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/* -*- mode: c++; tab-width: 4; indent-tabs-mode: nil; c-basic-offset: 4 -*- */
/*
Copyright (C) 2003, 2004 Ferdinando Ametrano
Copyright (C) 2003 StatPro Italia srl
Copyright (C) 2005 Gary Kennedy
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 "distributions.hpp"
#include "utilities.hpp"
#include <ql/math/distributions/normaldistribution.hpp>
#include <ql/math/distributions/bivariatenormaldistribution.hpp>
#include <ql/math/distributions/poissondistribution.hpp>
#include <ql/math/comparison.hpp>
using namespace QuantLib;
using namespace boost::unit_test_framework;
QL_BEGIN_TEST_LOCALS(DistributionTest)
Real average = 1.0, sigma = 2.0;
Real gaussian(Real x) {
Real normFact = sigma*std::sqrt(2*M_PI);
Real dx = x-average;
return std::exp(-dx*dx/(2.0*sigma*sigma))/normFact;
}
Real gaussianDerivative(Real x) {
Real normFact = sigma*sigma*sigma*std::sqrt(2*M_PI);
Real dx = x-average;
return -dx*std::exp(-dx*dx/(2.0*sigma*sigma))/normFact;
}
struct BivariateTestData {
Real a;
Real b;
Real rho;
Real result;
};
template <class Bivariate>
void checkBivariate(const char* tag) {
BivariateTestData values[] = {
/* The data below are from
"Option pricing formulas", E.G. Haug, McGraw-Hill 1998
pag 193
*/
{ 0.0, 0.0, 0.0, 0.250000 },
{ 0.0, 0.0, -0.5, 0.166667 },
{ 0.0, 0.0, 0.5, 1.0/3 },
{ 0.0, -0.5, 0.0, 0.154269 },
{ 0.0, -0.5, -0.5, 0.081660 },
{ 0.0, -0.5, 0.5, 0.226878 },
{ 0.0, 0.5, 0.0, 0.345731 },
{ 0.0, 0.5, -0.5, 0.273122 },
{ 0.0, 0.5, 0.5, 0.418340 },
{ -0.5, 0.0, 0.0, 0.154269 },
{ -0.5, 0.0, -0.5, 0.081660 },
{ -0.5, 0.0, 0.5, 0.226878 },
{ -0.5, -0.5, 0.0, 0.095195 },
{ -0.5, -0.5, -0.5, 0.036298 },
{ -0.5, -0.5, 0.5, 0.163319 },
{ -0.5, 0.5, 0.0, 0.213342 },
{ -0.5, 0.5, -0.5, 0.145218 },
{ -0.5, 0.5, 0.5, 0.272239 },
{ 0.5, 0.0, 0.0, 0.345731 },
{ 0.5, 0.0, -0.5, 0.273122 },
{ 0.5, 0.0, 0.5, 0.418340 },
{ 0.5, -0.5, 0.0, 0.213342 },
{ 0.5, -0.5, -0.5, 0.145218 },
{ 0.5, -0.5, 0.5, 0.272239 },
{ 0.5, 0.5, 0.0, 0.478120 },
{ 0.5, 0.5, -0.5, 0.419223 },
{ 0.5, 0.5, 0.5, 0.546244 },
// known analytical values
{ 0.0, 0.0, std::sqrt(1/2.0), 3.0/8},
// { 0.0, big, any, 0.500000 },
{ 0.0, 30, -1.0, 0.500000 },
{ 0.0, 30, 0.0, 0.500000 },
{ 0.0, 30, 1.0, 0.500000 },
// { big, big, any, 1.000000 },
{ 30, 30, -1.0, 1.000000 },
{ 30, 30, 0.0, 1.000000 },
{ 30, 30, 1.0, 1.000000 },
// {-big, any, any, 0.000000 }
{ -30, -1.0, -1.0, 0.000000 },
{ -30, 0.0, -1.0, 0.000000 },
{ -30, 1.0, -1.0, 0.000000 },
{ -30, -1.0, 0.0, 0.000000 },
{ -30, 0.0, 0.0, 0.000000 },
{ -30, 1.0, 0.0, 0.000000 },
{ -30, -1.0, 1.0, 0.000000 },
{ -30, 0.0, 1.0, 0.000000 },
{ -30, 1.0, 1.0, 0.000000 }
};
for (Size i=0; i<LENGTH(values); i++) {
Bivariate bcd(values[i].rho);
Real value = bcd(values[i].a, values[i].b);
Real tolerance = 1.0e-6;
if (std::fabs(value-values[i].result) >= tolerance) {
BOOST_ERROR(tag << " bivariate cumulative distribution\n"
<< " case: " << i+1 << "\n"
<< QL_FIXED
<< " a: " << values[i].a << "\n"
<< " b: " << values[i].b << "\n"
<< " rho: " << values[i].rho <<"\n"
<< QL_SCIENTIFIC
<< " tabulated value: " << values[i].result << "\n"
<< " result: " << value);
}
}
}
template <class Bivariate>
void checkBivariateAtZero(const char* tag, Real tolerance) {
/*
BVN(0.0,0.0,rho) = 1/4 + arcsin(rho)/(2*M_PI)
