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/* -*- mode: c++; tab-width: 4; indent-tabs-mode: nil; c-basic-offset: 4 -*- */
/*
Copyright (C) 2008 Roland Lichters
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/credit/lossdistribution.hpp>
#include <ql/math/randomnumbers/mt19937uniformrng.hpp>
using namespace std;
namespace QuantLib {
//--------------------------------------------------------------------------
Real LossDist::binomialProbabilityOfNEvents(int n, vector<Real>& p) {
//--------------------------------------------------------------------------
BinomialDistribution binomial (p[0], p.size());
return binomial(n);
}
//--------------------------------------------------------------------------
Real LossDist::binomialProbabilityOfAtLeastNEvents(int n, vector<Real>& p) {
//--------------------------------------------------------------------------
CumulativeBinomialDistribution binomial(p[0], p.size());
return 1.0 - binomial(n-1);
/*
Real defp = 0;
for (Size i = n; i <= p.size(); i++)
defp += binomialProbabilityOfNEvents (i, p);
return defp;
*/
}
//--------------------------------------------------------------------------
vector<Real> LossDist::probabilityOfNEvents(vector<Real>& p) {
//--------------------------------------------------------------------------
Size n = p.size();
vector<Real> probability(n+1, 0.0);
vector<Real> prev;
probability[0] = 1.0;
for (Size j = 0; j < n; j++) {
prev = probability;
probability[0] = prev[0] * (1.0 - p[j]);
for (Size i = 1; i <= j; i++)
probability[i] = prev[i-1] * p[j] + prev[i] * (1.0 - p[j]);
probability[j+1] = prev[j] * p[j];
}
return probability;
}
//--------------------------------------------------------------------------
Real LossDist::probabilityOfNEvents(int k, vector<Real>& p) {
//--------------------------------------------------------------------------
return probabilityOfNEvents(p)[k];
vector<Real> w (p.size(), 0);
vector<Real> u (k+1, 0);
vector<Real> v (k+1, 0);
Real pZero = 1.0;
for (Size i = 0; i < w.size(); i++) {
pZero *= (1.0 - p[i]);
w[i] = p[i] / (1.0 - p[i]);
}
if (k == 0) return pZero;
int kk = k;
Real prodw = 1.0;
// Cumulated probability of up to n events:
// Cut off when the cumulated probability reaches 1,
// i.e. set all following probabilities of exactly n events to zero.
Real sum = 1.0;
u[0] = 1.0;
for (int i = 1; i <= kk; i++) {
v[i] = 0;
for (Size j = 0; j < w.size(); j++)
v[i] += pow (w[j], i);
u[i] = 0;
for (int j = 1; j <= i; j++)
u[i] += pow (-1.0, j+1) * v[j] * u[i-j];
u[i] /= i;
// cut off
if (sum * pZero >= 1.0 || u[i] < 0 || u[i] * pZero >= 1.0)
u[i] = 0;
sum += u[i];
}
return pZero * prodw * u[kk];
}
//--------------------------------------------------------------------------
Real LossDist::probabilityOfAtLeastNEvents (int k, vector<Real>& p) {
//--------------------------------------------------------------------------
vector<Real> probability = probabilityOfNEvents(p);
Real sum = 1.0;
for (int j = 0; j < k; j++)
sum -= probability[j];
return sum;
/*
Real sum = 0;
for (Size i = k; i <= p.size(); i++)
sum += probabilityOfNEvents (i, p);
return sum;
*/
}
//--------------------------------------------------------------------------
Real ProbabilityOfNEvents::operator()(vector<Real> p) const {
//--------------------------------------------------------------------------
return LossDist::probabilityOfNEvents (n_, p);
}
//--------------------------------------------------------------------------
Real ProbabilityOfAtLeastNEvents::operator()(vector<Real> p) const {
//--------------------------------------------------------------------------
return LossDist::probabilityOfAtLeastNEvents (n_, p);
}
//--------------------------------------------------------------------------
Real BinomialProbabilityOfAtLeastNEvents::operator()(vector<Real> p) {
//--------------------------------------------------------------------------
return LossDist::binomialProbabilityOfAtLeastNEvents(n_, p);
}
//--------------------------------------------------------------------------
Distribution LossDistBinomial::operator()(Size n, Real volume,
Real probability) const {
//--------------------------------------------------------------------------
n_ = n;
probability_.clear();
probability_.resize(n_+1, 0.0);
Distribution dist (nBuckets_, 0.0, maximum_);
BinomialDistribution binomial (probability, n);
for (Size i = 0; i <= n; i++) {
if (volume_ * i <= maximum_) {
probability_[i] = binomial(i);
Size bucket = dist.locate(volume * i);
dist.addDensity (bucket, probability_[i] / dist.dx(bucket));
dist.addAverage (bucket, volume * i);
}
}
excessProbability_.clear();
excessProbability_.resize(n_+1, 0.0);
excessProbability_[n_] = probability_[n_];
for (int k = n_-1; k >= 0; k--)
