qdfpamericanengine.cpp

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/* -*- mode: c++; tab-width: 4; indent-tabs-mode: nil; c-basic-offset: 4 -*- */
 
/*
 Copyright (C) 2022 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 qrfpamericanengine.cpp
*/
 
#include <ql/math/distributions/normaldistribution.hpp>
#include <ql/math/functional.hpp>
#include <ql/math/integrals/gaussianquadratures.hpp>
#include <ql/math/integrals/tanhsinhintegral.hpp>
#include <ql/math/interpolations/chebyshevinterpolation.hpp>
#include <ql/pricingengines/blackcalculator.hpp>
#include <ql/pricingengines/vanilla/qdfpamericanengine.hpp>
#include <utility>
#ifndef QL_BOOST_HAS_TANH_SINH
#    include <ql/math/integrals/gausslobattointegral.hpp>
#endif
 
namespace QuantLib {
 
    QdFpLegendreScheme::QdFpLegendreScheme(
        Size l, Size m, Size n, Size p):
        m_(m), n_(n),
        fpIntegrator_(ext::make_shared<GaussLegendreIntegrator>(l)),
        exerciseBoundaryIntegrator_(
            ext::make_shared<GaussLegendreIntegrator>(p)) {
 
        QL_REQUIRE(m_ > 0, "at least one fixed point iteration step is needed");
        QL_REQUIRE(n_ > 0, "at least one interpolation point is needed");
    }
 
    Size QdFpLegendreScheme::getNumberOfChebyshevInterpolationNodes()
        const {
        return n_;
    }
    Size QdFpLegendreScheme::getNumberOfNaiveFixedPointSteps() const {
        return m_-1;
    }
    Size QdFpLegendreScheme::getNumberOfJacobiNewtonFixedPointSteps()
        const {
        return Size(1);
    }
 
    ext::shared_ptr<Integrator>
        QdFpLegendreScheme::getFixedPointIntegrator() const {
        return fpIntegrator_;
    }
    ext::shared_ptr<Integrator>
        QdFpLegendreScheme::getExerciseBoundaryToPriceIntegrator()
        const {
        return exerciseBoundaryIntegrator_;
    }
 
    QdFpTanhSinhIterationScheme::QdFpTanhSinhIterationScheme(
        Size m, Size n, Real eps)
    : m_(m), n_(n),
#ifdef QL_BOOST_HAS_TANH_SINH
      integrator_(ext::make_shared<TanhSinhIntegral>(eps))
#else
      integrator_(ext::make_shared<GaussLobattoIntegral>(
          100000, QL_MAX_REAL, 0.1*eps))
#endif
    {}
 
    Size QdFpTanhSinhIterationScheme::getNumberOfChebyshevInterpolationNodes()
        const {
        return n_;
    }
    Size QdFpTanhSinhIterationScheme::getNumberOfNaiveFixedPointSteps() const {
        return m_-1;
    }
    Size QdFpTanhSinhIterationScheme::getNumberOfJacobiNewtonFixedPointSteps()
        const {
        return Size(1);
    }
 
    ext::shared_ptr<Integrator>
    QdFpTanhSinhIterationScheme::getFixedPointIntegrator() const {
        return integrator_;
    }
    ext::shared_ptr<Integrator>
    QdFpTanhSinhIterationScheme::getExerciseBoundaryToPriceIntegrator()
        const {
        return integrator_;
    }
 
    QdFpLegendreTanhSinhScheme::QdFpLegendreTanhSinhScheme(
        Size l, Size m, Size n, Real eps)
    : QdFpLegendreScheme(l, m, n, 1),
      eps_(eps) {}
 
    ext::shared_ptr<Integrator>
    QdFpLegendreTanhSinhScheme::getExerciseBoundaryToPriceIntegrator() const {
#ifdef QL_BOOST_HAS_TANH_SINH
            return ext::make_shared<TanhSinhIntegral>(eps_);
#else
            return ext::make_shared<GaussLobattoIntegral>(
                100000, QL_MAX_REAL, 0.1*eps_);
#endif
    }
 
 
    class DqFpEquation {
      public:
        DqFpEquation(Rate _r,
                     Rate _q,
                     Volatility _vol,
                     std::function<Real(Real)> B,
                     ext::shared_ptr<Integrator> _integrator)
        : r(_r), q(_q), vol(_vol), B(std::move(B)), integrator(std::move(_integrator)) {
            const auto legendreIntegrator =
                ext::dynamic_pointer_cast<GaussLegendreIntegrator>(integrator);
 
            if (legendreIntegrator != nullptr) {
                x_i = legendreIntegrator->getIntegration()->x();
                w_i = legendreIntegrator->getIntegration()->weights();
            }
        }
 