"Handbook of the Normal Distribution", J.K. Patel & C.B.Read, 2nd Ed, 1996
*/
const Real rho[] = { 0.0, 0.1, 0.2, 0.3, 0.4, 0.5,
0.6, 0.7, 0.8, 0.9, 0.99999 };
const Real x(0.0);
const Real y(0.0);
for (Size i=0;i<LENGTH(rho);i++) {
for (Integer sgn=-1; sgn < 2; sgn+=2) {
Bivariate bvn(sgn*rho[i]);
Real expected = 0.25 + std::asin(sgn*rho[i]) / (2*M_PI) ;
Real realised = bvn(x,y);
if (std::fabs(realised-expected)>=tolerance) {
BOOST_ERROR(tag << " bivariate cumulative distribution\n"
<< QL_SCIENTIFIC
<< " rho: " << sgn*rho[i] << "\n"
<< " expected: " << expected << "\n"
<< " realised: " << realised << "\n"
<< " tolerance: " << tolerance);
}
}
}
}
QL_END_TEST_LOCALS(DistributionTest)
void DistributionTest::testNormal() {
BOOST_MESSAGE("Testing normal distributions...");
InverseCumulativeNormal invCumStandardNormal;
Real check = invCumStandardNormal(0.5);
if (check != 0.0e0) {
BOOST_ERROR("C++ inverse cumulative of the standard normal at 0.5 is "
<< QL_SCIENTIFIC << check
<< "\n instead of zero: something is wrong!");
}
NormalDistribution normal(average,sigma);
CumulativeNormalDistribution cum(average,sigma);
InverseCumulativeNormal invCum(average,sigma);
Size numberOfStandardDeviation = 6;
Real xMin = average - numberOfStandardDeviation*sigma,
xMax = average + numberOfStandardDeviation*sigma;
Size N = 100001;
Real h = (xMax-xMin)/(N-1);
std::vector<Real> x(N), y(N), yd(N), temp(N), diff(N);
Size i;
for (i=0; i<N; i++)
x[i] = xMin+h*i;
std::transform(x.begin(),x.end(),y.begin(),std::ptr_fun(gaussian));
std::transform(x.begin(),x.end(),yd.begin(),
std::ptr_fun(gaussianDerivative));
// check that normal = Gaussian
std::transform(x.begin(),x.end(),temp.begin(),normal);
std::transform(y.begin(),y.end(),temp.begin(),diff.begin(),
std::minus<Real>());
Real e = norm(diff.begin(),diff.end(),h);
if (e > 1.0e-16) {
BOOST_ERROR("norm of C++ NormalDistribution minus analytic Gaussian: "
<< QL_SCIENTIFIC << e << "\n"
<< "tolerance exceeded");
}
// check that invCum . cum = identity
std::transform(x.begin(),x.end(),temp.begin(),cum);
std::transform(temp.begin(),temp.end(),temp.begin(),invCum);
std::transform(x.begin(),x.end(),temp.begin(),diff.begin(),
std::minus<Real>());
e = norm(diff.begin(),diff.end(),h);
if (e > 1.0e-7) {
BOOST_ERROR("norm of invCum . cum minus identity: "
<< QL_SCIENTIFIC << e << "\n"
<< "tolerance exceeded");
}
// check that cum.derivative = Gaussian
for (i=0; i<x.size(); i++)
temp[i] = cum.derivative(x[i]);
std::transform(y.begin(),y.end(),temp.begin(),diff.begin(),
std::minus<Real>());
e = norm(diff.begin(),diff.end(),h);
if (e > 1.0e-16) {
BOOST_ERROR(
"norm of C++ Cumulative.derivative minus analytic Gaussian: "
<< QL_SCIENTIFIC << e << "\n"
<< "tolerance exceeded");
}
// check that normal.derivative = gaussianDerivative
for (i=0; i<x.size(); i++)
temp[i] = normal.derivative(x[i]);
std::transform(yd.begin(),yd.end(),temp.begin(),diff.begin(),
std::minus<Real>());
e = norm(diff.begin(),diff.end(),h);
if (e > 1.0e-16) {
BOOST_ERROR("norm of C++ Normal.derivative minus analytic derivative: "
<< QL_SCIENTIFIC << e << "\n"
<< "tolerance exceeded");
}
}
void DistributionTest::testBivariate() {
BOOST_MESSAGE("Testing bivariate cumulative normal distribution...");
checkBivariateAtZero<BivariateCumulativeNormalDistributionDr78>(
