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/* -*- mode: c++; tab-width: 4; indent-tabs-mode: nil; c-basic-offset: 4 -*- */
/*
Copyright (C) 2001, 2002, 2003 Sadruddin Rejeb
Copyright (C) 2003 Ferdinando Ametrano
Copyright (C) 2005 StatPro Italia srl
Copyright (C) 2008 John Maiden
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/experimental/lattices/extendedbinomialtree.hpp>
#include <ql/math/distributions/binomialdistribution.hpp>
namespace QuantLib {
ExtendedJarrowRudd::ExtendedJarrowRudd(
const boost::shared_ptr<StochasticProcess1D>& process,
Time end, Size steps, Real)
: ExtendedEqualProbabilitiesBinomialTree<ExtendedJarrowRudd>(
process, end, steps) {
// drift removed
up_ = process->stdDeviation(0.0, x0_, dt_);
}
Real ExtendedJarrowRudd::upStep(Time stepTime) const {
return treeProcess_->stdDeviation(stepTime, x0_, dt_);
}
ExtendedCoxRossRubinstein::ExtendedCoxRossRubinstein(
const boost::shared_ptr<StochasticProcess1D>& process,
Time end, Size steps, Real)
: ExtendedEqualJumpsBinomialTree<ExtendedCoxRossRubinstein>(
process, end, steps) {
dx_ = process->stdDeviation(0.0, x0_, dt_);
pu_ = 0.5 + 0.5*this->driftStep(0.0)/dx_;
pd_ = 1.0 - pu_;
QL_REQUIRE(pu_<=1.0, "negative probability");
QL_REQUIRE(pu_>=0.0, "negative probability");
}
Real ExtendedCoxRossRubinstein::dxStep(Time stepTime) const {
return this->treeProcess_->stdDeviation(stepTime, x0_, dt_);
}
Real ExtendedCoxRossRubinstein::probUp(Time stepTime) const {
return 0.5 + 0.5*this->driftStep(stepTime)/dxStep(stepTime);
}
ExtendedAdditiveEQPBinomialTree::ExtendedAdditiveEQPBinomialTree(
const boost::shared_ptr<StochasticProcess1D>& process,
Time end, Size steps, Real)
: ExtendedEqualProbabilitiesBinomialTree<ExtendedAdditiveEQPBinomialTree>(
process, end, steps) {
up_ = - 0.5 * this->driftStep(0.0) + 0.5 *
std::sqrt(4.0*process->variance(0.0, x0_, dt_)-
3.0*this->driftStep(0.0)*this->driftStep(0.0));
}
Real ExtendedAdditiveEQPBinomialTree::upStep(Time stepTime) const {
return (- 0.5 * this->driftStep(stepTime) + 0.5 *
std::sqrt(4.0*this->treeProcess_->variance(stepTime, x0_, dt_)-
3.0*this->driftStep(stepTime)*this->driftStep(stepTime)));
}
ExtendedTrigeorgis::ExtendedTrigeorgis(
const boost::shared_ptr<StochasticProcess1D>& process,
Time end, Size steps, Real)
: ExtendedEqualJumpsBinomialTree<ExtendedTrigeorgis>(process, end, steps) {
dx_ = std::sqrt(process->variance(0.0, x0_, dt_)+
this->driftStep(0.0)*this->driftStep(0.0));
pu_ = 0.5 + 0.5*this->driftStep(0.0)/this->dxStep(0.0);
pd_ = 1.0 - pu_;
QL_REQUIRE(pu_<=1.0, "negative probability");
QL_REQUIRE(pu_>=0.0, "negative probability");
}
Real ExtendedTrigeorgis::dxStep(Time stepTime) const {
return std::sqrt(this->treeProcess_->variance(stepTime, x0_, dt_)+
this->driftStep(stepTime)*this->driftStep(stepTime));
}
Real ExtendedTrigeorgis::probUp(Time stepTime) const {