        virtual std::pair<Real, Real> NDd(Real tau, Real b) const = 0;
        virtual std::tuple<Real, Real, Real> f(Real tau, Real b) const = 0;
 
        virtual ~DqFpEquation() = default;
 
      protected:
        std::pair<Real, Real> d(Time t, Real z) const {
            const Real v = vol * std::sqrt(t);
            const Real m = (std::log(z) + (r-q)*t)/v + 0.5*v;
 
            return std::make_pair(m, m-v);
        }
 
        Array x_i, w_i;
        const Rate r, q;
        const Volatility vol;
 
        const std::function<Real(Real)> B;
        const ext::shared_ptr<Integrator> integrator;
 
        const NormalDistribution phi;
        const CumulativeNormalDistribution Phi;
    };
 
    class DqFpEquation_B: public DqFpEquation {
      public:
        DqFpEquation_B(Real K,
                       Rate _r,
                       Rate _q,
                       Volatility _vol,
                       std::function<Real(Real)> B,
                       ext::shared_ptr<Integrator> _integrator);
 
        std::pair<Real, Real> NDd(Real tau, Real b) const override;
        std::tuple<Real, Real, Real> f(Real tau, Real b) const override;
 
      private:
          const Real K;
    };
 
    class DqFpEquation_A: public DqFpEquation {
      public:
        DqFpEquation_A(Real K,
                       Rate _r,
                       Rate _q,
                       Volatility _vol,
                       std::function<Real(Real)> B,
                       ext::shared_ptr<Integrator> _integrator);
 
        std::pair<Real, Real> NDd(Real tau, Real b) const override;
        std::tuple<Real, Real, Real> f(Real tau, Real b) const override;
 
      private:
          const Real K;
    };
 
    DqFpEquation_A::DqFpEquation_A(Real K,
                                   Rate _r,
                                   Rate _q,
                                   Volatility _vol,
                                   std::function<Real(Real)> B,
                                   ext::shared_ptr<Integrator> _integrator)
    : DqFpEquation(_r, _q, _vol, std::move(B), std::move(_integrator)), K(K) {}
 
    std::tuple<Real, Real, Real> DqFpEquation_A::f(Real tau, Real b) const {
        const Real v = vol * std::sqrt(tau);
 
        Real N, D;
        if (tau < squared(QL_EPSILON)) {
            if (close_enough(b, K)) {
                N = 1/(M_SQRT2*M_SQRTPI * v);
                D = N + 0.5;
            }
            else {
                N = 0.0;
                D = (b > K)? 1.0: 0.0;
            }
        }
        else {
            const Real stv = std::sqrt(tau)/vol;
 
            Real K12, K3;
            if (!x_i.empty()) {
                K12 = K3 = 0.0;
 
                for (Integer i = x_i.size()-1; i >= 0; --i) {
                    const Real y = x_i[i];
                    const Real m = 0.25*tau*squared(1+y);
                    const std::pair<Real, Real> dpm = d(m, b/B(tau-m));
 
                    K12 += w_i[i] * std::exp(q*tau - q*m)
                        *(0.5*tau*(y+1)*Phi(dpm.first) + stv*phi(dpm.first));
                    K3 += w_i[i] * stv*std::exp(r*tau-r*m)*phi(dpm.second);
                }
            } else {
                K12 = (*integrator)([&, this](Real y) -> Real {
                    const Real m = 0.25*tau*squared(1+y);
                    const Real df = std::exp(q*tau - q*m);
 
                    if (y <= 5*QL_EPSILON - 1) {
                        if (close_enough(b, B(tau-m)))
                            return df*stv/(M_SQRT2*M_SQRTPI);
                        else
                            return 0.0;
                    }
                    else {
                        const Real dp = d(m, b/B(tau-m)).first;
                        return df*(0.5*tau*(y+1)*Phi(dp) + stv*phi(dp));
                    }
                }, -1, 1);
 
                K3 = (*integrator)([&, this](Real y) -> Real {
                    const Real m = 0.25*tau*squared(1+y);
                    const Real df = std::exp(r*tau-r*m);
 
                    if (y <= 5*QL_EPSILON - 1) {
                        if (close_enough(b, B(tau-m)))
                            return df*stv/(M_SQRT2*M_SQRTPI);
                        else
                            return 0.0;
                    }
                    else
                        return df*stv*phi(d(m, b/B(tau-m)).second);
                }, -1, 1);
            }
            const std::pair<Real, Real> dpm = d(tau, b/K);
            N = phi(dpm.second)/v + r*K3;
            D = phi(dpm.first)/v + Phi(dpm.first) + q*K12;
        }
 
