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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/markovfunctional.hpp>
#include <ql/experimental/models/smilesectionutils.hpp>
namespace QuantLib {
MarkovFunctional::MarkovFunctional(
const Handle<YieldTermStructure> &termStructure, const Real reversion,
const std::vector<Date> &volstepdates,
const std::vector<Real> &volatilities,
const Handle<SwaptionVolatilityStructure> &swaptionVol,
const std::vector<Date> &swaptionExpiries,
const std::vector<Period> &swaptionTenors,
const boost::shared_ptr<SwapIndex> &swapIndexBase,
const MarkovFunctional::ModelSettings &modelSettings)
: Gaussian1dModel(termStructure), CalibratedModel(1),
modelSettings_(modelSettings), capletCalibrated_(false),
reversion_(ConstantParameter(reversion, NoConstraint())),
sigma_(arguments_[0]), volstepdates_(volstepdates),
volatilities_(volatilities), swaptionVol_(swaptionVol),
capletVol_(Handle<OptionletVolatilityStructure>()),
swaptionExpiries_(swaptionExpiries),
capletExpiries_(std::vector<Date>()), swaptionTenors_(swaptionTenors),
swapIndexBase_(swapIndexBase),
iborIndex_(swapIndexBase->iborIndex()) {
QL_REQUIRE(swaptionExpiries.size() == swaptionTenors.size(),
"number of swaption expiries ("
<< swaptionExpiries.size()
<< ") is differnt from number of swaption tenors ("
<< swaptionTenors.size() << ")");
QL_REQUIRE(swaptionExpiries.size() >= 1,
"need at least one swaption expiry to calibrate numeraire");
QL_REQUIRE(!termStructure.empty(),
"yield term structure handle is empty");
QL_REQUIRE(!swaptionVol.empty(),
"swaption volatility structure is empty");
modelSettings_.validate();
initialize();
}
MarkovFunctional::MarkovFunctional(
const Handle<YieldTermStructure> &termStructure, const Real reversion,
const std::vector<Date> &volstepdates,
const std::vector<Real> &volatilities,
const Handle<OptionletVolatilityStructure> &capletVol,
const std::vector<Date> &capletExpiries,
const boost::shared_ptr<IborIndex> &iborIndex,
const MarkovFunctional::ModelSettings &modelSettings)
: Gaussian1dModel(termStructure), CalibratedModel(1),
modelSettings_(modelSettings), capletCalibrated_(true),
reversion_(ConstantParameter(reversion, NoConstraint())),
sigma_(arguments_[0]), volstepdates_(volstepdates),
volatilities_(volatilities),
swaptionVol_(Handle<SwaptionVolatilityStructure>()),
capletVol_(capletVol), swaptionExpiries_(std::vector<Date>()),
capletExpiries_(capletExpiries),
swaptionTenors_(std::vector<Period>()), iborIndex_(iborIndex) {
QL_REQUIRE(capletExpiries.size() >= 1,
"need at least one caplet expiry to calibrate numeraire");
QL_REQUIRE(!termStructure.empty(),
"yield term structure handle is empty");
QL_REQUIRE(!capletVol.empty(), "caplet volatility structure is empty");
modelSettings_.validate();
initialize();
}
void MarkovFunctional::initialize() {
QL_MFMESSAGE(modelOutputs_, "initializing");
modelOutputs_.dirty_ = true;
modelOutputs_.settings_ = modelSettings_;
GaussHermiteIntegration gaussHermite(
modelSettings_.gaussHermitePoints_);
normalIntegralX_ = gaussHermite.x();
normalIntegralW_ = gaussHermite.weights();
for (Size i = 0; i < normalIntegralX_.size(); i++) {
normalIntegralW_[i] *=
exp(-normalIntegralX_[i] * normalIntegralX_[i]) * M_1_SQRTPI;
normalIntegralX_[i] *= M_SQRT2;
}
volsteptimes_.clear();
volsteptimesArray_ = Array(volstepdates_.size());
int j = 0;
for (std::vector<Date>::const_iterator i = volstepdates_.begin();
i != volstepdates_.end(); i++, j++) {
volsteptimes_.push_back(termStructure()->timeFromReference(*i));
volsteptimesArray_[j] = volsteptimes_[j];
if (j == 0)
QL_REQUIRE(volsteptimes_[0] > 0.0,
"volsteptimes must be positive (" << volsteptimes_[0]
<< ")");
else
QL_REQUIRE(volsteptimes_[j] > volsteptimes_[j - 1],
"volsteptimes must be strictly increasing ("
<< volsteptimes_[j - 1] << "@" << (j - 1) << ", "
<< volsteptimes_[j] << "@" << j << ")");
}
if (capletCalibrated_) {
for (std::vector<Date>::const_iterator i = capletExpiries_.begin(); i != capletExpiries_.end(); i++) {
makeCapletCalibrationPoint(*i);
}
} else {
std::vector<Date>::const_iterator i;
std::vector<Period>::const_iterator j;
for (i = swaptionExpiries_.begin(), j = swaptionTenors_.begin();
i != swaptionExpiries_.end(); i++, j++) {
makeSwaptionCalibrationPoint(*i, *j);
}
}
bool done;
numeraireDate_ = Date::minDate();
do {
