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/* -*- mode: c++; tab-width: 4; indent-tabs-mode: nil; c-basic-offset: 4 -*- */
/*
Copyright (C) 2006 Banca Profilo S.p.A.
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/processes/g2process.hpp>
#include <ql/processes/eulerdiscretization.hpp>
namespace QuantLib {
G2Process::G2Process(Real a, Real sigma, Real b, Real eta, Real rho)
: x0_(0.0), y0_(0.0), a_(a), sigma_(sigma), b_(b), eta_(eta), rho_(rho),
xProcess_(new QuantLib::OrnsteinUhlenbeckProcess(a, sigma, 0.0)),
yProcess_(new QuantLib::OrnsteinUhlenbeckProcess(b, eta, 0.0)) {}
Size G2Process::size() const {
return 2;
}
Disposable<Array> G2Process::initialValues() const {
Array tmp(2);
tmp[0] = x0_;
tmp[1] = y0_;
return tmp;
}
Disposable<Array> G2Process::drift(Time t, const Array& x) const {
Array tmp(2);
tmp[0] = xProcess_->drift(t, x[0]);
tmp[1] = yProcess_->drift(t, x[1]);
return tmp;
}
Disposable<Matrix> G2Process::diffusion(Time, const Array&) const {
/* the correlation matrix is
| 1 rho |
| rho 1 |
whose square root (which is used here) is
| 1 0 |
| rho sqrt(1-rho^2) |
*/
Matrix tmp(2,2);
Real sigma1 = sigma_;
Real sigma2 = eta_;
tmp[0][0] = sigma1; tmp[0][1] = 0.0;
tmp[1][0] = rho_*sigma1; tmp[1][1] = std::sqrt(1.0-rho_*rho_)*sigma2;
return tmp;
}
Disposable<Array> G2Process::expectation(Time t0, const Array& x0,
Time dt) const {
Array tmp(2);
tmp[0] = xProcess_->expectation(t0, x0[0], dt);
tmp[1] = yProcess_->expectation(t0, x0[1], dt);
return tmp;
}
Disposable<Matrix> G2Process::stdDeviation(Time t0, const Array& x0,
Time dt) const {
/* the correlation matrix is
| 1 rho |
| rho 1 |
whose square root (which is used here) is
| 1 0 |
| rho sqrt(1-rho^2) |
*/
Matrix tmp(2,2);
Real sigma1 = xProcess_->stdDeviation(t0, x0[0], dt);
Real sigma2 = yProcess_->stdDeviation(t0, x0[1], dt);
Real expa = std::exp(-a_*dt), expb = std::exp(-b_*dt);
Real H = (rho_*sigma_*eta_)/(a_+b_)*(1-expa*expb);
Real den =
(0.5*sigma_*eta_)*std::sqrt((1-expa*expa)*(1-expb*expb)/(a_*b_));
Real newRho = H/den;
tmp[0][0] = sigma1;
tmp[0][1] = 0.0;
tmp[1][0] = newRho*sigma2;
tmp[1][1] = std::sqrt(1.0-newRho*newRho)*sigma2;
return tmp;
}
Disposable<Matrix> G2Process::covariance(Time t0, const Array& x0,
Time dt) const {
Matrix sigma = stdDeviation(t0, x0, dt);
Matrix result = sigma*transpose(sigma);
return result;
}
Real G2Process::x0() const {
return x0_;
}
Real G2Process::y0() const {
return y0_;
}
Real G2Process::a() const {
return a_;
}
Real G2Process::sigma() const {
return sigma_;
}
Real G2Process::b() const {
return b_;
}
Real G2Process::eta() const {
return eta_;
}
Real G2Process::rho() const {
return rho_;
}
G2ForwardProcess::G2ForwardProcess(Real a, Real sigma, Real b,
Real eta, Real rho)
: x0_(0.0), y0_(0.0), a_(a), sigma_(sigma), b_(b), eta_(eta), rho_(rho),
xProcess_(new QuantLib::OrnsteinUhlenbeckProcess(a, sigma, 0.0)),