        const Real alpha = K*std::exp(-(r-q)*tau);
        Real fv;
        if (tau < squared(QL_EPSILON)) {
            if (close_enough(b, K))
                fv = alpha;
            else if (b > K)
                fv = 0.0;
            else {
                if (close_enough(q, Real(0)))
                    fv = alpha*r*((q < 0)? -1.0 : 1.0)/QL_EPSILON;
                else
                    fv = alpha*r/q;
            }
        }
        else
            fv = alpha*N/D;
 
        return std::make_tuple(N, D, fv);
    }
 
    std::pair<Real, Real> DqFpEquation_A::NDd(Real tau, Real b) const {
        Real Dd, Nd;
 
        if (tau < squared(QL_EPSILON)) {
            if (close_enough(b, K)) {
                const Real sqTau = std::sqrt(tau);
                const Real vol2 = vol*vol;
                Dd = M_1_SQRTPI*M_SQRT_2*(
                    -(0.5*vol2 + r-q) / (b*vol*vol2*sqTau) + 1 / (b*vol*sqTau));
                Nd = M_1_SQRTPI*M_SQRT_2 * (-0.5*vol2 + r-q)  / (b*vol*vol2*sqTau);
            }
            else
                Dd = Nd = 0.0;
        }
        else {
            const std::pair<Real, Real> dpm = d(tau, b/K);
 
            Dd = -phi(dpm.first) * dpm.first / (b*vol*vol*tau) +
                    phi(dpm.first) / (b*vol * std::sqrt(tau));
            Nd = -phi(dpm.second) * dpm.second / (b*vol*vol*tau);
        }
 
        return std::make_pair(Nd, Dd);
    }
 
 
    DqFpEquation_B::DqFpEquation_B(Real K,
                                   Rate _r,
                                   Rate _q,
                                   Volatility _vol,
                                   std::function<Real(Real)> B,
                                   ext::shared_ptr<Integrator> _integrator)
    : DqFpEquation(_r, _q, _vol, std::move(B), std::move(_integrator)), K(K) {}
 
 
    std::tuple<Real, Real, Real> DqFpEquation_B::f(Real tau, Real b) const {
        Real N, D;
        if (tau < squared(QL_EPSILON)) {
            if (close_enough(b, K))
                N = D = 0.5;
            else if (b < K)
                N = D = 0.0;
            else
                N = D = 1.0;
        }
        else {
            Real ni, di;
            if (!x_i.empty()) {
                const Real c = 0.5*tau;
 
                ni = di = 0.0;
                for (Integer i = x_i.size()-1; i >= 0; --i) {
                    const Real u = c*x_i[i] + c;
                    const std::pair<Real, Real> dpm = d(tau - u, b/B(u));
                    ni += w_i[i] * std::exp(r*u)*Phi(dpm.second);
                    di += w_i[i] * std::exp(q*u)*Phi(dpm.first);
                }
                ni *= c;
                di *= c;
            } else {
                ni = (*integrator)([&, this](Real u) -> Real {
                    const Real df = std::exp(r*u);
                    if (u >= tau*(1 - 5*QL_EPSILON)) {
                        if (close_enough(b, B(u)))
                            return 0.5*df;
                        else
                            return df*((b < B(u)? 0.0: 1.0));
                    }
                    else
                        return df*Phi(d(tau - u, b/B(u)).second);
                }, 0, tau);
                di = (*integrator)([&, this](Real u) -> Real {
                    const Real df = std::exp(q*u);
                    if (u >= tau*(1 - 5*QL_EPSILON)) {
                        if (close_enough(b, B(u)))
                            return 0.5*df;
                        else
                            return df*((b < B(u)? 0.0: 1.0));
                    }
                    else
                        return df*Phi(d(tau - u, b/B(u)).first);
                    }, 0, tau);
            }
 
            const std::pair<Real, Real> dpm = d(tau, b/K);
 
            N = Phi(dpm.second) + r*ni;
            D = Phi(dpm.first) + q*di;
        }
 
        Real fv;
        const Real alpha = K*std::exp(-(r-q)*tau);
        if (tau < squared(QL_EPSILON)) {
            if (close_enough(b, K) || b > K)
                fv = alpha;
            else {
                if (close_enough(q, Real(0)))
                    fv = alpha*r*((q < 0)? -1.0 : 1.0)/QL_EPSILON;
                else
                    fv = alpha*r/q;
            }
        }
        else
            fv = alpha*N/D;
 
        return std::make_tuple(N, D, fv);
    }
 
    std::pair<Real, Real> DqFpEquation_B::NDd(Real tau, Real b) const {
        const std::pair<Real, Real> dpm = d(tau, b/K);
        return std::make_pair(
            phi(dpm.second) / (b*vol*std::sqrt(tau)),
            phi(dpm.first)  / (b*vol*std::sqrt(tau))
        );
    }
 