Date numeraireKnown = numeraireDate_;
done = true;
for (std::map<Date, CalibrationPoint>::reverse_iterator i =
calibrationPoints_.rbegin();
i != calibrationPoints_.rend() && done; i++) {
if (i->second.paymentDates_.back() > numeraireDate_) {
numeraireDate_ = i->second.paymentDates_.back();
numeraireKnown = i->second.paymentDates_.back();
if (i != calibrationPoints_.rbegin()) {
done = false;
}
}
// Inlining this into the loop condition causes
// a bogus compilation error wih g++ 4.0.1 on Mac OS X
std::vector<Date>::const_reverse_iterator rend =
i->second.paymentDates_.rend();
for (std::vector<Date>::const_reverse_iterator j =
i->second.paymentDates_.rbegin();
j != rend && done; j++) {
if (*j < numeraireKnown) {
if (capletCalibrated_) {
makeCapletCalibrationPoint(*j);
done = false;
break;
} else {
UpRounding rounder(0);
makeSwaptionCalibrationPoint(
*j,
Period(
static_cast<Integer>(rounder(
(swapIndexBase_->dayCounter()
.yearFraction(*j, numeraireKnown) -
0.5 / 365) *
12.0)),
Months));
done = false;
break;
}
}
}
if (done) {
numeraireKnown = i->first;
}
}
} while (!done);
numeraireTime_ = termStructure()->timeFromReference(numeraireDate_);
times_.clear();
times_.push_back(0.0);
modelOutputs_.expiries_.clear();
modelOutputs_.tenors_.clear();
for (std::map<Date, CalibrationPoint>::iterator k =
calibrationPoints_.begin();
k != calibrationPoints_.end(); k++) {
times_.push_back(termStructure()->timeFromReference(k->first));
modelOutputs_.expiries_.push_back(k->first);
modelOutputs_.tenors_.push_back(k->second.tenor_);
}
times_.push_back(numeraireTime_);
QL_REQUIRE(volatilities_.size() == volsteptimes_.size() + 1,
"there must be n+1 volatilities ("
<< volatilities_.size()
<< ") for n volatility step times ("
<< volsteptimes_.size() << ")");
sigma_ =
PiecewiseConstantParameter(volsteptimes_, PositiveConstraint());
for (Size i = 0; i < sigma_.size(); i++) {
sigma_.setParam(i, volatilities_[i]);
}
stateProcess_ = boost::shared_ptr<MfStateProcess>(new MfStateProcess(
reversion_(0.0), volsteptimesArray_, sigma_.params()));
y_ = yGrid(modelSettings_.yStdDevs_, modelSettings_.yGridPoints_);
discreteNumeraire_ = boost::shared_ptr<Matrix>(new Matrix(
times_.size(), 2 * modelSettings_.yGridPoints_ + 1, 1.0));
for (Size i = 0; i < times_.size(); i++) {
boost::shared_ptr<Interpolation> numInt(new CubicInterpolation(
y_.begin(), y_.end(), discreteNumeraire_->row_begin(i),
CubicInterpolation::Spline, true, CubicInterpolation::Lagrange,
0.0, CubicInterpolation::Lagrange, 0.0));
numInt->enableExtrapolation();
numeraire_.push_back(numInt);
}
LazyObject::registerWith(termStructure());
if (!swaptionVol_.empty())
LazyObject::registerWith(swaptionVol_);
if (!capletVol_.empty())
LazyObject::registerWith(capletVol_);
}
void MarkovFunctional::makeSwaptionCalibrationPoint(const Date &expiry,
const Period &tenor) {
QL_REQUIRE(calibrationPoints_.count(expiry) == 0,
"swaption expiry ("
<< expiry
<< ") occurs more than once in calibration set");
CalibrationPoint p;
p.isCaplet_ = false;
p.tenor_ = tenor;
SwapIndex tmpIndex(
swapIndexBase_->familyName(), tenor, swapIndexBase_->fixingDays(),
swapIndexBase_->currency(), swapIndexBase_->fixingCalendar(),
swapIndexBase_->fixedLegTenor(),
swapIndexBase_->fixedLegConvention(), swapIndexBase_->dayCounter(),
swapIndexBase_->iborIndex());
boost::shared_ptr<VanillaSwap> underlying =
tmpIndex.underlyingSwap(expiry);
Schedule sched = underlying->fixedSchedule();
Calendar cal = sched.calendar();
BusinessDayConvention bdc = underlying->paymentConvention();
for (unsigned int k = 1; k < sched.size(); k++) {
p.yearFractions_.push_back(
swapIndexBase_->dayCounter().yearFraction(
k == 1 ? expiry : sched.date(k - 1), sched.date(k)));
p.paymentDates_.push_back(cal.adjust(sched.date(k), bdc));
}
calibrationPoints_[expiry] = p;
}
void MarkovFunctional::makeCapletCalibrationPoint(const Date &expiry) {
QL_REQUIRE(
calibrationPoints_.count(expiry) == 0,
"caplet expiry (" << expiry
<< ") occurs more than once in calibration set");
CalibrationPoint p;
p.isCaplet_ = true;
// p.expiry_ = expiry;
p.tenor_ = iborIndex_->tenor();
Date valueDate = iborIndex_->valueDate(expiry);
Date endDate = iborIndex_->fixingCalendar().advance(
valueDate, iborIndex_->tenor(), iborIndex_->businessDayConvention(),
iborIndex_->endOfMonth());
// FIXME Here we should use a calculation date calendar ?