yProcess_(new QuantLib::OrnsteinUhlenbeckProcess(b, eta, 0.0)) {}
Size G2ForwardProcess::size() const {
return 2;
}
Disposable<Array> G2ForwardProcess::initialValues() const {
Array tmp(2);
tmp[0] = x0_;
tmp[1] = y0_;
return tmp;
}
Disposable<Array> G2ForwardProcess::drift(Time t, const Array& x) const {
Array tmp(2);
tmp[0] = xProcess_->drift(t, x[0]) + xForwardDrift(t, T_);
tmp[1] = yProcess_->drift(t, x[1]) + yForwardDrift(t, T_);
return tmp;
}
Disposable<Matrix> G2ForwardProcess::diffusion(Time, const Array&) const {
Matrix tmp(2,2);
Real sigma1 = sigma_;
Real sigma2 = eta_;
tmp[0][0] = sigma1; tmp[0][1] = 0.0;
tmp[1][0] = rho_*sigma1; tmp[1][1] = std::sqrt(1.0-rho_*rho_)*sigma2;
return tmp;
}
Disposable<Array> G2ForwardProcess::expectation(Time t0, const Array& x0,
Time dt) const {
Array tmp(2);
tmp[0] = xProcess_->expectation(t0, x0[0], dt) - Mx_T(t0, t0+dt, T_);
tmp[1] = yProcess_->expectation(t0, x0[1], dt) - My_T(t0, t0+dt, T_);
return tmp;
}
Disposable<Matrix> G2ForwardProcess::stdDeviation(Time t0, const Array& x0,
Time dt) const {
Matrix tmp(2,2);
Real sigma1 = xProcess_->stdDeviation(t0, x0[0], dt);
Real sigma2 = yProcess_->stdDeviation(t0, x0[1], dt);
Real expa = std::exp(-a_*dt), expb = std::exp(-b_*dt);
Real H = (rho_*sigma_*eta_)/(a_+b_)*(1-expa*expb);
Real den =
(0.5*sigma_*eta_)*std::sqrt((1-expa*expa)*(1-expb*expb)/(a_*b_));
Real newRho = H/den;
tmp[0][0] = sigma1;
tmp[0][1] = 0.0;
tmp[1][0] = newRho*sigma2;
tmp[1][1] = std::sqrt(1.0-newRho*newRho)*sigma2;
return tmp;
}
Disposable<Matrix> G2ForwardProcess::covariance(Time t0, const Array& x0,
Time dt) const {
Matrix sigma = stdDeviation(t0, x0, dt);
Matrix result = sigma*transpose(sigma);
return result;
}
Real G2ForwardProcess::xForwardDrift(Time t, Time T) const {
Real expatT = std::exp(-a_*(T-t));
Real expbtT = std::exp(-b_*(T-t));
return -(sigma_*sigma_/a_) * (1-expatT)
- (rho_*sigma_*eta_/b_) * (1-expbtT);
}
Real G2ForwardProcess::yForwardDrift(Time t, Time T) const {
Real expatT = std::exp(-a_*(T-t));
Real expbtT = std::exp(-b_*(T-t));
return -(eta_*eta_/b_) * (1-expbtT)
- (rho_*sigma_*eta_/a_) * (1-expatT);
}
Real G2ForwardProcess::Mx_T(Real s, Real t, Real T) const {
Real M;
M = ( (sigma_*sigma_)/(a_*a_) + (rho_*sigma_*eta_)/(a_*b_) )
* (1-std::exp(-a_*(t-s)));
M += -(sigma_*sigma_)/(2*a_*a_) *
(std::exp(-a_*(T-t))-std::exp(-a_*(T+t-2*s)));
M += -(rho_*sigma_*eta_)/(b_*(a_+b_))
* (std::exp(-b_*(T-t)) -std::exp(-b_*T-a_*t+(a_+b_)*s));
return M;
}
Real G2ForwardProcess::My_T(Real s, Real t, Real T) const {
Real M;
M = ( (eta_*eta_)/(b_*b_) + (rho_*sigma_*eta_)/(a_*b_) )
* (1-std::exp(-b_*(t-s)));
M += -(eta_*eta_)/(2*b_*b_) *
(std::exp(-b_*(T-t))-std::exp(-b_*(T+t-2*s)));
M += -(rho_*sigma_*eta_)/(a_*(a_+b_))
* (std::exp(-a_*(T-t))-std::exp(-a_*T-b_*t+(a_+b_)*s));
return M;
}
}
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