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/* -*- mode: c++; tab-width: 4; indent-tabs-mode: nil; c-basic-offset: 4 -*- */
/*
Copyright (C) 2010 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 analyticptdhestonengine.cpp
\brief analytic piecewise time dependent Heston-model engine
*/
#include <ql/math/functional.hpp>
#include <ql/instruments/payoffs.hpp>
#include <ql/pricingengines/vanilla/analyticptdhestonengine.hpp>
namespace QuantLib {
// helper class for integration
class AnalyticPTDHestonEngine::Fj_Helper
: public std::unary_function<Real, Real> {
public:
Fj_Helper(
const Handle<PiecewiseTimeDependentHestonModel>& model,
Time term, Real strike, Size j);
Real operator()(Real phi) const;
private:
const Size j_;
const Time term_;
const Real v0_, x_, sx_;
std::vector<Rate> r_, q_;
const boost::shared_ptr<YieldTermStructure> qTS_;
const Handle<PiecewiseTimeDependentHestonModel> model_;
const TimeGrid timeGrid_;
};
AnalyticPTDHestonEngine::Fj_Helper::Fj_Helper(
const Handle<PiecewiseTimeDependentHestonModel>& model,
Time term, Real strike, Size j)
: j_(j),
term_(term),
v0_(model->v0()),
x_ (std::log(model->s0())),
sx_(std::log(strike)),
r_(model->timeGrid().size()-1),
q_(model->timeGrid().size()-1),
model_(model),
timeGrid_(model->timeGrid()){
for (Size i=0; i <timeGrid_.size()-1; ++i) {
const Time begin = std::min(term_, timeGrid_[i]);
const Time end = std::min(term_, timeGrid_[i+1]);
r_[i] = model->riskFreeRate()
->forwardRate(begin, end, Continuous, NoFrequency).rate();
q_[i] = model->dividendYield()
->forwardRate(begin, end, Continuous, NoFrequency).rate();
}
QL_REQUIRE(term_ < model_->timeGrid().back(), "maturity is too large");
}
Real AnalyticPTDHestonEngine::Fj_Helper::operator()(Real phi) const {
// avoid numeric overflow for phi->0.
// todo: use l'Hospital's rule use to get lim_{phi->0}
phi = std::max(Real(std::numeric_limits<float>::epsilon()), phi);
std::complex<Real> D = 0.0;
std::complex<Real> C = 0.0;
for (Size i=timeGrid_.size()-1; i > 0; --i) {
const Time begin = timeGrid_[i-1];
if (begin < term_) {
const Time end = std::min(term_, timeGrid_[i]);
const Time tau = end-begin;
const Time t = 0.5*(end+begin);
const Real rho = model_->rho(t);
const Real sigma = model_->sigma(t);
const Real kappa = model_->kappa(t);
const Real theta = model_->theta(t);
const Real sigma2 = sigma*sigma;
const Real t0 = kappa - ((j_== 1)? rho*sigma : 0);
const Real rpsig = rho*sigma*phi;
const std::complex<Real> t1 = t0+std::complex<Real>(0, -rpsig);
const std::complex<Real> d = std::sqrt(t1*t1 - sigma2*phi
*std::complex<Real>(-phi, (j_== 1)? 1 : -1));
const std::complex<Real> g = (t1-d)/(t1+d);
const std::complex<Real> gt
= (t1-d - D*sigma2)/(t1+d - D*sigma2);
D = (t1+d)/sigma2*(g-gt*std::exp(-d*tau))
/(1.0-gt*std::exp(-d*tau));
const std::complex<Real> lng
= std::log((1.0 - gt*std::exp(-d*tau))/(1.0 - gt));
C =(kappa*theta)/sigma2*((t1-d)*tau-2.0*lng)
+ std::complex<Real>(0.0, phi*(r_[i-1]-q_[i-1])*tau) + C;
}
}
return std::exp(v0_*D+C+std::complex<Real>(0.0, phi*(x_ - sx_))).imag()
/phi;
}
AnalyticPTDHestonEngine::AnalyticPTDHestonEngine(
const boost::shared_ptr<PiecewiseTimeDependentHestonModel>& model,
Size integrationOrder)
: GenericModelEngine<PiecewiseTimeDependentHestonModel,
VanillaOption::arguments,
VanillaOption::results>(model),
integration_(new AnalyticHestonEngine::Integration(
AnalyticHestonEngine::Integration::gaussLaguerre(integrationOrder))) {
}
AnalyticPTDHestonEngine::AnalyticPTDHestonEngine(
const boost::shared_ptr<PiecewiseTimeDependentHestonModel>& model,
Real relTolerance, Size maxEvaluations)
: GenericModelEngine<PiecewiseTimeDependentHestonModel,
VanillaOption::arguments,
VanillaOption::results>(model),
integration_(new AnalyticHestonEngine::Integration(
AnalyticHestonEngine::Integration::gaussLobatto(
relTolerance, Null<Real>(), maxEvaluations))) {
}
void AnalyticPTDHestonEngine::calculate() const {
// this is an european option pricer
QL_REQUIRE(arguments_.exercise->type() == Exercise::European,
"not an European option");
// plain vanilla
boost::shared_ptr<PlainVanillaPayoff> payoff =
boost::dynamic_pointer_cast<PlainVanillaPayoff>(arguments_.payoff);
QL_REQUIRE(payoff, "non-striked payoff given");
const Real v0 = model_->v0();
const Real spotPrice = model_->s0();
QL_REQUIRE(spotPrice > 0.0, "negative or null underlying given");
const Real strike = payoff->strike();
const Real term
= model_->riskFreeRate()->dayCounter().yearFraction(
model_->riskFreeRate()->referenceDate(),
arguments_.exercise->lastDate());
const Real riskFreeDiscount = model_->riskFreeRate()->discount(
arguments_.exercise->lastDate());
const Real dividendDiscount = model_->dividendYield()->discount(
arguments_.exercise->lastDate());
//average values
const TimeGrid& timeGrid = model_->timeGrid();
const Size n = timeGrid.size()-1;
Real kappaAvg = 0.0, thetaAvg = 0.0, sigmaAvg=0.0, rhoAvg = 0.0;
for (Size i=1; i <= n; ++i) {
const Time t = 0.5*(timeGrid[i-1] + timeGrid[i]);
kappaAvg += model_->kappa(t);
thetaAvg += model_->theta(t);
sigmaAvg += model_->sigma(t);
rhoAvg += model_->rho(t);
}
kappaAvg/=n; thetaAvg/=n; sigmaAvg/=n; rhoAvg/=n;
const Real c_inf = std::min(10.0, std::max(0.0001,
std::sqrt(1.0-square<Real>()(rhoAvg))/sigmaAvg))
*(v0 + kappaAvg*thetaAvg*term);
const Real p1 = integration_->calculate(c_inf,
Fj_Helper(model_, term, strike, 1))/M_PI;
const Real p2 = integration_->calculate(c_inf,
Fj_Helper(model_, term, strike, 2))/M_PI;
switch (payoff->optionType())
{
case Option::Call:
results_.value = spotPrice*dividendDiscount*(p1+0.5)
- strike*riskFreeDiscount*(p2+0.5);
break;
case Option::Put:
results_.value = spotPrice*dividendDiscount*(p1-0.5)
- strike*riskFreeDiscount*(p2-0.5);
break;
default:
QL_FAIL("unknown option type");
}
}
}
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