p.paymentDates_.push_back(endDate);
p.yearFractions_.push_back(
iborIndex_->dayCounter().yearFraction(expiry, endDate));
// adjust the first period to start on expiry
calibrationPoints_[expiry] = p;
}
void MarkovFunctional::updateSmiles() const {
QL_MFMESSAGE(modelOutputs_, "updating smiles");
modelOutputs_.dirty_ = true;
for (std::map<Date, CalibrationPoint>::reverse_iterator i =
calibrationPoints_.rbegin();
i != calibrationPoints_.rend(); i++) {
boost::shared_ptr<SmileSection> smileSection;
if (i->second.isCaplet_) {
i->second.annuity_ =
i->second.yearFractions_[0] *
termStructure()->discount(i->second.paymentDates_[0], true);
i->second.atm_ = (termStructure()->discount(i->first, true) -
termStructure()->discount(
i->second.paymentDates_[0], true)) /
i->second.annuity_;
smileSection = capletVol_->smileSection(i->first, true);
} else {
Real annuity = 0.0;
for (unsigned int k = 0; k < i->second.paymentDates_.size();
k++) {
annuity += i->second.yearFractions_[k] *
termStructure()->discount(
i->second.paymentDates_[k], true);
}
i->second.annuity_ = annuity;
i->second.atm_ = (termStructure()->discount(i->first, true) -
termStructure()->discount(
i->second.paymentDates_.back(), true)) /
annuity;
smileSection = swaptionVol_->smileSection(
i->first, i->second.tenor_, true);
}
i->second.rawSmileSection_ = boost::shared_ptr<SmileSection>(
new AtmSmileSection(smileSection, i->second.atm_));
if (modelSettings_.adjustments_ & ModelSettings::KahaleSmile) {
i->second.smileSection_ = boost::shared_ptr<KahaleSmileSection>(
new KahaleSmileSection(
i->second.rawSmileSection_, i->second.atm_,
(modelSettings_.adjustments_ &
ModelSettings::KahaleInterpolation) != 0,
(modelSettings_.adjustments_ &
ModelSettings::SmileExponentialExtrapolation) != 0,
(modelSettings_.adjustments_ &
ModelSettings::SmileDeleteArbitragePoints) != 0,
modelSettings_.smileMoneynessCheckpoints_,
modelSettings_.digitalGap_));
} else {
if (modelSettings_.adjustments_ & ModelSettings::SabrSmile) {
SmileSectionUtils ssutils(
*i->second.rawSmileSection_,
modelSettings_.smileMoneynessCheckpoints_);
std::vector<Real> k = ssutils.strikeGrid();
k.erase(k.begin()); // the first strike is zero which we do
// not want in the sabr calibration
QL_REQUIRE(
k.size() >= 4,
"for sabr calibration at least 4 points are needed (is "
<< k.size() << ")");
std::vector<Real> v;
for (Size j = 0; j < k.size(); j++) {
v.push_back(
i->second.rawSmileSection_->volatility(k[j]));
}
// TODO should we fix beta to avoid numerical instabilities
// during calibration ?
boost::shared_ptr<SabrInterpolatedSmileSection>
sabrSection(new SabrInterpolatedSmileSection(
i->first, i->second.atm_, k, false,
i->second.rawSmileSection_->volatility(i->second.atm_),
v, 0.03, 0.80, 0.50, 0.00, false, false, false, false));
// we make the sabr section arbitrage free by superimposing
// a kahalesection
i->second.smileSection_ = boost::shared_ptr<
KahaleSmileSection>(new KahaleSmileSection(
sabrSection, i->second.atm_, false,
(modelSettings_.adjustments_ &
ModelSettings::SmileExponentialExtrapolation) != 0,
(modelSettings_.adjustments_ &
ModelSettings::SmileDeleteArbitragePoints) != 0,
modelSettings_.smileMoneynessCheckpoints_,
modelSettings_.digitalGap_));
} else { // no smile pretreatment
i->second.smileSection_ = i->second.rawSmileSection_;
}
}
i->second.minRateDigital_ =
i->second.smileSection_->digitalOptionPrice(
modelSettings_.lowerRateBound_, Option::Call,
i->second.annuity_, modelSettings_.digitalGap_);
i->second.maxRateDigital_ =
i->second.smileSection_->digitalOptionPrice(
modelSettings_.upperRateBound_, Option::Call,
i->second.annuity_, modelSettings_.digitalGap_);
// output smile for testing
// boost::shared_ptr<SmileSection> sec1 =
// i->second.rawSmileSection_;
// boost::shared_ptr<KahaleSmileSection> sec2 =
// boost::dynamic_pointer_cast<KahaleSmileSection>(i->second.smileSection_);
// const std::vector<double> &money =
// modelSettings_.smileMoneynessCheckpoints_;
// SmileSectionUtils sutils(*sec1, money);
// std::cout
// <<
// "-------------------------------------------------------------------"
