analyticvariancegammaengine.cpp

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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
<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.
*/
 
#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;
    }
 
}