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/* -*- mode: c++; tab-width: 4; indent-tabs-mode: nil; c-basic-offset: 4 -*- */
/*
Copyright (C) 2018 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 cevrndcalculator.cpp */
#include <ql/errors.hpp>
#include <ql/math/functional.hpp>
#include <ql/math/solvers1d/brent.hpp>
#include <ql/math/distributions/normaldistribution.hpp>
#include <ql/methods/finitedifferences/utilities/cevrndcalculator.hpp>
#include <boost/math/special_functions/gamma.hpp>
#include <boost/math/distributions/non_central_chi_squared.hpp>
namespace QuantLib {
CEVRNDCalculator::CEVRNDCalculator(Real f0, Real alpha, Real beta)
: f0_(f0),
alpha_(alpha),
beta_(beta),
delta_((1.0-2.0*beta)/(1.0-beta)),
x0_(X(f0)) {
QL_REQUIRE(beta != 1.0, "beta can not be one");
}
Real CEVRNDCalculator::massAtZero(Time t) const {
if (delta_ < 2.0)
return 1.0-boost::math::gamma_p(-0.5*delta_+1.0,x0_/(2.0*t));
else
return 0.0;
}
Real CEVRNDCalculator::X(Real f) const {
return std::pow(f, 2.0*(1.0-beta_))/squared(alpha_*(1.0-beta_));
}
Real CEVRNDCalculator::invX(Real x) const {
return std::pow(x*squared(alpha_*(1.0-beta_)),
1.0/(2.0*(1.0-beta_)));
}
Real CEVRNDCalculator::pdf(Real f, Time t) const {
const Real y = X(f);
if (delta_ < 2.0) {
return boost::math::pdf(
boost::math::non_central_chi_squared_distribution<Real>(
4.0-delta_, y/t), x0_/t)/t * 2.0*(1.0-beta_)*y/f;
}
else {
return boost::math::pdf(
boost::math::non_central_chi_squared_distribution<Real>(
delta_, x0_/t), y/t)/t * 2.0*(beta_-1.0)*y/f;
}
}
Real CEVRNDCalculator::cdf(Real f, Time t) const {
const Real y = X(f);
if (delta_ < 2.0)
return 1.0 - boost::math::cdf(
boost::math::non_central_chi_squared_distribution<Real>(
2.0-delta_, y/t), x0_/t);
else
return 1.0 - boost::math::cdf(
boost::math::non_central_chi_squared_distribution<Real>(
delta_, x0_/t), y/t);
}
Real CEVRNDCalculator::sankaranApprox(Real c, Time t, Real x) const {
const Real a = x0_/t;
const Real b = 2.0 - delta_;
c = std::max(c, -0.45*b);
const Real h = 1 - 2*(b+c)*(b+3*c)/(3*squared(b+2*c));
const Real p = (b+2*c)/squared(b+c);
const Real m = (h-1)*(1-3*h);
const Real u = (std::pow(a/(b+c), h) - (1 + h*p*(h-1-0.5*(2-h)*m*p)))/
(h*std::sqrt(2*p)*(1+0.5*m*p));
return u - x;
}
Real CEVRNDCalculator::invcdf(Real q, Time t) const {
if (delta_ < 2.0) {
if (f0_ < QL_EPSILON || q < massAtZero(t))
return 0.0;
const Real x = InverseCumulativeNormal()(1-q);
auto cdfApprox = [&](Real _c){ return sankaranApprox(_c, t, x); };
const Real y0 = X(f0_)/t;
try {
Brent brent;
brent.setMaxEvaluations(20);
const Real guess =
invX(brent.solve(cdfApprox, 1e-8, y0, 0.02*y0) * t);
return InvCDFHelper(this, guess, 1e-8, 100).inverseCDF(q, t);
}
catch (...) {
return InvCDFHelper(this, f0_, 1e-8, 100).inverseCDF(q, t);
}
}
else {
const Real x = t * boost::math::quantile(
boost::math::non_central_chi_squared_distribution<Real>(
delta_, x0_/t), 1-q);
return invX(x);
}
}
}
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