return 0.5 + 0.5*this->driftStep(stepTime)/dxStep(stepTime);
}
ExtendedTian::ExtendedTian(
const boost::shared_ptr<StochasticProcess1D>& process,
Time end, Size steps, Real)
: ExtendedBinomialTree<ExtendedTian>(process, end, steps) {
Real q = std::exp(process->variance(0.0, x0_, dt_));
Real r = std::exp(this->driftStep(0.0))*std::sqrt(q);
up_ = 0.5 * r * q * (q + 1 + std::sqrt(q * q + 2 * q - 3));
down_ = 0.5 * r * q * (q + 1 - std::sqrt(q * q + 2 * q - 3));
pu_ = (r - down_) / (up_ - down_);
pd_ = 1.0 - pu_;
// doesn't work
// treeCentering_ = (up_+down_)/2.0;
// up_ = up_-treeCentering_;
QL_REQUIRE(pu_<=1.0, "negative probability");
QL_REQUIRE(pu_>=0.0, "negative probability");
}
Real ExtendedTian::underlying(Size i, Size index) const {
Time stepTime = i*this->dt_;
Real q = std::exp(this->treeProcess_->variance(stepTime, x0_, dt_));
Real r = std::exp(this->driftStep(stepTime))*std::sqrt(q);
Real up = 0.5 * r * q * (q + 1 + std::sqrt(q * q + 2 * q - 3));
Real down = 0.5 * r * q * (q + 1 - std::sqrt(q * q + 2 * q - 3));
return x0_ * std::pow(down, Real(BigInteger(i)-BigInteger(index)))
* std::pow(up, Real(index));
}
Real ExtendedTian::probability(Size i, Size, Size branch) const {
Time stepTime = i*this->dt_;
Real q = std::exp(this->treeProcess_->variance(stepTime, x0_, dt_));
Real r = std::exp(this->driftStep(stepTime))*std::sqrt(q);
Real up = 0.5 * r * q * (q + 1 + std::sqrt(q * q + 2 * q - 3));
Real down = 0.5 * r * q * (q + 1 - std::sqrt(q * q + 2 * q - 3));
Real pu = (r - down) / (up - down);
Real pd = 1.0 - pu;
return (branch == 1 ? pu : pd);
}
ExtendedLeisenReimer::ExtendedLeisenReimer(
const boost::shared_ptr<StochasticProcess1D>& process,
Time end, Size steps, Real strike)
: ExtendedBinomialTree<ExtendedLeisenReimer>(process, end,
(steps%2 ? steps : steps+1)),
end_(end), oddSteps_(steps%2 ? steps : steps+1), strike_(strike) {
QL_REQUIRE(strike>0.0, "strike " << strike << "must be positive");
Real variance = process->variance(0.0, x0_, end);
Real ermqdt = std::exp(this->driftStep(0.0) + 0.5*variance/oddSteps_);
Real d2 = (std::log(x0_/strike) + this->driftStep(0.0)*oddSteps_ ) /
std::sqrt(variance);
pu_ = PeizerPrattMethod2Inversion(d2, oddSteps_);
pd_ = 1.0 - pu_;
Real pdash = PeizerPrattMethod2Inversion(d2+std::sqrt(variance),
oddSteps_);
up_ = ermqdt * pdash / pu_;
down_ = (ermqdt - pu_ * up_) / (1.0 - pu_);
}
Real ExtendedLeisenReimer::underlying(Size i, Size index) const {
Time stepTime = i*this->dt_;
Real variance = this->treeProcess_->variance(stepTime, x0_, end_);
Real ermqdt = std::exp(this->driftStep(stepTime) + 0.5*variance/oddSteps_);
Real d2 = (std::log(x0_/strike_) + this->driftStep(stepTime)*oddSteps_ ) /
std::sqrt(variance);
Real pu = PeizerPrattMethod2Inversion(d2, oddSteps_);
Real pdash = PeizerPrattMethod2Inversion(d2+std::sqrt(variance),
oddSteps_);
Real up = ermqdt * pdash / pu;
Real down = (ermqdt - pu * up) / (1.0 - pu);
