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/* -*- mode: c++; tab-width: 4; indent-tabs-mode: nil; c-basic-offset: 4 -*- */
/*
Copyright (C) 2013 Peter Caspers
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/experimental/models/gaussian1dmodel.hpp>
namespace QuantLib {
const Real
Gaussian1dModel::forwardRate(const Date &fixing, const Date &referenceDate,
const Real y,
boost::shared_ptr<IborIndex> iborIdx) const {
QL_REQUIRE(iborIdx != NULL, "no ibor index given");
if (fixing <=
((Date)Settings::instance().evaluationDate()) +
(Settings::instance().enforcesTodaysHistoricFixings() ? 0 : -1))
return iborIdx->fixing(fixing);
Handle<YieldTermStructure> yts =
iborIdx->forwardingTermStructure(); // might be empty, then use
// model curve
Date valueDate = iborIdx->valueDate(fixing);
Date endDate = iborIdx->fixingCalendar().advance(
valueDate, iborIdx->tenor(), iborIdx->businessDayConvention(),
iborIdx->endOfMonth());
// FIXME Here we should use the calculation date calendar ?
Real dcf = iborIdx->dayCounter().yearFraction(valueDate, endDate);
return (zerobond(valueDate, referenceDate, y, yts) -
zerobond(endDate, referenceDate, y, yts)) /
(dcf * zerobond(endDate, referenceDate, y, yts));
}
const Real
Gaussian1dModel::swapRate(const Date &fixing, const Period &tenor,
const Date &referenceDate, const Real y,
boost::shared_ptr<SwapIndex> swapIdx) const {
QL_REQUIRE(swapIdx != NULL, "no swap index given");
if (fixing <=
((Date)Settings::instance().evaluationDate()) +
(Settings::instance().enforcesTodaysHistoricFixings() ? 0 : -1))
return swapIdx->fixing(fixing);
Handle<YieldTermStructure> ytsf =
swapIdx->iborIndex()->forwardingTermStructure();
Handle<YieldTermStructure> ytsd =
swapIdx->discountingTermStructure(); // either might be empty, then
// use model curve
Schedule sched, floatSched;
SwapIndex tmpIdx =
SwapIndex(swapIdx->familyName(), tenor, swapIdx->fixingDays(),
swapIdx->currency(), swapIdx->fixingCalendar(),
swapIdx->fixedLegTenor(), swapIdx->fixedLegConvention(),
swapIdx->dayCounter(), swapIdx->iborIndex());
boost::shared_ptr<VanillaSwap> underlying =
tmpIdx.underlyingSwap(fixing);
sched = underlying->fixedSchedule();
boost::shared_ptr<OvernightIndexedSwapIndex> oisIdx =
boost::dynamic_pointer_cast<OvernightIndexedSwapIndex>(swapIdx);
if (oisIdx != NULL) {
floatSched = sched;
} else {
floatSched = underlying->floatingSchedule();
}
Real annuity = swapAnnuity(fixing, tenor, referenceDate, y,
swapIdx); // should be fine for
// overnightindexed swap indices as
// well
Rate floatleg = 0.0;
if (ytsf.empty() && ytsd.empty()) { // simple 100-formula can be used
// only in one curve setup
floatleg = (zerobond(sched.dates().front(), referenceDate, y) -
zerobond(sched.calendar().adjust(
sched.dates().back(),
underlying->paymentConvention()),
referenceDate, y));
} else {
for (Size i = 1; i < floatSched.size(); i++) {
floatleg +=
(zerobond(floatSched[i - 1], referenceDate, y, ytsf) /
zerobond(floatSched[i], referenceDate, y, ytsf) -
1.0) *
zerobond(
floatSched.calendar().adjust(
