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/* -*- mode: c++; tab-width: 4; indent-tabs-mode: nil; c-basic-offset: 4 -*- */
/*
Copyright (C) 2012 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
<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.
*/
/*! \file analytichestonengine.cpp
\brief analytic Heston-Hull-White engine based on the H1-HW approximation
*/
#include <ql/math/distributions/gammadistribution.hpp>
#include <ql/pricingengines/vanilla/analytich1hwengine.hpp>
namespace QuantLib {
// integration helper class
class AnalyticH1HWEngine::Fj_Helper
: public std::unary_function<std::complex<Real>, Real> {
public:
Fj_Helper(const Handle<HestonModel>& hestonModel,
const boost::shared_ptr<HullWhite>& hullWhiteModel,
Real rho_xr, Time term, Real strike, Size j);
std::complex<Real> operator()(Real u) const;
private:
Real c(Time t) const;
Real lambda(Time t) const;
Real Lambda(Time t) const;
Real LambdaApprox(Time t) const;
const Size j_;
const Real lambda_, eta_;
const Real v0_, kappa_, theta_, gamma_;
const Real d_;
const Real rhoSr_;
const Time term_;
};
AnalyticH1HWEngine::Fj_Helper::Fj_Helper(
const Handle<HestonModel>& hestonModel,
const boost::shared_ptr<HullWhite>& hullWhiteModel,
Real rhoSr, Time term, Real, Size j)
: j_ (j),
lambda_(hullWhiteModel->a()),
eta_ (hullWhiteModel->sigma()),
v0_ (hestonModel->v0()),
kappa_ (hestonModel->kappa()),
theta_ (hestonModel->theta()),
gamma_ (hestonModel->sigma()),
d_ (4.0*kappa_*theta_/(gamma_*gamma_)),
rhoSr_ (rhoSr),
term_ (term) {
}
Real AnalyticH1HWEngine::Fj_Helper::c(Time t) const {
return gamma_*gamma_/(4*kappa_)*(1.0-std::exp(-kappa_*t));
}
Real AnalyticH1HWEngine::Fj_Helper::lambda(Time t) const {
return 4.0*kappa_*v0_*std::exp(-kappa_*t)
/(gamma_*gamma_*(1.0-std::exp(-kappa_*t)));
}
Real AnalyticH1HWEngine::Fj_Helper::LambdaApprox(Time t) const {
return std::sqrt( c(t)*(lambda(t)-1.0)
+ c(t)*d_*(1.0 + 1.0/(2.0*(d_+lambda(t)))));
}
Real AnalyticH1HWEngine::Fj_Helper::Lambda(Time t) const {
const GammaFunction g = GammaFunction();
const Size maxIter = 1000;
const Real lambdaT = lambda(t);
Size i=0;
Real retVal = 0.0, s;
do {
Real k = static_cast<Real>(i);
s=std::exp(k*std::log(0.5*lambdaT) + g.logValue(0.5*(1+d_)+k)
- g.logValue(k+1) - g.logValue(0.5*d_+k));
retVal += s;
} while (s > std::numeric_limits<float>::epsilon() && ++i < maxIter);
QL_REQUIRE(i < maxIter, "can not calculate Lambda");
retVal *= std::sqrt(2*c(t)) * std::exp(-0.5*lambdaT);
return retVal;
}
std::complex<Real> AnalyticH1HWEngine::Fj_Helper::operator()(Real u) const {
const Real gamma2 = gamma_*gamma_;
Real a, b, c;
if (8.0*kappa_*theta_/gamma2 > 1.0) {
a = std::sqrt(theta_-gamma2/(8.0*kappa_));
b = std::sqrt(v0_) - a;
c =-std::log((LambdaApprox(1.0)-a)/b);
}
else {
a = std::sqrt(gamma2/(2.0*kappa_))
*std::exp( GammaFunction().logValue(0.5*(d_+1.0))
- GammaFunction().logValue(0.5*d_));
const Time t1 = 0.0;
const Time t2 = 1.0/kappa_;
const Real Lambda_t1 = std::sqrt(v0_);
const Real Lambda_t2 = Lambda(t2);
c = std::log((Lambda_t2-a)/(Lambda_t1-a))/(t1-t2);
b = std::exp(c*t1)*(Lambda_t1-a);
}
const std::complex<Real> I4 =
-1.0/lambda_* std::complex<Real>(u*u, ((j_ == 1u)? -u : u))
*( b/c*(1.0 - std::exp(-c*term_))
+ a*term_
+ a/lambda_*(std::exp(-lambda_*term_) - 1.0)
+ b/(c-lambda_)*std::exp(-c*term_)
*(1.0 - std::exp(-term_*(lambda_-c))) );
return eta_*rhoSr_*I4;
}
AnalyticH1HWEngine::AnalyticH1HWEngine(
const boost::shared_ptr<HestonModel>& model,
const boost::shared_ptr<HullWhite>& hullWhiteModel,
Real rhoSr, Size integrationOrder)
: AnalyticHestonHullWhiteEngine(model, hullWhiteModel, integrationOrder),
rhoSr_(rhoSr) {
QL_REQUIRE(rhoSr_ >= 0.0, "Fourier integration is not stable if "
"the equity interest rate correlation is negative");
}
AnalyticH1HWEngine::AnalyticH1HWEngine(
const boost::shared_ptr<HestonModel>& model,
const boost::shared_ptr<HullWhite>& hullWhiteModel,
Real rhoSr, Real relTolerance, Size maxEvaluations)
: AnalyticHestonHullWhiteEngine(model, hullWhiteModel,
relTolerance, maxEvaluations),
rhoSr_(rhoSr) {
}
std::complex<Real> AnalyticH1HWEngine::addOnTerm(Real u, Time t, Size j)
const {
return AnalyticHestonHullWhiteEngine::addOnTerm(u, t, j)
+ Fj_Helper(model_, hullWhiteModel_, rhoSr_, t, 0.0, j)(u);
}
}
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