"Drezner 1978", 1.0e-6);
checkBivariate<BivariateCumulativeNormalDistributionDr78>("Drezner 1978");
checkBivariateAtZero<BivariateCumulativeNormalDistributionWe04DP>(
"West 2004", 1.0e-15);
checkBivariate<BivariateCumulativeNormalDistributionWe04DP>("West 2004");
}
void DistributionTest::testPoisson() {
BOOST_MESSAGE("Testing Poisson distribution...");
for (Real mean=0.0; mean<=10.0; mean+=0.5) {
BigNatural i = 0;
PoissonDistribution pdf(mean);
Real calculated = pdf(i);
Real logHelper = -mean;
Real expected = std::exp(logHelper);
Real error = std::fabs(calculated-expected);
if (error > 1.0e-16)
BOOST_ERROR("Poisson pdf(" << mean << ")(" << i << ")\n"
<< std::setprecision(16)
<< " calculated: " << calculated << "\n"
<< " expected: " << expected << "\n"
<< " error: " << error);
for (i=1; i<25; i++) {
calculated = pdf(i);
if (mean == 0.0) {
expected = 0.0;
} else {
logHelper = logHelper+std::log(mean)-std::log(Real(i));
expected = std::exp(logHelper);
}
error = std::fabs(calculated-expected);
if (error>1.0e-13)
BOOST_ERROR("Poisson pdf(" << mean << ")(" << i << ")\n"
<< std::setprecision(13)
<< " calculated: " << calculated << "\n"
<< " expected: " << expected << "\n"
<< " error: " << error);
}
}
}
void DistributionTest::testCumulativePoisson() {
BOOST_MESSAGE("Testing cumulative Poisson distribution...");
for (Real mean=0.0; mean<=10.0; mean+=0.5) {
BigNatural i = 0;
CumulativePoissonDistribution cdf(mean);
Real cumCalculated = cdf(i);
Real logHelper = -mean;
Real cumExpected = std::exp(logHelper);
Real error = std::fabs(cumCalculated-cumExpected);
if (error>1.0e-13)
BOOST_ERROR("Poisson cdf(" << mean << ")(" << i << ")\n"
<< std::setprecision(13)
<< " calculated: " << cumCalculated << "\n"
<< " expected: " << cumExpected << "\n"
<< " error: " << error);
for (i=1; i<25; i++) {
cumCalculated = cdf(i);
if (mean == 0.0) {
cumExpected = 1.0;
} else {
logHelper = logHelper+std::log(mean)-std::log(Real(i));
cumExpected += std::exp(logHelper);
}
error = std::fabs(cumCalculated-cumExpected);
if (error>1.0e-12)
BOOST_ERROR("Poisson cdf(" << mean << ")(" << i << ")\n"
<< std::setprecision(12)
<< " calculated: " << cumCalculated << "\n"
<< " expected: " << cumExpected << "\n"
<< " error: " << error);
}
}
}
void DistributionTest::testInverseCumulativePoisson() {
BOOST_MESSAGE("Testing inverse cumulative Poisson distribution...");
InverseCumulativePoisson icp(1.0);
Real data[] = { 0.2,
0.5,
0.9,
0.98,
0.99,
0.999,
0.9999,
0.99995,
0.99999,
0.999999,
0.9999999,
0.99999999
};
for (Size i=0; i<LENGTH(data); i++) {
if (!close(icp(data[i]), i)) {
BOOST_ERROR(std::setprecision(8)
<< "failed to reproduce known value for x = "
<< data[i] << "\n"
<< " calculated: " << icp(data[i]) << "\n"
<< " expected: " << Real(i));
}
}
}
test_suite* DistributionTest::suite() {
test_suite* suite = BOOST_TEST_SUITE("Distribution tests");
suite->add(BOOST_TEST_CASE(&DistributionTest::testNormal));
suite->add(BOOST_TEST_CASE(&DistributionTest::testBivariate));
suite->add(BOOST_TEST_CASE(&DistributionTest::testPoisson));
suite->add(BOOST_TEST_CASE(&DistributionTest::testCumulativePoisson));
suite->add(
BOOST_TEST_CASE(&DistributionTest::testInverseCumulativePoisson));
return suite;
}
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