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/* -*- mode: c++; tab-width: 4; indent-tabs-mode: nil; c-basic-offset: 4 -*- */
/*
Copyright (C) 2007, 2008 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 hybridhestonhullwhiteprocess.hpp
\brief hybrid equity (heston model)
with stochastic interest rates (hull white model)
*/
#include <ql/termstructures/yield/flatforward.hpp>
#include <ql/processes/hybridhestonhullwhiteprocess.hpp>
namespace QuantLib {
HybridHestonHullWhiteProcess::HybridHestonHullWhiteProcess(
const boost::shared_ptr<HestonProcess> & hestonProcess,
const boost::shared_ptr<HullWhiteForwardProcess> & hullWhiteProcess,
Real corrEquityShortRate,
HybridHestonHullWhiteProcess::Discretization discretization)
: hestonProcess_(hestonProcess),
hullWhiteProcess_(hullWhiteProcess),
hullWhiteModel_(new HullWhite(hestonProcess->riskFreeRate(),
hullWhiteProcess->a(),
hullWhiteProcess->sigma())),
corrEquityShortRate_(corrEquityShortRate),
discretization_(discretization),
maxRho_(std::sqrt(1-hestonProcess->rho()*hestonProcess->rho())
- std::sqrt(QL_EPSILON) /* reserve for rounding errors */),
T_(hullWhiteProcess->getForwardMeasureTime()),
endDiscount_(hestonProcess->riskFreeRate()->discount(T_)) {
QL_REQUIRE( corrEquityShortRate*corrEquityShortRate
+hestonProcess->rho()*hestonProcess->rho() <= 1.0,
"correlation matrix is not positive definite");
QL_REQUIRE(hullWhiteProcess->sigma() > 0.0,
"positive vol of Hull White process is required");
}
Size HybridHestonHullWhiteProcess::size() const {
return 3;
}
Disposable<Array> HybridHestonHullWhiteProcess::initialValues() const {
Array retVal(3);
retVal[0] = hestonProcess_->s0()->value();
retVal[1] = hestonProcess_->v0();
retVal[2] = hullWhiteProcess_->x0();
return retVal;
}
Disposable<Array>
HybridHestonHullWhiteProcess::drift(Time t, const Array& x) const {
Array retVal(3), x0(2);
x0[0] = x[0]; x0[1] = x[1];
Array y0 = hestonProcess_->drift(t, x0);
retVal[0] = y0[0]; retVal[1] = y0[1];
retVal[2] = hullWhiteProcess_->drift(t, x[2]);
return retVal;
}
Disposable<Array>
HybridHestonHullWhiteProcess::apply(const Array& x0,const Array& dx) const {
Array retVal(3), xt(2), dxt(2);
xt[0] = x0[0]; xt[1] = x0[1];
dxt[0] = dx[0]; dxt[1] = dx[1];
Array yt = hestonProcess_->apply(xt, dxt);
retVal[0] = yt[0]; retVal[1] = yt[1];
retVal[2] = hullWhiteProcess_->apply(x0[2], dx[2]);
return retVal;
}
Disposable<Matrix>
HybridHestonHullWhiteProcess::diffusion(Time t, const Array& x) const {
Matrix retVal(3,3);
Array xt(2); xt[0] = x[0]; xt[1] = x[1];
Matrix m = hestonProcess_->diffusion(t, xt);
retVal[0][0] = m[0][0]; retVal[0][1] = 0.0; retVal[0][2] = 0.0;
retVal[1][0] = m[1][0]; retVal[1][1] = m[1][1]; retVal[1][2] = 0.0;
const Real sigma = hullWhiteProcess_->sigma();
retVal[2][0] = corrEquityShortRate_ * sigma;
retVal[2][1] = - retVal[2][0]*retVal[1][0] / retVal[1][1];
retVal[2][2] = std::sqrt( sigma*sigma - retVal[2][1]*retVal[2][1]
- retVal[2][0]*retVal[2][0] );
return retVal;
}
Disposable<Array>
HybridHestonHullWhiteProcess::evolve(Time t0, const Array& x0,
Time dt, const Array& dw) const {
const Rate r = x0[2];
const Real a = hullWhiteProcess_->a();
const Real sigma = hullWhiteProcess_->sigma();
const Real rho = corrEquityShortRate_;
const Real xi = hestonProcess_->rho();