// << std::endl;
// std::cout << "Smile for expiry " << i->first << " tenor " <<
// i->second.tenor_
// << " atm is " << i->second.atm_ << std::endl;
// std::cout << "Arbitrage free region " <<
// sutils.arbitragefreeRegion().first
// << " ... " << sutils.arbitragefreeRegion().second <<
// std::endl;
// if (sec2)
// std::cout << "Kahale core region " <<
// sec2->leftCoreStrike()
// << " ... " << sec2->rightCoreStrike() << std::endl;
// std::cout <<
// "strike;rawVol;rawVar;rawCall;Call;rawDigial;Digital;"
// "rawDensity;Density;callDiff;Arb" << std::endl;
// Real strike = 0.00001;
// while (strike <= 0.20 + 1E-8) {
// std::cout << strike << ";" << sec1->volatility(strike) << ";"
// << sec1->variance(strike) << ";" <<
// sec1->optionPrice(strike)
// << ";" << (sec2 ? sec2->optionPrice(strike) : 0.0)
// << ";"
// << sec1->digitalOptionPrice(strike) << ";"
// << (sec2 ? sec2->digitalOptionPrice(strike) : 0.0)
// << ";"
// << sec1->density(strike) << ";"
// << (sec2 ? sec2->density(strike) : 0.0) << ";"
// << (sec2
// ? sec1->optionPrice(strike) -
// sec2->optionPrice(strike)
// : 0.0) << ";" << ((sec2 ? sec2->density(strike)
// : sec1->density(strike)) <
// 0.0
// ? "**********"
// : "") << std::endl;
// strike += 0.0010;
// }
// std::cout
// <<
// "-------------------------------------------------------------------"
// << std::endl;
// end output smile
}
}
void MarkovFunctional::updateNumeraireTabulation() const {
QL_MFMESSAGE(modelOutputs_, "updating numeraire tabulation");
modelOutputs_.dirty_ = true;
modelOutputs_.adjustmentFactors_.clear();
modelOutputs_.digitalsAdjustmentFactors_.clear();
int idx = times_.size() - 2;
for (std::map<Date, CalibrationPoint>::reverse_iterator
i = calibrationPoints_.rbegin();
i != calibrationPoints_.rend(); i++, idx--) {
Array discreteDeflatedAnnuities(y_.size(), 0.0);
Array deflatedFinalPayments;
Real numeraire0 = termStructure()->discount(numeraireTime_, true);
Real normalization =
termStructure()->discount(times_[idx], true) / numeraire0;
for (unsigned int k = 0; k < i->second.paymentDates_.size(); k++) {
deflatedFinalPayments =
deflatedZerobondArray(termStructure()->timeFromReference(
i->second.paymentDates_[k]),
times_[idx], y_);
discreteDeflatedAnnuities +=
deflatedFinalPayments * i->second.yearFractions_[k];
}
CubicInterpolation deflatedAnnuities(
y_.begin(), y_.end(), discreteDeflatedAnnuities.begin(),
CubicInterpolation::Spline, true, CubicInterpolation::Lagrange,
0.0, CubicInterpolation::Lagrange, 0.0);
deflatedAnnuities.enableExtrapolation();
Real digitalsCorrectionFactor = 1.0;
modelOutputs_.digitalsAdjustmentFactors_.insert(
modelOutputs_.digitalsAdjustmentFactors_.begin(),
digitalsCorrectionFactor);
Real digital = 0.0, swapRate, swapRate0;
for (int c = 0;
c == 0 || (c == 1 && (modelSettings_.adjustments_ &
ModelSettings::AdjustDigitals));
c++) {
if (c == 1) {
digitalsCorrectionFactor = i->second.annuity_ / digital;
modelOutputs_.digitalsAdjustmentFactors_.front() =
digitalsCorrectionFactor;
}
digital = 0.0;
swapRate0 =
modelSettings_.upperRateBound_ / 2.0; // initial guess
for (int j = y_.size() - 1; j >= 0; j--) {
Real integral = 0.0;
if (j == (int)(y_.size() - 1)) {
if ((modelSettings_.adjustments_ &
ModelSettings::NoPayoffExtrapolation) == 0) {
if ((modelSettings_.adjustments_ &
ModelSettings::ExtrapolatePayoffFlat) != 0) {
integral = gaussianShiftedPolynomialIntegral(
0.0, 0.0, 0.0, 0.0,
discreteDeflatedAnnuities[j - 1], y_[j - 1],
y_[j], 100.0);
} else {
Real ca =
deflatedAnnuities.aCoefficients()[j - 1];
Real cb =
deflatedAnnuities.bCoefficients()[j - 1];
Real cc =
deflatedAnnuities.cCoefficients()[j - 1];
integral = gaussianShiftedPolynomialIntegral(
0.0, cc, cb, ca,
discreteDeflatedAnnuities[j - 1], y_[j - 1],
y_[j], 100.0);
}
}
} else {
Real ca = deflatedAnnuities.aCoefficients()[j];
Real cb = deflatedAnnuities.bCoefficients()[j];
Real cc = deflatedAnnuities.cCoefficients()[j];
integral = gaussianShiftedPolynomialIntegral(
0.0, cc, cb, ca, discreteDeflatedAnnuities[j],
y_[j], y_[j], y_[j + 1]);
}
if (integral < 0) {
QL_MFMESSAGE(modelOutputs_,