return x0_ * std::pow(down, Real(BigInteger(i)-BigInteger(index)))
* std::pow(up, Real(index));
}
Real ExtendedLeisenReimer::probability(Size i, Size, Size branch) const {
Time stepTime = i*this->dt_;
Real variance = this->treeProcess_->variance(stepTime, x0_, end_);
Real d2 = (std::log(x0_/strike_) + this->driftStep(stepTime)*oddSteps_ ) /
std::sqrt(variance);
Real pu = PeizerPrattMethod2Inversion(d2, oddSteps_);
Real pd = 1.0 - pu;
return (branch == 1 ? pu : pd);
}
Real ExtendedJoshi4::computeUpProb(Real k, Real dj) const {
Real alpha = dj/(std::sqrt(8.0));
Real alpha2 = alpha*alpha;
Real alpha3 = alpha*alpha2;
Real alpha5 = alpha3*alpha2;
Real alpha7 = alpha5*alpha2;
Real beta = -0.375*alpha-alpha3;
Real gamma = (5.0/6.0)*alpha5 + (13.0/12.0)*alpha3
+(25.0/128.0)*alpha;
Real delta = -0.1025 *alpha- 0.9285 *alpha3
-1.43 *alpha5 -0.5 *alpha7;
Real p =0.5;
Real rootk= std::sqrt(k);
p+= alpha/rootk;
p+= beta /(k*rootk);
p+= gamma/(k*k*rootk);
// delete next line to get results for j three tree
p+= delta/(k*k*k*rootk);
return p;
}
ExtendedJoshi4::ExtendedJoshi4(
const boost::shared_ptr<StochasticProcess1D>& process,
Time end, Size steps, Real strike)
: ExtendedBinomialTree<ExtendedJoshi4>(process, end,
(steps%2 ? steps : steps+1)),
end_(end), oddSteps_(steps%2 ? steps : steps+1), strike_(strike) {
QL_REQUIRE(strike>0.0, "strike " << strike << "must be positive");
Real variance = process->variance(0.0, x0_, end);
Real ermqdt = std::exp(this->driftStep(0.0) + 0.5*variance/oddSteps_);
Real d2 = (std::log(x0_/strike) + this->driftStep(0.0)*oddSteps_ ) /
std::sqrt(variance);
pu_ = computeUpProb((oddSteps_-1.0)/2.0,d2 );
pd_ = 1.0 - pu_;
Real pdash = computeUpProb((oddSteps_-1.0)/2.0,d2+std::sqrt(variance));
up_ = ermqdt * pdash / pu_;
down_ = (ermqdt - pu_ * up_) / (1.0 - pu_);
}
Real ExtendedJoshi4::underlying(Size i, Size index) const {
Time stepTime = i*this->dt_;
Real variance = this->treeProcess_->variance(stepTime, x0_, end_);
Real ermqdt = std::exp(this->driftStep(stepTime) + 0.5*variance/oddSteps_);
Real d2 = (std::log(x0_/strike_) + this->driftStep(stepTime)*oddSteps_ ) /
std::sqrt(variance);
Real pu = computeUpProb((oddSteps_-1.0)/2.0,d2 );
Real pdash = computeUpProb((oddSteps_-1.0)/2.0,d2+std::sqrt(variance));
Real up = ermqdt * pdash / pu;
Real down = (ermqdt - pu * up) / (1.0 - pu);
return x0_ * std::pow(down, Real(BigInteger(i)-BigInteger(index)))
* std::pow(up, Real(index));
}
Real ExtendedJoshi4::probability(Size i, Size, Size branch) const {
Time stepTime = i*this->dt_;
Real variance = this->treeProcess_->variance(stepTime, x0_, end_);
Real d2 = (std::log(x0_/strike_) + this->driftStep(stepTime)*oddSteps_ ) /
std::sqrt(variance);
Real pu = computeUpProb((oddSteps_-1.0)/2.0,d2 );
Real pd = 1.0 - pu;
return (branch == 1 ? pu : pd);
}
}
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