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/* -*- mode: c++; tab-width: 4; indent-tabs-mode: nil; c-basic-offset: 4 -*- */
/*
Copyright (C) 2011 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 exponentialjump1dmesher.cpp
\brief mesher for a exponential jump mesher with high
mean reversion rate and low jump intensity
*/
#include <ql/math/incompletegamma.hpp>
#include <ql/math/integrals/gausslobattointegral.hpp>
#include <ql/math/distributions/gammadistribution.hpp>
#include <ql/methods/finitedifferences/meshers/exponentialjump1dmesher.hpp>
#include <boost/bind.hpp>
namespace QuantLib {
ExponentialJump1dMesher::ExponentialJump1dMesher(
Size steps, Real beta, Real jumpIntensity, Real eta, Real eps)
: Fdm1dMesher(steps),
beta_(beta), jumpIntensity_(jumpIntensity), eta_(eta)
{
QL_REQUIRE(eps > 0.0 && eps < 1.0, "eps > 0.0 and eps < 1.0");
QL_REQUIRE(steps > 1, "minimum number of steps is two");
const Real start = 0.0;
const Real end = 1.0-eps;
const Real dx = (end-start)/(steps-1);
for (Size i=0; i < steps; ++i) {
const Real p = start + i*dx;
locations_[i] = -1.0/eta*std::log(1.0-p);
}
for (Size i=0; i < steps-1; ++i) {
dminus_[i+1] = dplus_[i] = locations_[i+1]-locations_[i];
}
dplus_.back() = dminus_.front() = Null<Real>();
}
Real ExponentialJump1dMesher::jumpSizeDensity(Real x, Time t) const {
const Real a = 1.0-jumpIntensity_/beta_;
const Real norm = 1.0-std::exp(-jumpIntensity_*t);
const Real gammaValue
= std::exp(GammaFunction().logValue(1.0-jumpIntensity_/beta_));
return jumpIntensity_*gammaValue/norm
*( incompleteGammaFunction(a, x*std::exp(beta_*t)*eta_)
-incompleteGammaFunction(a, x*eta_))
*std::pow(eta_, jumpIntensity_/beta_)
/(beta_*std::pow(x, a));
}
Real ExponentialJump1dMesher::jumpSizeDensity(Real x) const {
const Real a = 1.0-jumpIntensity_/beta_;
const Real gammaValue
= std::exp(GammaFunction().logValue(jumpIntensity_/beta_));
return std::exp(-x*eta_)*std::pow(x, -a) * std::pow(eta_, 1.0-a)
/ gammaValue;
}
Real ExponentialJump1dMesher::jumpSizeDistribution(Real x, Time t) const {
const Real xmin = std::min(x, 1.0e-100);
return GaussLobattoIntegral(1000000, 1e-12)(
boost::bind(&ExponentialJump1dMesher::jumpSizeDensity, this, _1, t),
xmin, std::max(x, xmin));
}
Real ExponentialJump1dMesher::jumpSizeDistribution(Real x) const {
const Real a = jumpIntensity_/beta_;
const Real xmin = std::min(x, QL_EPSILON);
const Real gammaValue
= std::exp(GammaFunction().logValue(jumpIntensity_/beta_));
const Real lowerEps =
(std::pow(xmin, a)/a - std::pow(xmin, a+1)/(a+1))/gammaValue;
return lowerEps + GaussLobattoIntegral(10000, 1e-12)(
boost::bind(&ExponentialJump1dMesher::jumpSizeDensity, this, _1),
xmin/eta_, std::max(x, xmin/eta_));
}
}
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