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/* -*- mode: c++; tab-width: 4; indent-tabs-mode: nil; c-basic-offset: 4 -*- */
/*
Copyright (C) 2010 Adrian O' Neill
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/exercise.hpp>
#include <ql/experimental/variancegamma/analyticvariancegammaengine.hpp>
#include <ql/math/distributions/gammadistribution.hpp>
#include <ql/math/integrals/gausslobattointegral.hpp>
#include <ql/math/integrals/kronrodintegral.hpp>
#include <ql/math/integrals/segmentintegral.hpp>
#include <ql/pricingengines/blackscholescalculator.hpp>
#include <utility>
namespace QuantLib {
namespace {
class Integrand {
public:
Integrand(ext::shared_ptr<StrikedTypePayoff> payoff,
Real s0,
Real t,
Real riskFreeDiscount,
Real dividendDiscount,
Real sigma,
Real nu,
Real theta)
: payoff_(std::move(payoff)), s0_(s0), t_(t), riskFreeDiscount_(riskFreeDiscount),
dividendDiscount_(dividendDiscount), sigma_(sigma), nu_(nu), theta_(theta) {
omega_ = std::log(1.0 - theta_ * nu_ - (sigma_ * sigma_ * nu_) / 2.0) / nu_;
// We can precompute the denominator of the gamma pdf (does not depend on x)
// shape = t_/nu_, scale = nu_
GammaFunction gf;
gammaDenom_ = std::exp(gf.logValue(t_ / nu_)) * std::pow(nu_, t_ / nu_);
}
Real operator()(Real x) const {
// Compute adjusted black scholes price
Real s0_adj = s0_ * std::exp(theta_ * x + omega_ * t_ + (sigma_ * sigma_ * x) / 2.0);
Real vol_adj = sigma_ * std::sqrt(x / t_);
vol_adj *= std::sqrt(t_);
BlackScholesCalculator bs(payoff_, s0_adj, dividendDiscount_, vol_adj, riskFreeDiscount_);
Real bsprice = bs.value();
// Multiply by gamma distribution
Real gamp = (std::pow(x, t_ / nu_ - 1.0) * std::exp(-x / nu_)) / gammaDenom_;
Real result = bsprice * gamp;
return result;
}
private:
ext::shared_ptr<StrikedTypePayoff> payoff_;
Real s0_;
Real t_;
Real riskFreeDiscount_;
Real dividendDiscount_;
Rate sigma_;
Real nu_;
Real theta_;
Real omega_;
Real gammaDenom_;
};
}
VarianceGammaEngine::VarianceGammaEngine(ext::shared_ptr<VarianceGammaProcess> process,
Real absoluteError)
: process_(std::move(process)), absErr_(absoluteError) {
QL_REQUIRE(absErr_ > 0, "absolute error must be positive");
registerWith(process_);
}
void VarianceGammaEngine::calculate() const {
QL_REQUIRE(arguments_.exercise->type() == Exercise::European,
"not an European Option");
ext::shared_ptr<StrikedTypePayoff> payoff =
ext::dynamic_pointer_cast<StrikedTypePayoff>(arguments_.payoff);
QL_REQUIRE(payoff, "non-striked payoff given");
DiscountFactor dividendDiscount =
process_->dividendYield()->discount(
arguments_.exercise->lastDate());
DiscountFactor riskFreeDiscount =
process_->riskFreeRate()->discount(arguments_.exercise->lastDate());
DayCounter rfdc = process_->riskFreeRate()->dayCounter();
Time t = rfdc.yearFraction(process_->riskFreeRate()->referenceDate(),
arguments_.exercise->lastDate());
Integrand f(payoff,
process_->x0(),
t, riskFreeDiscount, dividendDiscount,
process_->sigma(), process_->nu(), process_->theta());
Real infinity = 15.0 * std::sqrt(process_->nu() * t);
Real target = absErr_*1e-4;
Real val = f(infinity);
while (std::abs(val)>target){
infinity*=1.5;
val = f(infinity);
}
// the integration is split due to occasional singularities at 0
Real split = 0.1;
GaussKronrodNonAdaptive integrator1(absErr_, 1000, 0);
Real pvA = integrator1(f, 0, split);
GaussLobattoIntegral integrator2(2000, absErr_);
Real pvB = integrator2(f, split, infinity);
results_.value = pvA + pvB;
}
}
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