    QdFpAmericanEngine::QdFpAmericanEngine(
        ext::shared_ptr<GeneralizedBlackScholesProcess> bsProcess,
        ext::shared_ptr<QdFpIterationScheme> iterationScheme,
        FixedPointEquation fpEquation)
    : detail::QdPutCallParityEngine(std::move(bsProcess)),
      iterationScheme_(std::move(iterationScheme)),
      fpEquation_(fpEquation) {
    }
 
    ext::shared_ptr<QdFpIterationScheme>
    QdFpAmericanEngine::fastScheme() {
        static auto scheme = ext::make_shared<QdFpLegendreScheme>(7, 2, 7, 27);
        return scheme;
    }
 
    ext::shared_ptr<QdFpIterationScheme>
    QdFpAmericanEngine::accurateScheme() {
        static auto scheme = ext::make_shared<QdFpLegendreTanhSinhScheme>(25, 5, 13, 1e-8);
        return scheme;
    }
 
    ext::shared_ptr<QdFpIterationScheme>
    QdFpAmericanEngine::highPrecisionScheme() {
        static auto scheme = ext::make_shared<QdFpTanhSinhIterationScheme>(10, 30, 1e-10);
        return scheme;
    }
 
    Real QdFpAmericanEngine::calculatePut(
            Real S, Real K, Rate r, Rate q, Volatility vol, Time T) const {
 
        if (r < 0.0 && q < r)
            QL_FAIL("double-boundary case q<r<0 for a put option is given");
 
        const Real xmax =  QdPlusAmericanEngine::xMax(K, r, q);
        const Size n = iterationScheme_->getNumberOfChebyshevInterpolationNodes();
 
        const ext::shared_ptr<ChebyshevInterpolation> interp =
            QdPlusAmericanEngine(
                    process_, n+1, QdPlusAmericanEngine::Halley, 1e-8)
                .getPutExerciseBoundary(S, K, r, q, vol, T);
 
        const Array z = interp->nodes();
        const Array x = 0.5*std::sqrt(T)*(1.0+z);
 
        const auto B = [xmax, T, &interp](Real tau) -> Real {
            const Real z = 2*std::sqrt(std::abs(tau)/T)-1;
            return xmax*std::exp(-std::sqrt(std::max(Real(0), (*interp)(z, true))));
        };
 
        const auto h = [=](Real fv) -> Real {
            return squared(std::log(fv/xmax));
        };
 
        const ext::shared_ptr<DqFpEquation> eqn
            = (fpEquation_ == FP_A
               || (fpEquation_ == Auto && std::abs(r-q) < 0.001))?
              ext::shared_ptr<DqFpEquation>(new DqFpEquation_A(
                  K, r, q, vol, B,
                  iterationScheme_->getFixedPointIntegrator()))
            : ext::shared_ptr<DqFpEquation>(new DqFpEquation_B(
                    K, r, q, vol, B,
                    iterationScheme_->getFixedPointIntegrator()));
 
        Array y(x.size());
        y[0] = 0.0;
 
        const Size n_newton
            = iterationScheme_->getNumberOfJacobiNewtonFixedPointSteps();
        for (Size k=0; k < n_newton; ++k) {
            for (Size i=1; i < x.size(); ++i) {
                const Real tau = squared(x[i]);
                const Real b = B(tau);
 
                const std::tuple<Real, Real, Real> results = eqn->f(tau, b);
                const Real N = std::get<0>(results);
                const Real D = std::get<1>(results);
                const Real fv = std::get<2>(results);
 
                if (tau < QL_EPSILON)
                    y[i] = h(fv);
                else {
                    const std::pair<Real, Real> ndd = eqn->NDd(tau, b);
                    const Real Nd = std::get<0>(ndd);
                    const Real Dd = std::get<1>(ndd);
 
                    const Real fd = K*std::exp(-(r-q)*tau) * (Nd/D - Dd*N/(D*D));
 
                    y[i] = h(b - (fv - b)/ (fd-1));
                }
            }
            interp->updateY(y);
        }
 
        const Size n_fp = iterationScheme_->getNumberOfNaiveFixedPointSteps();
        for (Size k=0; k < n_fp; ++k) {
            for (Size i=1; i < x.size(); ++i) {
                const Real tau = squared(x[i]);
                const Real fv = std::get<2>(eqn->f(tau, B(tau)));
 
                y[i] = h(fv);
            }
            interp->updateY(y);
        }
 
        const detail::QdPlusAddOnValue aov(T, S, K, r, q, vol, xmax, interp);
        const Real addOn =
           (*iterationScheme_->getExerciseBoundaryToPriceIntegrator())(
               aov, 0.0, std::sqrt(T));
 
        const Real europeanValue = BlackCalculator(
            Option::Put, K, S*std::exp((r-q)*T),
            vol*std::sqrt(T), std::exp(-r*T)).value();
 
        return std::max(europeanValue, 0.0) + std::max(0.0, addOn);
    }
 
}