g2.cpp

/* -*- mode: c++; tab-width: 4; indent-tabs-mode: nil; c-basic-offset: 4 -*- */
 
/*
 Copyright (C) 2001, 2002, 2003 Sadruddin Rejeb
 Copyright (C) 2004 Mike Parker
 Copyright (C) 2021 Magnus Mencke
 
 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/math/distributions/normaldistribution.hpp>
#include <ql/math/integrals/segmentintegral.hpp>
#include <ql/math/solvers1d/brent.hpp>
#include <ql/models/shortrate/twofactormodels/g2.hpp>
#include <ql/pricingengines/blackformula.hpp>
#include <utility>
 
namespace QuantLib {
 
    G2::G2(const Handle<YieldTermStructure>& termStructure,
           Real a, Real sigma, Real b, Real eta, Real rho)
    : TwoFactorModel(5), TermStructureConsistentModel(termStructure),
      a_(arguments_[0]), sigma_(arguments_[1]), b_(arguments_[2]),
      eta_(arguments_[3]), rho_(arguments_[4]) {
 
        a_     = ConstantParameter(a,     PositiveConstraint());
        sigma_ = ConstantParameter(sigma, PositiveConstraint());
        b_     = ConstantParameter(b,     PositiveConstraint());
        eta_   = ConstantParameter(eta,   PositiveConstraint());
        rho_   = ConstantParameter(rho,   BoundaryConstraint(-1.0, 1.0));
 
        G2::generateArguments();
 
        registerWith(termStructure);
    }
 
    ext::shared_ptr<TwoFactorModel::ShortRateDynamics> G2::dynamics() const {
        return ext::shared_ptr<ShortRateDynamics>(new
            Dynamics(phi_, a(), sigma(), b(), eta(), rho()));
    }
 
    void G2::generateArguments() {
 
        phi_ = FittingParameter(termStructure(),
            a(), sigma(), b(), eta(), rho());
    }
 
    Real G2::sigmaP(Time t, Time s) const {
        Real temp = 1.0 - std::exp(-(a()+b())*t);
        Real temp1 = 1.0 - std::exp(-a()*(s-t));
        Real temp2 = 1.0 - std::exp(-b()*(s-t));
        Real a3 = a()*a()*a();
        Real b3 = b()*b()*b();
        Real sigma2 = sigma()*sigma();
        Real eta2 = eta()*eta();
        Real value =
            0.5*sigma2*temp1*temp1*(1.0 - std::exp(-2.0*a()*t))/a3 +
            0.5*eta2*temp2*temp2*(1.0 - std::exp(-2.0*b()*t))/b3 +
            2.0*rho()*sigma()*eta()/(a()*b()*(a()+b()))*
            temp1*temp2*temp;
        return std::sqrt(value);
    }
 
    Real G2::discountBond(Time t, Time T, Real x, Real y) const {
        return A(t,T) * std::exp(-B(a(),(T-t))*x-B(b(),(T-t))*y);
    }
 
    Real G2::discountBondOption(Option::Type type, Real strike, Time maturity,
        Time bondMaturity) const {
 
        Real v = sigmaP(maturity, bondMaturity);
        Real f = termStructure()->discount(bondMaturity);
        Real k = termStructure()->discount(maturity)*strike;
 
        return blackFormula(type, k, f, v);
    }
 
    Real G2::V(Time t) const {
        Real expat = std::exp(-a()*t);
        Real expbt = std::exp(-b()*t);
        Real cx = sigma()/a();
        Real cy = eta()/b();
        Real valuex = cx*cx*(t + (2.0*expat-0.5*expat*expat-1.5)/a());
        Real valuey = cy*cy*(t + (2.0*expbt-0.5*expbt*expbt-1.5)/b());
        Real value = 2.0*rho()*cx*cy* (t + (expat - 1.0)/a()
                                         + (expbt - 1.0)/b()
                                         - (expat*expbt-1.0)/(a()+b()));
        return valuex + valuey + value;
    }
 
    Real G2::A(Time t, Time T) const {
        return termStructure()->discount(T)/termStructure()->discount(t)*
            std::exp(0.5*(V(T-t) - V(T) + V(t)));
    }
 
    Real G2::B(Real x, Time t) const {
        return (1.0 - std::exp(-x*t))/x;
    }
 
    class G2::SwaptionPricingFunction {
      public:
        SwaptionPricingFunction(Real a,
                                Real sigma,
                                Real b,
                                Real eta,
                                Real rho,
                                Real w,
                                Real start,
                                std::vector<Time> payTimes,
                                Rate fixedRate,
                                const G2& model)
        : a_(a), sigma_(sigma), b_(b), eta_(eta), rho_(rho), w_(w), T_(start),
          t_(std::move(payTimes)), rate_(fixedRate), size_(t_.size()), A_(size_), Ba_(size_),
          Bb_(size_) {
 
 
            sigmax_ = sigma_*std::sqrt(0.5*(1.0-std::exp(-2.0*a_*T_))/a_);
            sigmay_ =   eta_*std::sqrt(0.5*(1.0-std::exp(-2.0*b_*T_))/b_);
            rhoxy_ = rho_*eta_*sigma_*(1.0 - std::exp(-(a_+b_)*T_))/
                ((a_+b_)*sigmax_*sigmay_);
 