excessProbability_[k] = excessProbability_[k+1] + probability_[k];
dist.normalize();
return dist;
}
//--------------------------------------------------------------------------
Distribution LossDistBinomial::operator()(const vector<Real>& nominals,
const vector<Real>& probabilities) const {
//--------------------------------------------------------------------------
return operator()(nominals.size(), nominals[0], probabilities[0]);
}
//--------------------------------------------------------------------------
Distribution LossDistHomogeneous::operator()(Real volume,
const vector<Real>& p) const {
//--------------------------------------------------------------------------
volume_ = volume;
n_ = p.size();
probability_.clear();
probability_.resize(n_+1, 0.0);
vector<Real> prev;
probability_[0] = 1.0;
for (Size k = 0; k < n_; k++) {
prev = probability_;
probability_[0] = prev[0] * (1.0 - p[k]);
for (Size i = 1; i <= k; i++)
probability_[i] = prev[i-1] * p[k] + prev[i] * (1.0 - p[k]);
probability_[k+1] = prev[k] * p[k];
}
excessProbability_.clear();
excessProbability_.resize(n_+1, 0.0);
excessProbability_[n_] = probability_[n_];
for (int k = n_ - 1; k >= 0; k--)
excessProbability_[k] = excessProbability_[k+1] + probability_[k];
Distribution dist (nBuckets_, 0.0, maximum_);
for (Size i = 0; i <= n_; i++) {
if (volume * i <= maximum_) {
Size bucket = dist.locate(volume * i);
dist.addDensity (bucket, probability_[i] / dist.dx(bucket));
dist.addAverage (bucket, volume*i);
}
}
dist.normalize();
return dist;
}
//--------------------------------------------------------------------------
Distribution LossDistHomogeneous::operator()(const vector<Real>& nominals,
const vector<Real>& probabilities) const {
//--------------------------------------------------------------------------
return operator()(nominals[0], probabilities);
}
//--------------------------------------------------------------------------
Distribution LossDistBucketing::operator()(const vector<Real>& nominals,
const vector<Real>& probabilities) const {
//--------------------------------------------------------------------------
QL_REQUIRE (nominals.size() == probabilities.size(), "sizes differ: "
<< nominals.size() << " vs " << probabilities.size());
vector<Real> p (nBuckets_, 0.0);
vector<Real> a (nBuckets_, 0.0);
vector<Real> ap (nBuckets_, 0.0);
p[0] = 1.0;
a[0] = 0.0;
Real dx = maximum_ / nBuckets_;
for (Size k = 1; k < nBuckets_; k++)
a[k] = dx * k + dx/2;
for (Size i = 0; i < nominals.size(); i++) {
Real L = nominals[i];
Real P = probabilities[i];
for (int k = a.size()-1; k >= 0; k--) {
if (p[k] > 0) {
int u = locateTargetBucket (a[k] + L, k);
QL_REQUIRE (u >= 0, "u=" << u << " at i=" << i << " k=" << k);
QL_REQUIRE (u >= k, "u=" << u << "<k=" << k << " at i=" << i);
Real dp = p[k] * P;
if (u == k)
a[k] += P * L;
else {
// no update of a[u] and p[u] if u is beyond grid end
if (u < int(nBuckets_)) {
// a[u] remains unchanged, if dp = 0
if (dp > 0.0) {
// on Windows, p[u]/dp could cause a NaN for
// some very small values of p[k].
// Writing the above as (p[u]/p[k])/P prevents
// the NaN. What can I say?
Real f = 1.0 / (1.0 + (p[u]/p[k]) / P);
a[u] = (1.0 - f) * a[u] + f * (a[k] + L);
}
/* formulation of Hull-White:
if (p[u] + dp > 0)
a[u] = (p[u] * a[u] + dp * (a[k] + L))
/ (p[u] + dp);
*/
p[u] += dp;
}
p[k] -= dp;
}
}
QL_REQUIRE(a[k] + epsilon_ >= dx * k && a[k] < dx * (k+1),
"a out of range at k=" << k << ", contract " << i);
}
}
Distribution dist (nBuckets_, 0.0, maximum_);
for (Size i = 0; i < nBuckets_; i++) {
dist.addDensity (i, p[i] / dx);
dist.addAverage (i, a[i]);
}
return dist;
}
//--------------------------------------------------------------------------
int LossDistBucketing::locateTargetBucket (Real loss, Size i0) const {
//--------------------------------------------------------------------------
QL_REQUIRE (loss >= 0, "loss " << loss << " must be >= 0");
Real dx = maximum_ / nBuckets_;
for (Size i = i0; i < nBuckets_; i++)
if (dx * i > loss + epsilon_) return i - 1;
return nBuckets_;
}
//--------------------------------------------------------------------------
Distribution LossDistMonteCarlo::operator()(const vector<Real>& nominals,
const vector<Real>& probabilities) const {
//--------------------------------------------------------------------------
Distribution dist (nBuckets_, 0.0, maximum_);
// KnuthUniformRng rng(seed_);
// LecuyerUniformRng rng;
MersenneTwisterUniformRng rng;
for (Size i = 0; i < simulations_; i++) {
double e = 0;
for (Size j = 0; j < nominals.size(); j++) {
Real r = rng.next().value;
if (r <= probabilities[j])
e += nominals[j];
}
dist.add (e + epsilon_);
}
dist.normalize();
return dist;
}
}
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