floatSched[i], underlying->paymentConvention()),
referenceDate, y, ytsd);
}
}
return floatleg / annuity;
}
const Real
Gaussian1dModel::swapAnnuity(const Date &fixing, const Period &tenor,
const Date &referenceDate, const Real y,
boost::shared_ptr<SwapIndex> swapIdx) const {
QL_REQUIRE(swapIdx != NULL, "no swap index given");
Handle<YieldTermStructure> ytsd =
swapIdx->discountingTermStructure(); // might be empty, then use
// model curve
SwapIndex tmpIdx =
SwapIndex(swapIdx->familyName(), tenor, swapIdx->fixingDays(),
swapIdx->currency(), swapIdx->fixingCalendar(),
swapIdx->fixedLegTenor(), swapIdx->fixedLegConvention(),
swapIdx->dayCounter(), swapIdx->iborIndex());
boost::shared_ptr<VanillaSwap> underlying =
tmpIdx.underlyingSwap(fixing);
Schedule sched = underlying->fixedSchedule();
Real annuity = 0.0;
for (unsigned int j = 1; j < sched.size(); j++) {
annuity +=
zerobond(sched.calendar().adjust(
sched.date(j), underlying->paymentConvention()),
referenceDate, y, ytsd) *
swapIdx->dayCounter().yearFraction(sched.date(j - 1),
sched.date(j));
}
return annuity;
}
const Real Gaussian1dModel::zerobondOption(
const Option::Type &type, const Date &expiry, const Date &valueDate,
const Date &maturity, const Rate strike, const Date &referenceDate,
const Real y, const Handle<YieldTermStructure> &yts,
const Real yStdDevs, const Size yGridPoints,
const bool extrapolatePayoff,
const bool flatPayoffExtrapolation) const {
Time fixingTime = termStructure()->timeFromReference(expiry);
Time referenceTime =
referenceDate == Null<Date>()
? 0.0
: termStructure()->timeFromReference(referenceDate);
Array yg = yGrid(yStdDevs, yGridPoints, fixingTime, referenceTime, y);
Array z = yGrid(yStdDevs, yGridPoints);
Array p(yg.size());
for (Size i = 0; i < yg.size(); i++) {
Real expValDsc = zerobond(valueDate, expiry, yg[i], yts);
Real discount = zerobond(maturity, expiry, yg[i], yts) / expValDsc;
p[i] = std::max((type == Option::Call ? 1.0 : -1.0) *
(discount - strike),
0.0) /
numeraire(fixingTime, yg[i], yts) * expValDsc;
}
CubicInterpolation payoff(z.begin(), z.end(), p.begin(),
CubicInterpolation::Spline, true,
CubicInterpolation::Lagrange, 0.0,
CubicInterpolation::Lagrange, 0.0);
Real price = 0.0;
for (Size i = 0; i < z.size() - 1; i++) {
price += gaussianShiftedPolynomialIntegral(
0.0, payoff.cCoefficients()[i], payoff.bCoefficients()[i],
payoff.aCoefficients()[i], p[i], z[i], z[i], z[i + 1]);
}
if (extrapolatePayoff) {
if (flatPayoffExtrapolation) {
price += gaussianShiftedPolynomialIntegral(
0.0, 0.0, 0.0, 0.0, p[z.size() - 2], z[z.size() - 2],
z[z.size() - 1], 100.0);
price += gaussianShiftedPolynomialIntegral(
0.0, 0.0, 0.0, 0.0, p[0], z[0], -100.0, z[0]);
} else {
if (type == Option::Call)
price += gaussianShiftedPolynomialIntegral(
0.0, payoff.cCoefficients()[z.size() - 2],
payoff.bCoefficients()[z.size() - 2],
payoff.aCoefficients()[z.size() - 2], p[z.size() - 2],
z[z.size() - 2], z[z.size() - 1], 100.0);
if (type == Option::Put)
price += gaussianShiftedPolynomialIntegral(
