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/* -*- mode: c++; tab-width: 4; indent-tabs-mode: nil; c-basic-offset: 4 -*- */
/*
Copyright (C) 2005, 2007 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.
*/
#include <ql/math/distributions/normaldistribution.hpp>
#include <ql/math/distributions/chisquaredistribution.hpp>
#include <ql/quotes/simplequote.hpp>
#include <ql/processes/hestonprocess.hpp>
#include <ql/processes/eulerdiscretization.hpp>
namespace QuantLib {
HestonProcess::HestonProcess(
const Handle<YieldTermStructure>& riskFreeRate,
const Handle<YieldTermStructure>& dividendYield,
const Handle<Quote>& s0,
double v0, double kappa,
double theta, double sigma, double rho,
Discretization d)
: StochasticProcess(boost::shared_ptr<discretization>(
new EulerDiscretization)),
riskFreeRate_(riskFreeRate), dividendYield_(dividendYield), s0_(s0),
v0_(v0), kappa_(kappa), theta_(theta), sigma_(sigma), rho_(rho),
discretization_(d) {
registerWith(riskFreeRate_);
registerWith(dividendYield_);
registerWith(s0_);
}
Size HestonProcess::size() const {
return 2;
}
Disposable<Array> HestonProcess::initialValues() const {
Array tmp(2);
tmp[0] = s0_->value();
tmp[1] = v0_;
return tmp;
}
Disposable<Array> HestonProcess::drift(Time t, const Array& x) const {
Array tmp(2);
const Real vol = (x[1] > 0.0) ? std::sqrt(x[1])
: (discretization_ == Reflection) ? -sqrt(-x[1])
: 0.0;
tmp[0] = riskFreeRate_->forwardRate(t, t, Continuous)
- dividendYield_->forwardRate(t, t, Continuous)
- 0.5 * vol * vol;
tmp[1] = kappa_*
(theta_-((discretization_==PartialTruncation) ? x[1] : vol*vol));
return tmp;
}
Disposable<Matrix> HestonProcess::diffusion(Time, const Array& x) 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);
const Real vol = (x[1] > 0.0) ? std::sqrt(x[1])
: (discretization_ == Reflection) ? -sqrt(-x[1])
: 1e-8; // set vol to (almost) zero but still
// expose some correlation information
const Real sigma2 = sigma_ * vol;
const Real sqrhov = std::sqrt(1.0 - rho_*rho_);
tmp[0][0] = vol; tmp[0][1] = 0.0;
tmp[1][0] = rho_*sigma2; tmp[1][1] = sqrhov*sigma2;
return tmp;
}
Disposable<Array> HestonProcess::apply(const Array& x0,
const Array& dx) const {
Array tmp(2);
tmp[0] = x0[0] * std::exp(dx[0]);
tmp[1] = x0[1] + dx[1];
return tmp;
}
Disposable<Array> HestonProcess::evolve(Time t0, const Array& x0,
Time dt, const Array& dw) const {
Array retVal(2);
Real ncp, df, p, dy;
Real vol, vol2, mu, nu;
const Real sdt = std::sqrt(dt);
const Real sqrhov = std::sqrt(1.0 - rho_*rho_);
switch (discretization_) {
// For the definition of PartialTruncation, FullTruncation
// and Reflection see Lord, R., R. Koekkoek and D. van Dijk (2006),
// "A Comparison of biased simulation schemes for
// stochastic volatility models",
// Working Paper, Tinbergen Institute
case PartialTruncation:
vol = (x0[1] > 0.0) ? std::sqrt(x0[1]) : 0.0;
vol2 = sigma_ * vol;
mu = riskFreeRate_->forwardRate(t0, t0+dt, Continuous)
- dividendYield_->forwardRate(t0, t0+dt, Continuous)
- 0.5 * vol * vol;
nu = kappa_*(theta_ - x0[1]);
retVal[0] = x0[0] * std::exp(mu*dt+vol*dw[0]*sdt);
retVal[1] = x0[1] + nu*dt + vol2*sdt*(rho_*dw[0] + sqrhov*dw[1]);
break;
case FullTruncation:
vol = (x0[1] > 0.0) ? std::sqrt(x0[1]) : 0.0;
vol2 = sigma_ * vol;
mu = riskFreeRate_->forwardRate(t0, t0+dt, Continuous)
- dividendYield_->forwardRate(t0, t0+dt, Continuous)
- 0.5 * vol * vol;
nu = kappa_*(theta_ - vol*vol);
retVal[0] = x0[0] * std::exp(mu*dt+vol*dw[0]*sdt);
retVal[1] = x0[1] + nu*dt + vol2*sdt*(rho_*dw[0] + sqrhov*dw[1]);
break;
case Reflection:
vol = std::sqrt(std::fabs(x0[1]));
vol2 = sigma_ * vol;
mu = riskFreeRate_->forwardRate(t0, t0+dt, Continuous)
- dividendYield_->forwardRate(t0, t0+dt, Continuous)
- 0.5 * vol*vol;
nu = kappa_*(theta_ - vol*vol);
retVal[0] = x0[0]*std::exp(mu*dt+vol*dw[0]*sdt);
retVal[1] = vol*vol
+nu*dt + vol2*sdt*(rho_*dw[0] + sqrhov*dw[1]);
break;
case ExactVariance:
// use Alan Lewis trick to decorrelate the equity and the variance
// process by using y(t)=x(t)-\frac{rho}{sigma}\nu(t)
// and Ito's Lemma. Then use exact sampling for the variance
// process. For further details please read the wilmott thread
// "QuantLib code is very high quality"
vol = (x0[1] > 0.0) ? std::sqrt(x0[1]) : 0.0;
mu = riskFreeRate_->forwardRate(t0, t0+dt, Continuous)
- dividendYield_->forwardRate(t0, t0+dt, Continuous)
- 0.5 * vol*vol;
df = 4*theta_*kappa_/(sigma_*sigma_);
ncp = 4*kappa_*std::exp(-kappa_*dt)
/(sigma_*sigma_*(1-std::exp(-kappa_*dt)))*x0[1];
p = CumulativeNormalDistribution()(dw[1]);
if (p<0.0)
p = 0.0;
else if (p >= 1.0)
p = 1.0-QL_EPSILON;
retVal[1] = sigma_*sigma_*(1-std::exp(-kappa_*dt))/(4*kappa_)
*InverseNonCentralChiSquareDistribution(df, ncp, 100)(p);
dy = (mu - rho_/sigma_*kappa_
*(theta_-vol*vol)) * dt + vol*sqrhov*dw[0]*sdt;
retVal[0] = x0[0]*std::exp(dy + rho_/sigma_*(retVal[1]-x0[1]));
break;
default:
QL_FAIL("unknown discretization schema");
}
return retVal;
}
const Handle<Quote>& HestonProcess::s0() const {
return s0_;
}
const Handle<YieldTermStructure>& HestonProcess::dividendYield() const {
return dividendYield_;
}
const Handle<YieldTermStructure>& HestonProcess::riskFreeRate() const {
return riskFreeRate_;
}
Time HestonProcess::time(const Date& d) const {
return riskFreeRate_->dayCounter().yearFraction(
riskFreeRate_->referenceDate(), d);
}
}
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