"WARNING: integral for digitalPrice is "
"negative for j="
<< j << " (" << integral
<< ") --- reset it to zero.");
integral = 0.0;
}
digital += integral * numeraire0 * digitalsCorrectionFactor;
if (digital >= i->second.minRateDigital_)
swapRate = modelSettings_.lowerRateBound_;
else {
if (digital <= i->second.maxRateDigital_)
swapRate = modelSettings_.upperRateBound_;
else {
swapRate = marketSwapRate(i->first, i->second,
digital, swapRate0);
if (j < (int)y_.size() - 1 &&
swapRate > swapRate0) {
QL_MFMESSAGE(
modelOutputs_,
"WARNING: swap rate is decreasing in y for "
"t="
<< times_[idx] << ", j=" << j
<< " (y, swap rate) is (" << y_[j]
<< "," << swapRate
<< ") but for j=" << j + 1 << " it is ("
<< y_[j + 1] << "," << swapRate0
<< ") --- reset rate to " << swapRate0
<< " in node j=" << j);
swapRate = swapRate0;
}
}
}
swapRate0 = swapRate;
Real numeraire =
1.0 / (swapRate * discreteDeflatedAnnuities[j] +
deflatedFinalPayments[j]);
(*discreteNumeraire_)[idx][j] = numeraire * normalization;
}
}
if (modelSettings_.adjustments_ & ModelSettings::AdjustYts) {
numeraire_[idx]->update();
Real modelDeflatedZerobond = deflatedZerobond(times_[idx], 0.0);
Real marketDeflatedZerobond =
termStructure()->discount(times_[idx], true) /
termStructure()->discount(numeraireTime_, true);
for (int j = y_.size() - 1; j >= 0; j--) {
(*discreteNumeraire_)[idx][j] *=
modelDeflatedZerobond / marketDeflatedZerobond;
}
modelOutputs_.adjustmentFactors_.insert(
modelOutputs_.adjustmentFactors_.begin(),
modelDeflatedZerobond / marketDeflatedZerobond);
} else {
modelOutputs_.adjustmentFactors_.insert(
modelOutputs_.adjustmentFactors_.begin(), 1.0);
}
numeraire_[idx]->update();
}
}
const MarkovFunctional::ModelOutputs &
MarkovFunctional::modelOutputs() const {
if (modelOutputs_.dirty_) {
calculate();
// yield term structure
modelOutputs_.marketZerorate_.clear();
modelOutputs_.modelZerorate_.clear();
for (Size i = 1; i < times_.size() - 1; i++) {
modelOutputs_.marketZerorate_.push_back(
termStructure()->zeroRate(times_[i], QuantLib::Continuous,
QuantLib::Annual));
modelOutputs_.modelZerorate_.push_back(
-std::log(zerobond(times_[i])) / times_[i]);
}
// volatility surface
modelOutputs_.smileStrikes_.clear();
modelOutputs_.marketCallPremium_.clear();
modelOutputs_.marketPutPremium_.clear();
modelOutputs_.modelCallPremium_.clear();
modelOutputs_.modelPutPremium_.clear();
modelOutputs_.marketVega_.clear();
modelOutputs_.marketRawCallPremium_.clear();
modelOutputs_.marketRawPutPremium_.clear();
for (std::map<Date, CalibrationPoint>::iterator i =
calibrationPoints_.begin();
i != calibrationPoints_.end(); i++) {
modelOutputs_.atm_.push_back(i->second.atm_);
modelOutputs_.annuity_.push_back(i->second.annuity_);
boost::shared_ptr<SmileSection> sec = i->second.smileSection_;
boost::shared_ptr<SmileSection> rawSec =
i->second.rawSmileSection_;
SmileSectionUtils ssutils(
*sec, modelSettings_.smileMoneynessCheckpoints_,
i->second.atm_);
std::vector<Real> money = ssutils.moneyGrid();
std::vector<Real> strikes, marketCall, marketPut, modelCall,
modelPut, marketVega, marketRawCall, marketRawPut;
for (Size j = 0; j < money.size(); j++) {
strikes.push_back(money[j] * i->second.atm_);
try {
marketRawCall.push_back(rawSec->optionPrice(
strikes[j], Option::Call, i->second.annuity_));
marketRawPut.push_back(rawSec->optionPrice(
strikes[j], Option::Put, i->second.annuity_));
}
catch (QuantLib::Error) {
// the smile section might not be able to output an
// option price because it has no atm level
marketRawCall.push_back(0.0);
marketRawPut.push_back(0.0);
}
marketCall.push_back(sec->optionPrice(
strikes[j], Option::Call, i->second.annuity_));
marketPut.push_back(sec->optionPrice(
strikes[j], Option::Put, i->second.annuity_));
modelCall.push_back(
i->second.isCaplet_
? capletPriceInternal(Option::Call, i->first, strikes[j],
Null<Date>(), 0.0, true)
: swaptionPriceInternal(Option::Call, i->first,
i->second.tenor_, strikes[j],
Null<Date>(), 0.0, true));
modelPut.push_back(
i->second.isCaplet_
? capletPriceInternal(Option::Put, i->first, strikes[j],
Null<Date>(), 0.0, true)