0.0, payoff.cCoefficients()[0],
payoff.bCoefficients()[0], payoff.aCoefficients()[0],
p[0], z[0], -100.0, z[0]);
}
}
return numeraire(referenceTime, y, yts) * price;
}
const Real Gaussian1dModel::gaussianPolynomialIntegral(
const Real a, const Real b, const Real c, const Real d, const Real e,
const Real y0, const Real y1) {
#ifdef GAUSS1D_ENABLE_NTL
const boost::math::ntl::RR aa = 4.0 * a, ba = 2.0 * M_SQRT2 * b,
ca = 2.0 * c, da = M_SQRT2 * d;
const boost::math::ntl::RR x0 = y0 * M_SQRT1_2, x1 = y1 * M_SQRT1_2;
const boost::math::ntl::RR res =
(0.125 * (3.0 * aa + 2.0 * ca + 4.0 * e) * boost::math::erf(x1) -
1.0 / (4.0 * M_SQRTPI) * exp(-x1 * x1) *
(2.0 * aa * x1 * x1 * x1 + 3.0 * aa * x1 +
2.0 * ba * (x1 * x1 + 1.0) + 2.0 * ca * x1 + 2.0 * da)) -
(0.125 * (3.0 * aa + 2.0 * ca + 4.0 * e) * boost::math::erf(x0) -
1.0 / (4.0 * M_SQRTPI) * exp(-x0 * x0) *
(2.0 * aa * x0 * x0 * x0 + 3.0 * aa * x0 +
2.0 * ba * (x0 * x0 + 1.0) + 2.0 * ca * x0 + 2.0 * da));
return NTL::to_double(res.value());
#else
const Real aa = 4.0 * a, ba = 2.0 * M_SQRT2 * b, ca = 2.0 * c,
da = M_SQRT2 * d;
const Real x0 = y0 * M_SQRT1_2, x1 = y1 * M_SQRT1_2;
return (0.125 * (3.0 * aa + 2.0 * ca + 4.0 * e) * boost::math::erf(x1) -
1.0 / (4.0 * M_SQRTPI) * exp(-x1 * x1) *
(2.0 * aa * x1 * x1 * x1 + 3.0 * aa * x1 +
2.0 * ba * (x1 * x1 + 1.0) + 2.0 * ca * x1 + 2.0 * da)) -
(0.125 * (3.0 * aa + 2.0 * ca + 4.0 * e) * boost::math::erf(x0) -
1.0 / (4.0 * M_SQRTPI) * exp(-x0 * x0) *
(2.0 * aa * x0 * x0 * x0 + 3.0 * aa * x0 +
2.0 * ba * (x0 * x0 + 1.0) + 2.0 * ca * x0 + 2.0 * da));
#endif
}
const Real Gaussian1dModel::gaussianShiftedPolynomialIntegral(
const Real a, const Real b, const Real c, const Real d, const Real e,
const Real h, const Real x0, const Real x1) {
return gaussianPolynomialIntegral(
a, -4.0 * a * h + b, 6.0 * a * h * h - 3.0 * b * h + c,
-4 * a * h * h * h + 3.0 * b * h * h - 2.0 * c * h + d,
a * h * h * h * h - b * h * h * h + c * h * h - d * h + e, x0, x1);
}
const Disposable<Array> Gaussian1dModel::yGrid(const Real stdDevs,
const int gridPoints,
const Real T, const Real t,
const Real y) const {
// we use that the standard deviation is independent of $x$ here !
QL_REQUIRE(stateProcess_ != NULL, "state process not set");
Array result(2 * gridPoints + 1, 0.0);
Real x_t, e_0_t, e_t_T, stdDev_0_t, stdDev_t_T;
Real stdDev_0_T = stateProcess_->stdDeviation(0.0, 0.0, T);
Real e_0_T = stateProcess_->expectation(0.0, 0.0, T);
if (t < QL_EPSILON) {
stdDev_0_t = 0.0;
stdDev_t_T = stdDev_0_T;
e_0_t = 0.0;
x_t = 0.0;
e_t_T = e_0_T;
} else {
stdDev_0_t = stateProcess_->stdDeviation(0.0, 0.0, t);
stdDev_t_T = stateProcess_->stdDeviation(t, 0.0, T - t);
e_0_t = stateProcess_->expectation(0.0, 0.0, t);
x_t = y * stdDev_0_t + e_0_t;
e_t_T = stateProcess_->expectation(t, x_t, T - t);
}
Real h = stdDevs / ((Real)gridPoints);
for (int j = -gridPoints; j <= gridPoints; j++) {
result[j + gridPoints] =
(e_t_T + stdDev_t_T * ((Real)j) * h - e_0_T) / stdDev_0_T;
}
return result;
}
}
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