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/* -*- mode: c++; tab-width: 4; indent-tabs-mode: nil; c-basic-offset: 4 -*- */
/*
Copyright (C) 2015 Johannes Göttker-Schnetmann
Copyright (C) 2015 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.
*/
#include <ql/math/functional.hpp>
#include <ql/math/integrals/gausslobattointegral.hpp>
#include <ql/methods/finitedifferences/utilities/bsmrndcalculator.hpp>
#include <ql/methods/finitedifferences/utilities/hestonrndcalculator.hpp>
#include <ql/processes/blackscholesprocess.hpp>
#include <ql/processes/hestonprocess.hpp>
#include <ql/termstructures/volatility/equityfx/blackconstantvol.hpp>
#include <ql/time/calendars/nullcalendar.hpp>
#include <complex>
#include <utility>
namespace QuantLib {
namespace {
struct HestonParams {
Real v0, kappa, theta, sigma, rho;
};
HestonParams getHestonParams(
const ext::shared_ptr<HestonProcess>& process) {
const HestonParams p = { process->v0(), process->kappa(),
process->theta(), process->sigma(),
process->rho() };
return p;
}
std::complex<Real> gamma(const HestonParams& p, Real p_x) {
return std::complex<Real>(p.kappa, p.rho*p.sigma*p_x);
}
std::complex<Real> omega(const HestonParams& p, Real p_x) {
const std::complex<Real> g = gamma(p, p_x);
return std::sqrt(g*g
+ p.sigma*p.sigma*std::complex<Real>(p_x*p_x, -p_x));
}
class CpxPv_Helper {
public:
CpxPv_Helper(const HestonParams& p, Real x, Time t)
: p_(p), t_(t), x_(x),
c_inf_(std::min(10.0, std::max(0.0001,
std::sqrt(1.0-squared(p_.rho))/p_.sigma))
*(p_.v0 + p_.kappa*p_.theta*t)) {}
Real operator()(Real x) const {
return std::real(transformPhi(x));
}
Real p0(Real p_x) const {
if (p_x < QL_EPSILON) {
return 0.0;
}
const Real u_x = std::max(QL_EPSILON, -std::log(p_x)/c_inf_);
return std::real(phi(u_x)
/((p_x*c_inf_)*std::complex<Real>(0.0, u_x)));
}
private:
std::complex<Real> transformPhi(Real x) const {
if (x < QL_EPSILON) {
return std::complex<Real>(0.0, 0.0);
}
const Real u_x = -std::log(x)/c_inf_;
return phi(u_x)/(x*c_inf_);
}
std::complex<Real> phi(Real p_x) const {
const Real sigma2 = p_.sigma*p_.sigma;
const std::complex<Real> g = gamma(p_, p_x);
const std::complex<Real> o = omega(p_, p_x);
const std::complex<Real> gamma = (g-o)/(g+o);
return 2.0*std::exp(std::complex<Real>(0.0, p_x*x_)
- p_.v0*std::complex<Real>(p_x*p_x, -p_x)
/(g+o*(1.0+std::exp(-o*t_))/(1.0-std::exp(-o*t_)))
+p_.kappa*p_.theta/sigma2*(
(g-o)*t_ - 2.0*std::log((1.0-gamma*std::exp(-o*t_))
/(1.0-gamma))));
}
const HestonParams p_;
const Time t_;
const Real x_, c_inf_;
};
}
HestonRNDCalculator::HestonRNDCalculator(ext::shared_ptr<HestonProcess> hestonProcess,
Real integrationEps,
Size maxIntegrationIterations)
: hestonProcess_(std::move(hestonProcess)), x0_(std::log(hestonProcess_->s0()->value())),
integrationEps_(integrationEps), maxIntegrationIterations_(maxIntegrationIterations) {}
Real HestonRNDCalculator::x_t(Real x, Time t) const {
const DiscountFactor dr = hestonProcess_->riskFreeRate()->discount(t);
const DiscountFactor dq = hestonProcess_->dividendYield()->discount(t);
return x - x0_ + std::log(dr/dq);
}
Real HestonRNDCalculator::pdf(Real x, Time t) const {
return GaussLobattoIntegral(
maxIntegrationIterations_, 0.1*integrationEps_)(
CpxPv_Helper(getHestonParams(hestonProcess_), x_t(x, t), t),
0.0, 1.0)/M_TWOPI;
}
Real HestonRNDCalculator::cdf(Real x, Time t) const {
CpxPv_Helper helper(getHestonParams(hestonProcess_), x_t(x, t), t);
return GaussLobattoIntegral(maxIntegrationIterations_, 0.1*integrationEps_)(
[&](Real p_x){ return helper.p0(p_x); },
0.0, 1.0)/M_TWOPI + 0.5;
}
Real HestonRNDCalculator::invcdf(Real p, Time t) const {
const Real v0 = hestonProcess_->v0();
const Real kappa = hestonProcess_->kappa();
const Real theta = hestonProcess_->theta();
const Volatility expVol
= std::sqrt(theta + (v0-theta)*(1-std::exp(-kappa*t))/(t*kappa));
const ext::shared_ptr<BlackScholesMertonProcess> bsmProcess(
ext::make_shared<BlackScholesMertonProcess>(
hestonProcess_->s0(),
hestonProcess_->dividendYield(),
hestonProcess_->riskFreeRate(),
Handle<BlackVolTermStructure>(
ext::make_shared<BlackConstantVol>(
hestonProcess_->riskFreeRate()->referenceDate(),
NullCalendar(),
expVol,
hestonProcess_->riskFreeRate()->dayCounter()))));
const Real guess = BSMRNDCalculator(bsmProcess).invcdf(p, t);
return RiskNeutralDensityCalculator::InvCDFHelper(
this, guess, 0.1*integrationEps_, maxIntegrationIterations_)
.inverseCDF(p, t);
}
}
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