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/* -*- mode: c++; tab-width: 4; indent-tabs-mode: nil; c-basic-offset: 4 -*- */
/*
Copyright (C) 2015 Johannes Göttker-Schnetmann
Copyright (C) 2015 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 hestonslvprocess.cpp
\brief Heston stochastic local volatility process
*/
#include <ql/math/functional.hpp>
#include <ql/math/distributions/normaldistribution.hpp>
#include <ql/experimental/processes/hestonslvprocess.hpp>
#include <ql/methods/finitedifferences/utilities/squarerootprocessrndcalculator.hpp>
namespace QuantLib {
HestonSLVProcess::HestonSLVProcess(
const ext::shared_ptr<HestonProcess>& hestonProcess,
const ext::shared_ptr<LocalVolTermStructure>& leverageFct,
const Real mixingFactor)
: mixingFactor_(mixingFactor),
hestonProcess_(hestonProcess),
leverageFct_(leverageFct) {
registerWith(hestonProcess);
update();
};
void HestonSLVProcess::update() {
v0_ = hestonProcess_->v0();
kappa_ = hestonProcess_->kappa();
theta_ = hestonProcess_->theta();
sigma_ = hestonProcess_->sigma();
rho_ = hestonProcess_->rho();
mixedSigma_ = mixingFactor_ * sigma_;
}
Disposable<Array> HestonSLVProcess::drift(Time t, const Array& x) const {
Array tmp(2);
const Volatility vol =
std::max(1e-8, std::sqrt(x[1])*leverageFct_->localVol(t, x[0], true));
tmp[0] = riskFreeRate()->forwardRate(t, t, Continuous)
- dividendYield()->forwardRate(t, t, Continuous)
- 0.5*vol*vol;
tmp[1] = kappa_*(theta_ - x[1]);
return tmp;
}
Disposable<Matrix> HestonSLVProcess::diffusion(Time t, const Array& x)
const {
const Real vol =
std::max(1e-8, std::sqrt(x[1])*leverageFct_->localVol(t, x[0], true));
const Real sigma2 = mixedSigma_ * std::sqrt(x[1]);
const Real sqrhov = std::sqrt(1.0 - rho_*rho_);
Matrix tmp(2,2);
tmp[0][0] = vol; tmp[0][1] = 0.0;
tmp[1][0] = rho_*sigma2; tmp[1][1] = sqrhov*sigma2;
return tmp;
}
Disposable<Array> HestonSLVProcess::evolve(
Time t0, const Array& x0, Time dt, const Array& dw) const {
Array retVal(2);
const Real ex = std::exp(-kappa_*dt);
const Real m = theta_+(x0[1]-theta_)*ex;
const Real s2 = x0[1]*mixedSigma_*mixedSigma_*ex/kappa_*(1-ex)
+ theta_*mixedSigma_*mixedSigma_/(2*kappa_)*(1-ex)*(1-ex);
const Real psi = s2/(m*m);
if (psi < 1.5) {
const Real b2 = 2/psi-1+std::sqrt(2/psi*(2/psi-1));
const Real b = std::sqrt(b2);
const Real a = m/(1+b2);
retVal[1] = a*(b+dw[1])*(b+dw[1]);
}
else {
const Real p = (psi-1)/(psi+1);
const Real beta = (1-p)/m;
const Real u = CumulativeNormalDistribution()(dw[1]);
retVal[1] = ((u <= p) ? 0.0 : std::log((1-p)/(1-u))/beta);
}
const Real mu = riskFreeRate()->forwardRate(t0, t0+dt, Continuous)
- dividendYield()->forwardRate(t0, t0+dt, Continuous);
const Real rho1 = std::sqrt(1-rho_*rho_);
const Volatility l_0 = leverageFct_->localVol(t0, x0[0], true);
const Real v_0 = 0.5*(x0[1]+retVal[1])*l_0*l_0;
retVal[0] = x0[0]*std::exp(mu*dt - 0.5*v_0*dt
+ rho_/mixedSigma_*l_0 * (
retVal[1] - kappa_*theta_*dt
+ 0.5*(x0[1]+retVal[1])*kappa_*dt - x0[1])
+ rho1*std::sqrt(v_0*dt)*dw[0]);
return retVal;
}
}
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