" << n_ << ")");
exposureSum_ = 0.0;
el_ = 0.0;
el2_ = 0.0;
for (Size i = 0; i < m_; ++i) {
QL_REQUIRE(exposure_[i] >= 0.0, "exposure #"
<< i << " is negative ("
<< exposure_[i] << ")");
QL_REQUIRE(pd_[i] > 0.0, "pd #" << i << " is negative (" << pd_[i]
<< ")");
QL_REQUIRE(sector_[i] < n_, "sector #" << i << " (" << sector_[i]
<< ") is out of range 0..."
<< (n_ - 1));
exposureSum_ += exposure_[i];
el_ += pd_[i] * exposure_[i];
el2_ += pd_[i] * exposure_[i]*exposure_[i];
}
QL_REQUIRE(unit_ > 0.0, "loss unit (" << unit_ << ") must be positive");
compute();
}
Real CreditRiskPlus::lossQuantile(const Real p) {
Size i = 0;
Real sum = loss_[0];
while(i < upperIndex_-1 && sum < p) {
++i;
sum += loss_[i];
}
if(loss_[0] >= p)
return 0.0;
Real p1 = sum - loss_[i];
Real p2 = sum >= p ? sum : 1.0;
Real l1 = (i - 1) * unit_;
Real l2 = i * unit_;
return l1 + (p - p1) / (p2 - p1) * (l2 - l1);
}
void CreditRiskPlus::compute() {
std::vector<Real> sectorPdSum_, sectorSpecTerms_;
sectorPdSum_ = std::vector<Real>(n_, 0.0);
sectorExposure_ = std::vector<Real>(n_, 0.0);
sectorEl_ = std::vector<Real>(n_, 0.0);
sectorSpecTerms_ = std::vector<Real>(n_, 0.0);
sectorUl_ = std::vector<Real>(n_, 0.0);
marginalLoss_ = std::vector<Real>(m_, 0.0);
std::vector<Real> pdAdj(m_, 0.0);
// compute exposure bands
unsigned long maxNu_ = 0;
upperIndex_ = 0;
// map of nuC_ to expected loss
std::map<unsigned long, Real, std::less<unsigned long> > epsNuC_;
std::map<unsigned long, Real, std::less<unsigned long> >::iterator iter;
for (Size k = 0; k < m_; ++k) {
unsigned long exUnit = (unsigned long)(std::floor(0.5 + exposure_[k] / unit_)); // round
if (exposure_[k] > 0 && exUnit == 0)
exUnit = 1; // but avoid zero exposure
if (exUnit > maxNu_)
maxNu_ = exUnit;
pdAdj[k] = exposure_[k] > 0.0
? exposure_[k] * pd_[k] / (exUnit * unit_)
: 0.0; // adjusted pd
Real el = exUnit * pdAdj[k];
if (exUnit > 0) {
iter = epsNuC_.find(exUnit);
if (iter == epsNuC_.end()) {
epsNuC_.insert(std::pair<unsigned long, Real>(exUnit, el));
} else {
(*iter).second += el;
}
upperIndex_ += exUnit;
}
}
// compute per sector figures
Real pdSum_ = 0;
for (Size k = 0; k < m_; ++k) {
pdSum_ += pdAdj[k];
sectorPdSum_[sector_[k]] += pd_[k];
sectorExposure_[sector_[k]] += exposure_[k];
sectorEl_[sector_[k]] += exposure_[k] * pd_[k];
}
for (Size i = 0; i < n_; ++i) {
// precompute sector specific terms (formula 15 in [1])
sectorSpecTerms_[i] += relativeDefaultVariance_[i] * sectorEl_[i];
for (Size j = 0; j < n_; ++j) {
if (j != i) {
sectorSpecTerms_[i] +=
correlation_[i][j] *
std::sqrt(relativeDefaultVariance_[i] *
relativeDefaultVariance_[j]) *
sectorEl_[j];
}
}
}
// compute synthetic standard deviation (formula 12 in [1])
ul_ = 0.0;
for (Size i = 0; i < n_; ++i) {
sectorUl_[i] =
relativeDefaultVariance_[i] * sectorEl_[i] * sectorEl_[i];
ul_ += sectorUl_[i];
for (Size j = 0; j < n_; ++j) {
if (j != i) {
ul_ += correlation_[i][j] *
std::sqrt(relativeDefaultVariance_[i] *
relativeDefaultVariance_[j]) *
sectorEl_[i] * sectorEl_[j];
}
}
}
Real matchUl_ = ul_; // formula 13 in [1], rhs
for (Size k = 0; k < m_; ++k) {
Real tmp = pd_[k] * exposure_[k] * exposure_[k];
sectorUl_[sector_[k]] += tmp;
ul_ += tmp;
}
ul_ = std::sqrt(ul_);
for (Size i = 0; i < n_; ++i)
sectorUl_[i] = std::sqrt(sectorUl_[i]);
// compute risk contributions (formula 15 in [1])
for (Size k = 0; k < m_; ++k) {
marginalLoss_[k] = pd_[k] * exposure_[k] / ul_ *
(sectorSpecTerms_[sector_[k]] + exposure_[k]);
}
// compute sigmaC_ and deduced figures
Real sigmaC_ = pdSum_ * sqrt(matchUl_ / (el_ * el_));
Real alphaC_ = pdSum_ * pdSum_ / (sigmaC_ * sigmaC_);
Real betaC_ = sigmaC_ * sigmaC_ / pdSum_;
Real pC_ = betaC_ / (1.0 + betaC_);
// compute loss distribution
loss_.clear();
loss_.push_back(std::pow(1.0 - pC_, alphaC_)); // A(0)
Real res;
for (unsigned long n = 0; n < upperIndex_ - 1; ++n) { // compute A(n+1)
// recursively
res = 0.0;
for (unsigned long j = 0;
j <= std::min<unsigned long>(maxNu_ - 1, n); ++j) {
iter = epsNuC_.find(j + 1);
if (iter != epsNuC_.end()) {
res += (*iter).second * loss_[n - j] * alphaC_;
if (j <= n - 1)
res += (*iter).second / ((Real)(j + 1)) *
((Real)(n - j)) * loss_[n - j];
}
}
loss_.push_back(res * pC_ / (pdSum_ * ((Real)(n + 1))));
}
}
}
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