: swaptionPriceInternal(Option::Put, i->first,
i->second.tenor_, strikes[j],
Null<Date>(), 0.0, true));
marketVega.push_back(
sec->vega(strikes[j], i->second.annuity_));
}
modelOutputs_.smileStrikes_.push_back(strikes);
modelOutputs_.marketCallPremium_.push_back(marketCall);
modelOutputs_.marketPutPremium_.push_back(marketPut);
modelOutputs_.modelCallPremium_.push_back(modelCall);
modelOutputs_.modelPutPremium_.push_back(modelPut);
modelOutputs_.marketVega_.push_back(marketVega);
modelOutputs_.marketRawCallPremium_.push_back(marketRawCall);
modelOutputs_.marketRawPutPremium_.push_back(marketRawPut);
}
modelOutputs_.dirty_ = false;
}
return modelOutputs_;
}
const Disposable<Array>
MarkovFunctional::numeraireArray(const Time t, const Array &y) const {
calculate();
Array res(y.size(), termStructure()->discount(numeraireTime_, true));
if (t < QL_EPSILON)
return res;
Real inverseNormalization =
termStructure()->discount(numeraireTime_, true) /
termStructure()->discount(t, true);
Time tz = std::min(t, times_.back());
Size i = std::min<Size>(
std::upper_bound(times_.begin(), times_.end() - 1, t) -
times_.begin(),
times_.size() - 1);
Real ta = times_[i - 1];
Real tb = times_[i];
Real dt = tb - ta;
for (Size j = 0; j < y.size(); j++) {
Real yv = y[j];
if (yv < y_.front())
yv = y_.front();
// FIXME flat extrapolation should be incoperated into interpolation
// object, see above
if (yv > y_.back())
yv = y_.back();
Real na = (*numeraire_[i - 1])(yv);
Real nb = (*numeraire_[i])(yv);
res[j] =
inverseNormalization / ((tz - ta) / nb + (tb - tz) / na) * dt;
// linear in reciprocal of normalized numeraire
}
return res;
}
const Disposable<Array>
MarkovFunctional::zerobondArray(const Time T, const Time t,
const Array &y) const {
return deflatedZerobondArray(T, t, y) * numeraireArray(t, y);
}
const Disposable<Array>
MarkovFunctional::deflatedZerobondArray(const Time T, const Time t,
const Array &y) const {
calculate();
Array result(y.size(), 0.0);
// Gauss Hermite
Real stdDev_0_t = stateProcess_->stdDeviation(0.0, 0.0, t);
// we use that the standard deviation is independent of $x$ here
Real stdDev_0_T = stateProcess_->stdDeviation(0.0, 0.0, T);
Real stdDev_t_T = stateProcess_->stdDeviation(t, 0.0, T - t);
for (Size j = 0; j < y.size(); j++) {
Array ya(modelSettings_.gaussHermitePoints_);
for (Size i = 0; i < modelSettings_.gaussHermitePoints_; i++) {
ya[i] = (y[j] * stdDev_0_t + stdDev_t_T * normalIntegralX_[i]) /
stdDev_0_T;
}
Array res = numeraireArray(T, ya);
for (Size i = 0; i < modelSettings_.gaussHermitePoints_; i++) {
result[j] += normalIntegralW_[i] / res[i];
}
}
return result;
}
const Real MarkovFunctional::numeraireImpl(
const Time t, const Real y,
const Handle<YieldTermStructure> &yts) const {
if (t == 0)
return yts.empty()
? this->termStructure()->discount(numeraireTime(), true)
: yts->discount(numeraireTime());
Array ya(1, y);
return numeraireArray(t, ya)[0] *
(yts.empty() ? 1.0
: (yts->discount(numeraireTime()) /
yts->discount(t) * termStructure()->discount(t) /
termStructure()->discount(numeraireTime())));
}
const Real MarkovFunctional::zerobondImpl(
const Time T, const Time t, const Real y,
const Handle<YieldTermStructure> &yts) const {
if (t == 0.0)
return yts.empty() ? this->termStructure()->discount(T, true)
: yts->discount(T, true);
Array ya(1, y);
return zerobondArray(T, t, ya)[0] *
(yts.empty() ? 1.0 : (yts->discount(T) / yts->discount(t) *
termStructure()->discount(t) /
termStructure()->discount(T)));
}
const Real MarkovFunctional::deflatedZerobond(Time T, Time t,
Real y) const {
Array ya(1, y);
return deflatedZerobondArray(T, t, ya)[0];
}
const Real MarkovFunctional::marketSwapRate(const Date &expiry,
const CalibrationPoint &p,
const Real digitalPrice,
const Real guess) const {
ZeroHelper z(this, expiry, p, digitalPrice);
Brent b;
Real solution = b.solve(
z, modelSettings_.marketRateAccuracy_,
std::max(std::min(guess, modelSettings_.upperRateBound_ - 0.00001),
modelSettings_.lowerRateBound_ + 0.00001),
modelSettings_.lowerRateBound_, modelSettings_.upperRateBound_);
return solution;
}
