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/* -*- mode: c++; tab-width: 4; indent-tabs-mode: nil; c-basic-offset: 4 -*- */
/*
Copyright (C) 2008 Andreas Gaida
Copyright (C) 2008 Ralph Schreyer
Copyright (C) 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.
*/
#include <ql/math/functional.hpp>
#include <ql/math/distributions/chisquaredistribution.hpp>
#include <ql/math/interpolations/linearinterpolation.hpp>
#include <ql/math/integrals/gausslobattointegral.hpp>
#include <ql/methods/finitedifferences/meshers/fdmhestonvariancemesher.hpp>
#include <set>
#include <algorithm>
namespace QuantLib {
FdmHestonVarianceMesher::FdmHestonVarianceMesher(
Size size,
const boost::shared_ptr<HestonProcess> & process,
Time maturity, Size tAvgSteps, Real epsilon)
: Fdm1dMesher(size) {
std::vector<Real> vGrid(size, 0.0), pGrid(size, 0.0);
const Real df = 4*process->theta()*process->kappa()/
square<Real>()(process->sigma());
try {
std::multiset<std::pair<Real, Real> > grid;
for (Size l=1; l<=tAvgSteps; ++l) {
const Real t = (maturity*l)/tAvgSteps;
const Real ncp = 4*process->kappa()*std::exp(-process->kappa()*t)
/(square<Real>()(process->sigma())
*(1-std::exp(-process->kappa()*t)))*process->v0();
const Real k = square<Real>()(process->sigma())
*(1-std::exp(-process->kappa()*t))/(4*process->kappa());
const Real qMin = 0.0; // v_min = 0.0;
const Real qMax = std::max(process->v0(),
k*InverseNonCentralChiSquareDistribution(
df, ncp, 1000, 1e-8)(1-epsilon));
const Real minVStep=(qMax-qMin)/(50*size);
Real ps,p = 0.0;
Real vTmp = qMin;
grid.insert(std::pair<Real, Real>(qMin, epsilon));
for (Size i=1; i < size; ++i) {
ps = (1 - epsilon - p)/(size-i);
p += ps;
const Real tmp = k*InverseNonCentralChiSquareDistribution(
df, ncp, 1000, 1e-8)(p);
const Real vx = std::max(vTmp+minVStep, tmp);
p = NonCentralChiSquareDistribution(df, ncp)(vx/k);
vTmp=vx;
grid.insert(std::pair<Real, Real>(vx, p));
}
}
QL_REQUIRE(grid.size() == size*tAvgSteps,
"something wrong with the grid size");
std::vector<std::pair<Real, Real> > tp(grid.begin(), grid.end());
for (Size i=0; i < size; ++i) {
const Size b = (i*tp.size())/size;
const Size e = ((i+1)*tp.size())/size;
for (Size j=b; j < e; ++j) {
vGrid[i]+=tp[j].first/(e-b);
pGrid[i]+=tp[j].second/(e-b);
}
}
}
catch (const Error&) {
// use default mesh
const Real vol = process->sigma()*
std::sqrt(process->theta()/(2*process->kappa()));
const Real mean = process->theta();
const Real upperBound = std::max(process->v0()+4*vol, mean+4*vol);
const Real lowerBound
= std::max(0.0, std::min(process->v0()-4*vol, mean-4*vol));
for (Size i=0; i < size; ++i) {
pGrid[i] = i/(size-1.0);
vGrid[i] = lowerBound + i*(upperBound-lowerBound)/(size-1.0);
}
}
Real skewHint = ((process->kappa() != 0.0)
? std::max(1.0, process->sigma()/process->kappa()) : 1.0);
std::sort(pGrid.begin(), pGrid.end());
volaEstimate_ = GaussLobattoIntegral(100000, 1e-4)(
boost::function1<Real, Real>(
compose(std::ptr_fun<Real, Real>(std::sqrt),
LinearInterpolation(pGrid.begin(), pGrid.end(),
vGrid.begin()))),
pGrid.front(), pGrid.back())*std::pow(skewHint, 1.5);
const Real v0 = process->v0();
for (Size i=1; i<vGrid.size(); ++i) {
if (vGrid[i-1] <= v0 && vGrid[i] >= v0) {
if (std::fabs(vGrid[i-1] - v0) < std::fabs(vGrid[i] - v0))
vGrid[i-1] = v0;
else
vGrid[i] = v0;
}
}
std::copy(vGrid.begin(), vGrid.end(), locations_.begin());
for (Size i=0; i < size-1; ++i) {
dminus_[i+1] = dplus_[i] = vGrid[i+1] - vGrid[i];
}
dplus_.back() = dminus_.front() = Null<Real>();
}
}
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