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/* -*- mode: c++; tab-width: 4; indent-tabs-mode: nil; c-basic-offset: 4 -*- */
/*
Copyright (C) 2008 Yee Man Chan
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/gjrgarchprocess.hpp>
#include <ql/processes/eulerdiscretization.hpp>
namespace QuantLib {
GJRGARCHProcess::GJRGARCHProcess(
const Handle<YieldTermStructure>& riskFreeRate,
const Handle<YieldTermStructure>& dividendYield,
const Handle<Quote>& s0, Real v0,
Real omega, Real alpha, Real beta,
Real gamma, Real lambda, Real daysPerYear, Discretization d)
: StochasticProcess(ext::shared_ptr<discretization>(
new EulerDiscretization)),
riskFreeRate_(riskFreeRate), dividendYield_(dividendYield), s0_(s0),
v0_(v0), omega_(omega), alpha_(alpha),
beta_(beta), gamma_(gamma), lambda_(lambda), daysPerYear_(daysPerYear),
discretization_(d) {
registerWith(riskFreeRate_);
registerWith(dividendYield_);
registerWith(s0_);
}
Size GJRGARCHProcess::size() const {
return 2;
}
Disposable<Array> GJRGARCHProcess::initialValues() const {
Array tmp(2);
tmp[0] = s0_->value();
tmp[1] = daysPerYear_*v0_;
return tmp;
}
Disposable<Array> GJRGARCHProcess::drift(Time t, const Array& x) const {
Array tmp(2);
const Real N = CumulativeNormalDistribution()(lambda_);
const Real n = std::exp(-lambda_*lambda_/2.0)/std::sqrt(2*M_PI);
const Real q2 = 1.0 + lambda_*lambda_;
const Real q3 = lambda_*n + N + lambda_*lambda_*N;
const Real vol = (x[1] > 0.0) ? std::sqrt(x[1])
: (discretization_ == Reflection) ? - std::sqrt(-x[1])
: 0.0;
tmp[0] = riskFreeRate_->forwardRate(t, t, Continuous)
- dividendYield_->forwardRate(t, t, Continuous)
- 0.5 * vol * vol;
tmp[1] = daysPerYear_*daysPerYear_*omega_ + daysPerYear_*(beta_
+ alpha_*q2 + gamma_*q3 - 1.0) *
((discretization_==PartialTruncation) ? x[1] : vol*vol);
return tmp;
}
Disposable<Matrix> GJRGARCHProcess::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 std::sqrt(1-rho^2) |
*/
Matrix tmp(2,2);
const Real N = CumulativeNormalDistribution()(lambda_);
const Real n = std::exp(-lambda_*lambda_/2.0)/std::sqrt(2*M_PI);
const Real sigma2 = 2.0 + 4.0*lambda_*lambda_;
const Real q3 = lambda_*n + N + lambda_*lambda_*N;
const Real Eml_e4 = lambda_*lambda_*lambda_*n + 5.0*lambda_*n
+ 3.0*N + lambda_*lambda_*lambda_*lambda_*N
+ 6.0*lambda_*lambda_*N;
const Real sigma3 = Eml_e4 - q3*q3;
const Real sigma12 = -2.0*lambda_;
const Real sigma13 = -2.0*n - 2*lambda_*N;
const Real sigma23 = 2.0*N + sigma12*sigma13;
const Real vol = (x[1] > 0.0) ? std::sqrt(x[1])
: (discretization_ == Reflection) ? - std::sqrt(-x[1])
: 1e-8; // set vol to (almost) zero but still
// expose some correlation information
const Real rho1 = std::sqrt(daysPerYear_)*(alpha_*sigma12
+ gamma_*sigma13) * vol * vol;
const Real rho2 = vol*vol*std::sqrt(daysPerYear_)
*std::sqrt(alpha_*alpha_*(sigma2 - sigma12*sigma12)
+ gamma_*gamma_*(sigma3 - sigma13*sigma13)