const Real MarkovFunctional::marketDigitalPrice(const Date &expiry,
const CalibrationPoint &p,
const Option::Type &type,
const Real strike) const {
return p.smileSection_->digitalOptionPrice(strike, type, p.annuity_,
modelSettings_.digitalGap_);
}
std::ostream &operator<<(std::ostream &out,
const MarkovFunctional::ModelOutputs &m) {
out << "Markov functional model trace output " << std::endl;
out << "Model settings" << std::endl;
out << "Grid points y : " << m.settings_.yGridPoints_
<< std::endl;
out << "Std devs y : " << m.settings_.yStdDevs_ << std::endl;
out << "Lower rate bound : " << m.settings_.lowerRateBound_
<< std::endl;
out << "Upper rate bound : " << m.settings_.upperRateBound_
<< std::endl;
out << "Gauss Hermite points : " << m.settings_.gaussHermitePoints_
<< std::endl;
out << "Digital gap : " << m.settings_.digitalGap_
<< std::endl;
out << "Adjustments : "
<< (m.settings_.adjustments_ &
MarkovFunctional::ModelSettings::AdjustDigitals
? "Digitals "
: "") << (m.settings_.adjustments_ &
MarkovFunctional::ModelSettings::AdjustYts
? "Yts "
: "")
<< (m.settings_.adjustments_ &
MarkovFunctional::ModelSettings::ExtrapolatePayoffFlat
? "FlatPayoffExt "
: "")
<< (m.settings_.adjustments_ &
MarkovFunctional::ModelSettings::NoPayoffExtrapolation
? "NoPayoffExt "
: "")
<< (m.settings_.adjustments_ &
MarkovFunctional::ModelSettings::KahaleSmile
? "Kahale "
: "")
<< (m.settings_.adjustments_ & MarkovFunctional::ModelSettings::
SmileExponentialExtrapolation
? "SmileExp "
: "")
<< (m.settings_.adjustments_ &
MarkovFunctional::ModelSettings::KahaleInterpolation
? "KahaleInt "
: "")
<< (m.settings_.adjustments_ & MarkovFunctional::ModelSettings::
SmileDeleteArbitragePoints
? "SmileDelArb "
: "") << (m.settings_.adjustments_ &
MarkovFunctional::ModelSettings::SabrSmile
? "Sabr"
: "") << std::endl;
out << "Smile moneyness checkpoints: ";
for (Size i = 0; i < m.settings_.smileMoneynessCheckpoints_.size(); i++)
out << m.settings_.smileMoneynessCheckpoints_[i]
<< (i < m.settings_.smileMoneynessCheckpoints_.size() - 1 ? ";"
: "");
out << std::endl;
QL_REQUIRE(!m.dirty_, "model outputs are dirty");
if (m.expiries_.size() == 0)
return out; // no trace information was collected so no output
out << std::endl;
out << "Messages:" << std::endl;
for (std::vector<std::string>::const_iterator i = m.messages_.begin();
i != m.messages_.end(); i++)
out << (*i) << std::endl;
out << std::endl << std::setprecision(16);
out << "Yield termstructure fit:" << std::endl;
out << "expiry;tenor;atm;annuity;digitalAdj;ytsAdj;marketzerorate;"
"modelzerorate;diff(bp)" << std::endl;
for (Size i = 0; i < m.expiries_.size(); i++) {
out << m.expiries_[i] << ";" << m.tenors_[i] << ";" << m.atm_[i]
<< ";" << m.annuity_[i] << ";"
<< m.digitalsAdjustmentFactors_[i] << ";"
<< m.adjustmentFactors_[i] << ";" << m.marketZerorate_[i] << ";"
<< m.modelZerorate_[i] << ";"
<< (m.marketZerorate_[i] - m.modelZerorate_[i]) * 10000.0
<< std::endl;
}
out << std::endl;
out << "Volatility smile fit:" << std::endl;
for (Size i = 0; i < m.expiries_.size(); i++) {
std::ostringstream os;
os << m.expiries_[i] << "/" << m.tenors_[i];
std::string p = os.str();
out << "strike(" << p << ");marketCallRaw(" << p << ";marketCall("
<< p << ");modelCall(" << p << ");marketPutRaw(" << p
<< ");marketPut(" << p << ");modelPut(" << p << ");marketVega("
<< p << ")" << (i < m.expiries_.size() - 1 ? ";" : "");
}
out << std::endl;
for (Size j = 0; j < m.smileStrikes_[0].size(); j++) {
for (Size i = 0; i < m.expiries_.size(); i++) {
out << m.smileStrikes_[i][j] << ";"
<< m.marketRawCallPremium_[i][j] << ";"
<< m.marketCallPremium_[i][j] << ";"
<< m.modelCallPremium_[i][j] << ";"
<< m.marketRawPutPremium_[i][j] << ";"
<< m.marketPutPremium_[i][j] << ";"
<< m.modelPutPremium_[i][j] << ";" << m.marketVega_[i][j]
<< (i < m.expiries_.size() - 1 ? ";" : "");
}
out << std::endl;
}
return out;
}
const Real MarkovFunctional::forwardRateInternal(
const Date &fixing, const Date &referenceDate, const Real y,
const bool zeroFixingDays, boost::shared_ptr<IborIndex> iborIdx) const {
calculate();
if (!iborIdx)
iborIdx = iborIndex_;
