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/* -*- mode: c++; tab-width: 4; indent-tabs-mode: nil; c-basic-offset: 4 -*- */
/*
Copyright (C) 2012, 2013 Klaus Spanderen
Copyright (C) 2014 Johannes Göttker-Schnetmann
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.
*/
/*! \file fdmhestonfwdop.cpp
*/
#include <ql/methods/finitedifferences/operators/fdmhestonfwdop.hpp>
#include <ql/methods/finitedifferences/operators/modtriplebandlinearop.hpp>
#include <ql/methods/finitedifferences/meshers/fdmmesher.hpp>
#include <ql/methods/finitedifferences/operators/fdmlinearoplayout.hpp>
#include <ql/methods/finitedifferences/operators/firstderivativeop.hpp>
#include <ql/methods/finitedifferences/operators/secondderivativeop.hpp>
#include <ql/methods/finitedifferences/operators/secondordermixedderivativeop.hpp>
#include <ql/processes/hestonprocess.hpp>
#include <cmath>
#include <utility>
using std::exp;
namespace QuantLib {
FdmHestonFwdOp::FdmHestonFwdOp(const ext::shared_ptr<FdmMesher>& mesher,
const ext::shared_ptr<HestonProcess>& process,
FdmSquareRootFwdOp::TransformationType type,
ext::shared_ptr<LocalVolTermStructure> leverageFct,
const Real mixingFactor)
: type_(type), kappa_(process->kappa()), theta_(process->theta()), sigma_(process->sigma()),
rho_(process->rho()), v0_(process->v0()), mixedSigma_(mixingFactor * sigma_),
rTS_(process->riskFreeRate().currentLink()), qTS_(process->dividendYield().currentLink()),
varianceValues_(0.5 * mesher->locations(1)),
dxMap_(ext::make_shared<FirstDerivativeOp>(0, mesher)),
dxxMap_(ext::make_shared<ModTripleBandLinearOp>(TripleBandLinearOp(
type == FdmSquareRootFwdOp::Log ?
SecondDerivativeOp(0, mesher).mult(0.5 * Exp(mesher->locations(1))) :
SecondDerivativeOp(0, mesher).mult(0.5 * mesher->locations(1))))),
boundary_(ext::make_shared<ModTripleBandLinearOp>(TripleBandLinearOp(
SecondDerivativeOp(0, mesher).mult(Array(mesher->locations(0).size(), 0.0))))),
mapX_(ext::make_shared<TripleBandLinearOp>(0, mesher)),
mapY_(ext::make_shared<FdmSquareRootFwdOp>(mesher, kappa_, theta_, mixedSigma_, 1, type)),
correlation_(ext::make_shared<NinePointLinearOp>(
type == FdmSquareRootFwdOp::Log ?
SecondOrderMixedDerivativeOp(0, 1, mesher)
.mult(Array(mesher->layout()->size(), rho_ * mixedSigma_)) :
SecondOrderMixedDerivativeOp(0, 1, mesher)
.mult(rho_ * mixedSigma_ * mesher->locations(1)))),
leverageFct_(std::move(leverageFct)), mesher_(mesher) {
// zero flux boundary condition
const Size n = mesher->layout()->dim()[1];
const Real lowerBoundaryFactor = mapY_->lowerBoundaryFactor(type);
const Real upperBoundaryFactor = mapY_->upperBoundaryFactor(type);
const Real logFacLow = type == FdmSquareRootFwdOp::Log ? Real(exp(mapY_->v(0))) : 1.0;
const Real logFacUpp = type == FdmSquareRootFwdOp::Log ? Real(exp(mapY_->v(n+1))) : 1.0;
const Real alpha = -2*rho_/mixedSigma_*lowerBoundaryFactor*logFacLow;
const Real beta = -2*rho_/mixedSigma_*upperBoundaryFactor*logFacUpp;
ModTripleBandLinearOp fDx(FirstDerivativeOp(0, mesher));
for (const auto& iter : *mesher->layout()) {
if (iter.coordinates()[1] == 0) {
const Size idx = iter.index();
if (!leverageFct_) {
dxxMap_->upper(idx) += alpha*fDx.upper(idx);
dxxMap_->diag(idx) += alpha*fDx.diag(idx);
dxxMap_->lower(idx) += alpha*fDx.lower(idx);
}
boundary_->upper(idx)= alpha*fDx.upper(idx);
boundary_->diag(idx) = alpha*fDx.diag(idx);
boundary_->lower(idx) = alpha*fDx.lower(idx);
}
else if (iter.coordinates()[1] == n-1) {
const Size idx = iter.index();
if (!leverageFct_) {
dxxMap_->upper(idx)+= beta*fDx.upper(idx);
dxxMap_->diag(idx) += beta*fDx.diag(idx);