+ 2.0*alpha_*gamma_*(sigma23 - sigma12*sigma13));
// tmp[0][0], tmp[0][1] are the coefficients of dW_1 and dW_2
// in asset return stochastic process
tmp[0][0] = vol; tmp[0][1] = 0.0;
tmp[1][0] = rho1; tmp[1][1] = rho2;
return tmp;
}
Disposable<Array> GJRGARCHProcess::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> GJRGARCHProcess::evolve(Time t0, const Array& x0,
Time dt, const Array& dw) const {
Array retVal(2);
Real vol, mu, nu;
const Real sdt = std::sqrt(dt);
const Real N = CumulativeNormalDistribution()(lambda_);
const Real n = std::exp(-lambda_*lambda_/2.0)/std::sqrt(2*M_PI);
const Real sigma2 = 2.0 + 4.0*lambda_*lambda_;
const Real q2 = 1.0 + lambda_*lambda_;
const Real q3 = lambda_*n + N + lambda_*lambda_*N;
const Real Eml_e4 = lambda_*lambda_*lambda_*n + 5.0*lambda_*n
+ 3.0*N + lambda_*lambda_*lambda_*lambda_*N
+ 6.0*lambda_*lambda_*N;
const Real sigma3 = Eml_e4 - q3*q3;
const Real sigma12 = -2.0*lambda_;
const Real sigma13 = -2.0*n - 2*lambda_*N;
const Real sigma23 = 2.0*N + sigma12*sigma13;
const Real rho1 = std::sqrt(daysPerYear_)*(alpha_*sigma12 + gamma_*sigma13);
const Real rho2 = std::sqrt(daysPerYear_)
*std::sqrt(alpha_*alpha_*(sigma2 - sigma12*sigma12)
+ gamma_*gamma_*(sigma3 - sigma13*sigma13)
+ 2.0*alpha_*gamma_*(sigma23 - sigma12*sigma13));
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;
mu = riskFreeRate_->forwardRate(t0, t0+dt, Continuous)
- dividendYield_->forwardRate(t0, t0+dt, Continuous)
- 0.5 * vol * vol;
nu = daysPerYear_*daysPerYear_*omega_
+ daysPerYear_*(beta_ + alpha_*q2 + gamma_*q3 - 1.0) * x0[1];
retVal[0] = x0[0] * std::exp(mu*dt+vol*dw[0]*sdt);
retVal[1] = x0[1] + nu*dt + sdt*vol*vol*(rho1*dw[0] + rho2*dw[1]);
break;
case FullTruncation:
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;
nu = daysPerYear_*daysPerYear_*omega_
+ daysPerYear_*(beta_ + alpha_*q2 + gamma_*q3 - 1.0) * vol *vol;
retVal[0] = x0[0] * std::exp(mu*dt+vol*dw[0]*sdt);
retVal[1] = x0[1] + nu*dt + sdt*vol*vol*(rho1*dw[0] + rho2*dw[1]);
break;
case Reflection:
vol = std::sqrt(std::fabs(x0[1]));
mu = riskFreeRate_->forwardRate(t0, t0+dt, Continuous)
- dividendYield_->forwardRate(t0, t0+dt, Continuous)
- 0.5 * vol*vol;
nu = daysPerYear_*daysPerYear_*omega_
+ daysPerYear_*(beta_ + alpha_*q2 + gamma_*q3 - 1.0) * vol * vol;
retVal[0] = x0[0]*std::exp(mu*dt+vol*dw[0]*sdt);
retVal[1] = vol*vol
+nu*dt + sdt*vol*vol*(rho1*dw[0] + rho2*dw[1]);
break;
default:
QL_FAIL("unknown discretization schema");
}
return retVal;
}
const Handle<Quote>& GJRGARCHProcess::s0() const {
return s0_;
}
const Handle<YieldTermStructure>& GJRGARCHProcess::dividendYield() const {
return dividendYield_;
}
const Handle<YieldTermStructure>& GJRGARCHProcess::riskFreeRate() const {
return riskFreeRate_;
}
Time GJRGARCHProcess::time(const Date& d) const {
return riskFreeRate_->dayCounter().yearFraction(
riskFreeRate_->referenceDate(), d);
}
}
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