const Volatility eta = (x0[1] > 0.0) ? std::sqrt(x0[1]) : 0.0;
const Time s = t0;
const Time t = t0 + dt;
const Time T = T_;
const Rate dy
= hestonProcess_->dividendYield()->forwardRate(s, t, Continuous,
NoFrequency);
const Real df
= std::log( hestonProcess_->riskFreeRate()->discount(t)
/ hestonProcess_->riskFreeRate()->discount(s));
const Real eaT=std::exp(-a*T);
const Real eat=std::exp(-a*t);
const Real eas=std::exp(-a*s);
const Real iat=1.0/eat;
const Real ias=1.0/eas;
const Real m1 = -(dy+0.5*eta*eta)*dt - df;
const Real m2 = -rho*sigma*eta/a*(dt-1/a*eaT*(iat-ias));
const Real m3 = (r - hullWhiteProcess_->alpha(s))
*hullWhiteProcess_->B(s,t);
const Real m4 = sigma*sigma/(2*a*a)
*(dt + 2/a*(eat-eas) - 1/(2*a)*(eat*eat-eas*eas));
const Real m5 = -sigma*sigma/(a*a)
*(dt - 1/a*(1-eat*ias) - 1/(2*a)*eaT*(iat-2*ias+eat*ias*ias));
const Real mu = m1 + m2 + m3 + m4 + m5;
Array retVal(3);
const Real eta2 = hestonProcess_->sigma() * eta;
const Real nu
= hestonProcess_->kappa()*(hestonProcess_->theta() - eta*eta);
retVal[1] = x0[1] + nu*dt + eta2*std::sqrt(dt)
*(xi*dw[0]+std::sqrt(1-xi*xi)*dw[1]);
if (discretization_ == BSMHullWhite) {
const Real v1 = eta*eta*dt
+ sigma*sigma/(a*a)*(dt - 2/a*(1 - eat*ias)
+ 1/(2*a)*(1 - eat*eat*ias*ias))
+ 2*sigma*eta/a*rho*(dt - 1/a*(1 - eat*ias));
const Real v2 = hullWhiteProcess_->variance(t0, r, dt);
const Real v12 = (1-eat*ias)*(sigma*eta/a*rho + sigma*sigma/(a*a))
- sigma*sigma/(2*a*a)*(1 - eat*eat*ias*ias);
QL_REQUIRE(v1 > 0.0 && v2 > 0.0, "zero or negative variance given");
// terminal rho must be between -maxRho and +maxRho
const Real rhoT
= std::min(maxRho_, std::max(-maxRho_, v12/std::sqrt(v1*v2)));
QL_REQUIRE( rhoT <= 1.0 && rhoT >= -1.0
&& 1-rhoT*rhoT/(1-xi*xi) >= 0.0,
"invalid terminal correlation");
const Real dw_0 = dw[0];
const Real dw_2 = rhoT*dw[0]- rhoT*xi/std::sqrt(1-xi*xi)*dw[1]
+ std::sqrt(1 - rhoT*rhoT/(1-xi*xi))*dw[2];
retVal[2] = hullWhiteProcess_->evolve(t0, r, dt, dw_2);
const Real vol = std::sqrt(v1)*dw_0;
retVal[0] = x0[0]*std::exp(mu + vol);
}
else if (discretization_ == Euler) {
const Real dw_2 = rho*dw[0]- rho*xi/std::sqrt(1-xi*xi)*dw[1]
+ std::sqrt(1 - rho*rho/(1-xi*xi))*dw[2];
retVal[2] = hullWhiteProcess_->evolve(t0, r, dt, dw_2);
const Real vol = eta*std::sqrt(dt)*dw[0];
retVal[0] = x0[0]*std::exp(mu + vol);
}
else
QL_FAIL("unknown discretization scheme");
return retVal;
}
DiscountFactor
HybridHestonHullWhiteProcess::numeraire(Time t, const Array& x) const {
return hullWhiteModel_->discountBond(t, T_, x[2]) / endDiscount_;
}
Real HybridHestonHullWhiteProcess::eta() const {
return corrEquityShortRate_;
}
const boost::shared_ptr<HestonProcess>&
HybridHestonHullWhiteProcess::hestonProcess() const {
return hestonProcess_;
}
const boost::shared_ptr<HullWhiteForwardProcess>&
HybridHestonHullWhiteProcess::hullWhiteProcess() const {
return hullWhiteProcess_;
}
HybridHestonHullWhiteProcess::Discretization
HybridHestonHullWhiteProcess::discretization() const {
return discretization_;
}
Time HybridHestonHullWhiteProcess::time(const Date& date) const {
return hestonProcess_->time(date);
}
void HybridHestonHullWhiteProcess::update() {
endDiscount_ = hestonProcess_->riskFreeRate()->discount(T_);
}
}
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