dxxMap_->lower(idx) += beta*fDx.lower(idx);
}
boundary_->upper(idx)= beta*fDx.upper(idx);
boundary_->diag(idx) = beta*fDx.diag(idx);
boundary_->lower(idx) = beta*fDx.lower(idx);
}
}
}
Size FdmHestonFwdOp::size() const {
return 2;
}
void FdmHestonFwdOp::setTime(Time t1, Time t2){
const Rate r = rTS_->forwardRate(t1, t2, Continuous).rate();
const Rate q = qTS_->forwardRate(t1, t2, Continuous).rate();
if (leverageFct_ != nullptr) {
L_ = getLeverageFctSlice(t1, t2);
Array Lsquare = L_*L_;
if (type_ == FdmSquareRootFwdOp::Plain) {
mapX_->axpyb( Array(1, -r + q), *dxMap_,
dxxMap_->multR(Lsquare).add(boundary_->multR(L_))
.add(dxMap_->multR(rho_*mixedSigma_*L_))
.add(dxMap_->mult(varianceValues_).multR(Lsquare)),
Array());
} else if (type_ == FdmSquareRootFwdOp::Power) {
mapX_->axpyb( Array(1, -r + q), *dxMap_,
dxxMap_->multR(Lsquare).add(boundary_->multR(L_))
.add(dxMap_->multR(rho_*2.0*kappa_*theta_/(mixedSigma_)*L_))
.add(dxMap_->mult(varianceValues_).multR(Lsquare)), Array());
} else if (type_ == FdmSquareRootFwdOp::Log) {
mapX_->axpyb( Array(1, -r + q), *dxMap_,
dxxMap_->multR(Lsquare).add(boundary_->multR(L_))
.add(dxMap_->mult(0.5*Exp(2.0*varianceValues_)).multR(Lsquare)),
Array());
}
} else {
if (type_ == FdmSquareRootFwdOp::Plain) {
mapX_->axpyb( - r + q + rho_*mixedSigma_ + varianceValues_, *dxMap_,
*dxxMap_, Array());
} else if (type_ == FdmSquareRootFwdOp::Power) {
mapX_->axpyb( - r + q + rho_*2.0*kappa_*theta_/(mixedSigma_) + varianceValues_,
*dxMap_, *dxxMap_, Array());
} else if (type_ == FdmSquareRootFwdOp::Log) {
mapX_->axpyb( - r + q + 0.5*Exp(2.0*varianceValues_), *dxMap_,
*dxxMap_, Array());
}
}
}
Array FdmHestonFwdOp::apply(const Array& u) const {
if (leverageFct_ != nullptr) {
return mapX_->apply(u)
+ mapY_->apply(u)
+ correlation_->apply(L_*u);
} else {
return mapX_->apply(u)
+ mapY_->apply(u)
+ correlation_->apply(u);
}
}
Array FdmHestonFwdOp::apply_mixed(const Array& u) const{
if (leverageFct_ != nullptr) {
return correlation_->apply(L_*u);
} else {
return correlation_->apply(u);
}
}
Array FdmHestonFwdOp::apply_direction(
Size direction, const Array& u) const {
if (direction == 0)
return mapX_->apply(u) ;
else if (direction == 1)
return mapY_->apply(u) ;
else
QL_FAIL("direction too large");
}
Array FdmHestonFwdOp::solve_splitting(
Size direction, const Array& u, Real s) const{
if (direction == 0) {
return mapX_->solve_splitting(u, s, 1.0);
}
else if (direction == 1) {
return mapY_->solve_splitting(1, u, s);
}
else
QL_FAIL("direction too large");
}
Array FdmHestonFwdOp::preconditioner(
const Array& u, Real dt) const{
return solve_splitting(1, u, dt);
}
Array FdmHestonFwdOp::getLeverageFctSlice(Time t1, Time t2) const {
Array v(mesher_->layout()->size(), 1.0);
if (!leverageFct_)
return v;
const Real t = 0.5*(t1+t2);
const Time time = std::min(leverageFct_->maxTime(), t);
//std::max(leverageFct_->minTime(), t));
for (const auto& iter : *mesher_->layout()) {
const Size nx = iter.coordinates()[0];
if (iter.coordinates()[1] == 0) {
const Real x = std::exp(mesher_->location(iter, 0));
const Real spot = std::min(leverageFct_->maxStrike(),
std::max(leverageFct_->minStrike(), x));
v[nx] = std::max(0.01, leverageFct_->localVol(time, spot, true));
}
else {
v[iter.index()] = v[nx];
}
}
return v;
}
std::vector<SparseMatrix> FdmHestonFwdOp::toMatrixDecomp() const {
std::vector<SparseMatrix> retVal(3);
retVal[0] = mapX_->toMatrix();
retVal[1] = mapY_->toMatrix();
retVal[2] = correlation_->toMatrix();
return retVal;
}
}
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