Date valueDate = zeroFixingDays ? fixing : iborIdx->valueDate(fixing);
Date endDate = iborIdx->fixingCalendar().advance(
iborIdx->valueDate(fixing), 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) -
zerobond(endDate, referenceDate, y)) /
(dcf * zerobond(endDate, referenceDate, y));
}
const Real
MarkovFunctional::swapRateInternal(const Date &fixing, const Period &tenor,
const Date &referenceDate, const Real y,
bool zeroFixingDays,
boost::shared_ptr<SwapIndex> swapIdx) const {
calculate();
if (!swapIdx)
swapIdx = swapIndexBase_;
QL_REQUIRE(swapIdx, "No swap index given");
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 = swapAnnuityInternal(fixing, tenor, referenceDate, y,
zeroFixingDays, swapIdx);
Rate atm =
(zerobond(zeroFixingDays ? fixing : sched.dates().front(),
referenceDate, y) -
zerobond(sched.calendar().adjust(sched.dates().back(),
underlying->paymentConvention()),
referenceDate, y)) /
annuity;
return atm;
}
const Real MarkovFunctional::swapAnnuityInternal(
const Date &fixing, const Period &tenor, const Date &referenceDate,
const Real y, const bool zeroFixingDays,
boost::shared_ptr<SwapIndex> swapIdx) const {
calculate();
if (!swapIdx)
swapIdx = swapIndexBase_;
QL_REQUIRE(swapIdx, "No swap index given");
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) *
swapIdx->dayCounter().yearFraction(
j == 1 && zeroFixingDays ? fixing : sched.date(j - 1),
sched.date(j));
}
return annuity;
}
const Real MarkovFunctional::swaptionPriceInternal(
const Option::Type &type, const Date &expiry, const Period &tenor,
const Rate strike, const Date &referenceDate, const Real y,
const bool zeroFixingDays, boost::shared_ptr<SwapIndex> swapIdx) const {
calculate();
Time fixingTime = termStructure()->timeFromReference(expiry);
Time referenceTime =
referenceDate == Null<Date>()
? 0.0
: termStructure()->timeFromReference(referenceDate);
Array yg = yGrid(modelSettings_.yStdDevs_, modelSettings_.yGridPoints_,
fixingTime, referenceTime, y);
Array z = yGrid(modelSettings_.yStdDevs_, modelSettings_.yGridPoints_);
Array p(yg.size());
for (Size i = 0; i < yg.size(); i++) {
Real annuity = swapAnnuityInternal(expiry, tenor, expiry, yg[i],
zeroFixingDays, swapIdx);
Rate atm = swapRateInternal(expiry, tenor, expiry, yg[i], zeroFixingDays,
swapIdx);
p[i] = annuity * std::max((type == Option::Call ? 1.0 : -1.0) *
(atm - strike),
0.0) /
numeraire(fixingTime, yg[i]);
}
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 ((modelSettings_.adjustments_ &
ModelSettings::NoPayoffExtrapolation) == 0) {
if ((modelSettings_.adjustments_ &
ModelSettings::ExtrapolatePayoffFlat) != 0) {
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) * price;
}
const Real MarkovFunctional::capletPriceInternal(
const Option::Type &type, const Date &expiry, const Rate strike,
const Date &referenceDate, const Real y, const bool zeroFixingDays,
boost::shared_ptr<IborIndex> iborIdx) const {
calculate();
if (!iborIdx)
iborIdx = iborIndex_;
Time fixingTime = termStructure()->timeFromReference(expiry);
Time referenceTime =
referenceDate == Null<Date>()
? 0.0
: termStructure()->timeFromReference(referenceDate);
Array yg = yGrid(modelSettings_.yStdDevs_, modelSettings_.yGridPoints_,
fixingTime, referenceTime, y);
Array z = yGrid(modelSettings_.yStdDevs_, modelSettings_.yGridPoints_);
Array p(yg.size());
Date valueDate = iborIdx->valueDate(expiry);
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(
zeroFixingDays ? expiry : valueDate, endDate);
for (Size i = 0; i < yg.size(); i++) {
Real annuity = zerobond(endDate, expiry, yg[i]) * dcf;
Rate atm =
forwardRateInternal(expiry, expiry, yg[i], zeroFixingDays, iborIdx);
p[i] = annuity * std::max((type == Option::Call ? 1.0 : -1.0) *
(atm - strike),
0.0) /
numeraire(fixingTime, yg[i]);
}
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 ((modelSettings_.adjustments_ &
ModelSettings::NoPayoffExtrapolation) == 0) {
if ((modelSettings_.adjustments_ &
ModelSettings::ExtrapolatePayoffFlat) != 0) {
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) * price;
}
}
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