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/* -*- mode: c++; tab-width: 4; indent-tabs-mode: nil; c-basic-offset: 4 -*- */
/*
Copyright (C) 2016 Klaus Spanderen
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
<https://www.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.
*/
/*! \file squarerootclvmodel.cpp
\brief CLV model with a square root kernel process
*/
#include <ql/processes/blackscholesprocess.hpp>
#include <ql/processes/squarerootprocess.hpp>
#include <ql/math/integrals/gaussianquadratures.hpp>
#include <ql/experimental/models/squarerootclvmodel.hpp>
#include <ql/methods/finitedifferences/utilities/gbsmrndcalculator.hpp>
#include <boost/math/distributions/non_central_chi_squared.hpp>
#include <utility>
namespace QuantLib {
SquareRootCLVModel::SquareRootCLVModel(
const ext::shared_ptr<GeneralizedBlackScholesProcess>& bsProcess,
ext::shared_ptr<SquareRootProcess> sqrtProcess,
std::vector<Date> maturityDates,
Size lagrangeOrder,
Real pMax,
Real pMin)
: pMax_(pMax), pMin_(pMin), bsProcess_(bsProcess), sqrtProcess_(std::move(sqrtProcess)),
maturityDates_(std::move(maturityDates)), lagrangeOrder_(lagrangeOrder),
rndCalculator_(ext::make_shared<GBSMRNDCalculator>(bsProcess)) {}
Real SquareRootCLVModel::cdf(const Date& d, Real k) const {
return rndCalculator_->cdf(k, bsProcess_->time(d));
}
Real SquareRootCLVModel::invCDF(const Date& d, Real q) const {
return rndCalculator_->invcdf(q, bsProcess_->time(d));
}
std::pair<Real, Real> SquareRootCLVModel::nonCentralChiSquaredParams(
const Date& d) const {
const Time t = bsProcess_->time(d);
const Real kappa = sqrtProcess_->a();
const Real theta = sqrtProcess_->b();
const Real sigma = sqrtProcess_->sigma();
const Real df = 4*theta*kappa/(sigma*sigma);
const Real ncp = 4*kappa*std::exp(-kappa*t)
/ (sigma*sigma*(1-std::exp(-kappa*t)))*sqrtProcess_->x0();
return std::make_pair(df, ncp);
}
Array SquareRootCLVModel::collocationPointsX(const Date& d) const {
const std::pair<Real, Real> p = nonCentralChiSquaredParams(d);
Array x = GaussianQuadrature(lagrangeOrder_,
GaussNonCentralChiSquaredPolynomial(p.first, p.second))
.x();
std::sort(x.begin(), x.end());
const boost::math::non_central_chi_squared_distribution<Real>
dist(p.first, p.second);
const Real xMin = std::max(x.front(),
(pMin_ == Null<Real>())
? 0.0 : boost::math::quantile(dist, pMin_));
const Real xMax = std::min(x.back(),
(pMax_ == Null<Real>())
? QL_MAX_REAL : boost::math::quantile(dist, pMax_));
const Real b = xMin - x.front();
const Real a = (xMax - xMin)/(x.back() - x.front());
for (Real& i : x) {
i = a * i + b;
}
return x;
}
Array SquareRootCLVModel::collocationPointsY(const Date& d) const {
const Array x = collocationPointsX(d);
const std::pair<Real, Real> params = nonCentralChiSquaredParams(d);
const boost::math::non_central_chi_squared_distribution<Real>
dist(params.first, params.second);
Array s(x.size());
for (Size i=0, n=s.size(); i < n; ++i) {
const Real q = boost::math::cdf(dist, x[i]);
s[i] = invCDF(d, q);
}
return s;
}
std::function<Real(Time, Real)> SquareRootCLVModel::g() const {
calculate();
return g_;
}
void SquareRootCLVModel::performCalculations() const {
g_ = std::function<Real(Time, Real)>(MappingFunction(*this));
}
SquareRootCLVModel::MappingFunction::MappingFunction(
const SquareRootCLVModel& model)
: s_(ext::make_shared<Matrix>(
model.maturityDates_.size(), model.lagrangeOrder_)),
x_(ext::make_shared<Matrix>(
model.maturityDates_.size(), model.lagrangeOrder_)) {
std::vector<Date> maturityDates = model.maturityDates_;
std::sort(maturityDates.begin(), maturityDates.end());
const ext::shared_ptr<GeneralizedBlackScholesProcess>&
bsProcess = model.bsProcess_;
for (Size i=0, n = maturityDates.size(); i < n; ++i) {
const Date maturityDate = maturityDates[i];
const Array x = model.collocationPointsX(maturityDate);
const Array y = model.collocationPointsY(maturityDate);
std::copy(x.begin(), x.end(), x_->row_begin(i));
std::copy(y.begin(), y.end(), s_->row_begin(i));
const Time maturity = bsProcess->time(maturityDate);
interpl.insert(
std::make_pair(maturity,
ext::make_shared<LagrangeInterpolation>(
x_->row_begin(i), x_->row_end(i),
s_->row_begin(i))));
}
}
Real SquareRootCLVModel::MappingFunction::operator()(Time t,Real x) const {
const auto ge = interpl.lower_bound(t);
if (close_enough(ge->first, t)) {
return (*ge->second)(x, true);
}
QL_REQUIRE(ge != interpl.end() && ge != interpl.begin(),
"extrapolation to large or small t is not allowed");
const Time t1 = ge->first;
const Real y1 = (*ge->second)(x, true);
interpl_type::const_iterator lt = ge;
std::advance(lt, -1);
const Time t0 = lt->first;
const Real y0 = (*lt->second)(x, true);
return y0 + (y1 - y0)/(t1 - t0)*(t - t0);
}
}
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