            Real temp = sigma_*sigma_/(a_*a_);
            mux_ = -((temp+rho_*sigma_*eta_/(a_*b_))*(1.0 - std::exp(-a*T_)) -
                     0.5*temp*(1.0 - std::exp(-2.0*a_*T_)) -
                     rho_*sigma_*eta_/(b_*(a_+b_))*
                     (1.0- std::exp(-(b_+a_)*T_)));
 
            temp = eta_*eta_/(b_*b_);
            muy_ = -((temp+rho_*sigma_*eta_/(a_*b_))*(1.0 - std::exp(-b*T_)) -
                     0.5*temp*(1.0 - std::exp(-2.0*b_*T_)) -
                     rho_*sigma_*eta_/(a_*(a_+b_))*
                     (1.0- std::exp(-(b_+a_)*T_)));
 
            for (Size i=0; i<size_; i++) {
                A_[i] = model.A(T_, t_[i]);
                Ba_[i] = model.B(a_, t_[i]-T_);
                Bb_[i] = model.B(b_, t_[i]-T_);
            }
        }
 
        Real mux() const { return mux_; }
        Real sigmax() const { return sigmax_; }
        Real operator()(Real x) const {
            CumulativeNormalDistribution phi;
            Real temp = (x - mux_)/sigmax_;
            Real txy = std::sqrt(1.0 - rhoxy_*rhoxy_);
 
            Array lambda(size_);
            Size i;
            for (i=0; i<size_; i++) {
                Real tau = (i==0 ? t_[0] - T_ : t_[i] - t_[i-1]);
                Real c = (i==size_-1 ? Real(1.0+rate_*tau) : rate_*tau);
                lambda[i] = c*A_[i]*std::exp(-Ba_[i]*x);
            }
 
            SolvingFunction function(lambda, Bb_) ;
            Brent s1d;
            s1d.setMaxEvaluations(1000);
            Real searchBound = std::max(10.0*sigmay_, 1.0);
            Real yb = s1d.solve(function, 1e-6, 0.00, -searchBound, searchBound);
 
            Real h1 = (yb - muy_)/(sigmay_*txy) -
                rhoxy_*(x  - mux_)/(sigmax_*txy);
            Real value = phi(-w_*h1);
 
 
            for (i=0; i<size_; i++) {
                Real h2 = h1 +
                    Bb_[i]*sigmay_*std::sqrt(1.0-rhoxy_*rhoxy_);
                Real kappa = - Bb_[i] *
                    (muy_ - 0.5*txy*txy*sigmay_*sigmay_*Bb_[i] +
                     rhoxy_*sigmay_*(x-mux_)/sigmax_);
                value -= lambda[i] *std::exp(kappa)*phi(-w_*h2);
            }
 
            return std::exp(-0.5*temp*temp)*value/
                (sigmax_*std::sqrt(2.0*M_PI));
        }
 
 
      private:
        class SolvingFunction {
          public:
            SolvingFunction(const Array& lambda, const Array& Bb)
            : lambda_(lambda), Bb_(Bb) {}
            Real operator()(Real y) const {
                Real value = 1.0;
                for (Size i=0; i<lambda_.size(); i++) {
                    value -= lambda_[i]*std::exp(-Bb_[i]*y);
                }
                return value;
            }
          private:
            const Array& lambda_;
            const Array& Bb_;
        };
 
        Real a_, sigma_, b_, eta_, rho_, w_;
        Time T_;
        std::vector<Time> t_;
        Rate rate_;
        Size size_;
        Array A_, Ba_, Bb_;
        Real mux_, muy_, sigmax_, sigmay_, rhoxy_;
    };
 
    Real G2::swaption(const Swaption::arguments& arguments,
                      Rate fixedRate, Real range, Size intervals) const {
 
        QL_REQUIRE(arguments.nominal != Null<Real>(),
                   "non-constant nominals are not supported yet");
 
        Date settlement = termStructure()->referenceDate();
        DayCounter dayCounter = termStructure()->dayCounter();
        Time start = dayCounter.yearFraction(settlement,
                                             arguments.floatingResetDates[0]);
        Real w = (arguments.type==Swap::Payer ? 1 : -1 );
 
        std::vector<Time> fixedPayTimes(arguments.fixedPayDates.size());
        for (Size i=0; i<fixedPayTimes.size(); ++i)
            fixedPayTimes[i] =
                dayCounter.yearFraction(settlement,
                                        arguments.fixedPayDates[i]);
 
        SwaptionPricingFunction function(a(), sigma(), b(), eta(), rho(),
                                         w, start,
                                         fixedPayTimes,
                                         fixedRate, (*this));
 
        Real upper = function.mux() + range*function.sigmax();
        Real lower = function.mux() - range*function.sigmax();
        SegmentIntegral integrator(intervals);
        return arguments.nominal * w * termStructure()->discount(start) *
            integrator(function, lower, upper);
    }
 
}