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/* -*- mode: c++; tab-width: 4; indent-tabs-mode: nil; c-basic-offset: 4 -*- */
/*
Copyright (C) 2014, 2015, 2018 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
<https://www.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 lognormalcmsspreadpricer.cpp
*/
#include <ql/experimental/coupons/cmsspreadcoupon.hpp>
#include <ql/experimental/coupons/lognormalcmsspreadpricer.hpp>
#include <ql/math/integrals/kronrodintegral.hpp>
#include <ql/pricingengines/blackformula.hpp>
#include <ql/termstructures/volatility/swaption/swaptionvolcube.hpp>
#include <ql/optional.hpp>
#include <utility>
using std::sqrt;
namespace QuantLib {
class LognormalCmsSpreadPricer::integrand_f {
const LognormalCmsSpreadPricer* pricer;
public:
explicit integrand_f(const LognormalCmsSpreadPricer* pricer)
: pricer(pricer) {}
Real operator()(Real x) const {
return pricer->integrand(x);
}
};
LognormalCmsSpreadPricer::LognormalCmsSpreadPricer(
const ext::shared_ptr<CmsCouponPricer>& cmsPricer,
const Handle<Quote>& correlation,
Handle<YieldTermStructure> couponDiscountCurve,
const Size integrationPoints,
const ext::optional<VolatilityType>& volatilityType,
const Real shift1,
const Real shift2)
: CmsSpreadCouponPricer(correlation), cmsPricer_(cmsPricer),
couponDiscountCurve_(std::move(couponDiscountCurve)) {
registerWith(correlation);
if (!couponDiscountCurve_.empty())
registerWith(couponDiscountCurve_);
registerWith(cmsPricer_);
QL_REQUIRE(integrationPoints >= 4,
"at least 4 integration points should be used ("
<< integrationPoints << ")");
integrator_ =
ext::make_shared<GaussHermiteIntegration>(integrationPoints);
cnd_ = ext::make_shared<CumulativeNormalDistribution>(0.0, 1.0);
if (!volatilityType) {
QL_REQUIRE(shift1 == Null<Real>() && shift2 == Null<Real>(),
"if volatility type is inherited, no shifts should be "
"specified");
inheritedVolatilityType_ = true;
volType_ = cmsPricer->swaptionVolatility()->volatilityType();
} else {
shift1_ = shift1 == Null<Real>() ? 0.0 : shift1;
shift2_ = shift2 == Null<Real>() ? 0.0 : shift2;
inheritedVolatilityType_ = false;
volType_ = *volatilityType;
}
}
Real LognormalCmsSpreadPricer::integrand(const Real x) const {
// this is Brigo, 13.16.2 with x = v / sqrt(2)
Real v = M_SQRT2 * x;
Real h =
k_ - b_ * s2_ * std::exp((m2_ - 0.5 * v2_ * v2_) * fixingTime_ +
v2_ * std::sqrt(fixingTime_) * v);
Real phi1, phi2;
phi1 = (*cnd_)(
phi_ * (std::log(a_ * s1_ / h) +
(m1_ + (0.5 - rho_ * rho_) * v1_ * v1_) * fixingTime_ +
rho_ * v1_ * std::sqrt(fixingTime_) * v) /
(v1_ * std::sqrt(fixingTime_ * (1.0 - rho_ * rho_))));
phi2 = (*cnd_)(
phi_ * (std::log(a_ * s1_ / h) +
(m1_ - 0.5 * v1_ * v1_) * fixingTime_ +
rho_ * v1_ * std::sqrt(fixingTime_) * v) /
(v1_ * std::sqrt(fixingTime_ * (1.0 - rho_ * rho_))));
Real f = a_ * phi_ * s1_ *
std::exp(m1_ * fixingTime_ -
0.5 * rho_ * rho_ * v1_ * v1_ * fixingTime_ +
rho_ * v1_ * std::sqrt(fixingTime_) * v) *
phi1 -
phi_ * h * phi2;
return std::exp(-x * x) * f;
}
Real LognormalCmsSpreadPricer::integrand_normal(const Real x) const {
// this is http://ssrn.com/abstract=2686998, 3.20 with x = s / sqrt(2)
Real s = M_SQRT2 * x;
Real beta =
phi_ *
(gearing1_ * adjustedRate1_ + gearing2_ * adjustedRate2_ - k_ +
std::sqrt(fixingTime_) *
(rho_ * gearing1_ * vol1_ + gearing2_ * vol2_) * s);
Real f =
close_enough(alpha_, 0.0)
? Real(std::max(beta, 0.0))
: psi_ * alpha_ / (M_SQRTPI * M_SQRT2) *
std::exp(-beta * beta / (2.0 * alpha_ * alpha_)) +
beta * (1.0 - (*cnd_)(-psi_ * beta / alpha_));
return std::exp(-x * x) * f;
}
void
LognormalCmsSpreadPricer::initialize(const FloatingRateCoupon &coupon) {
coupon_ = dynamic_cast<const CmsSpreadCoupon *>(&coupon);
QL_REQUIRE(coupon_, "CMS spread coupon needed");
index_ = coupon_->swapSpreadIndex();
gearing_ = coupon_->gearing();
spread_ = coupon_->spread();
fixingDate_ = coupon_->fixingDate();
paymentDate_ = coupon_->date();
// if no coupon discount curve is given just use the discounting curve
// from the _first_ swap index.
// for rate calculation this curve cancels out in the computation, so
// e.g. the discounting
// swap engine will produce correct results, even if the
// couponDiscountCurve is not set here.
// only the price member function in this class will be dependent on the
// coupon discount curve.
today_ = QuantLib::Settings::instance().evaluationDate();
if (couponDiscountCurve_.empty())
couponDiscountCurve_ =
index_->swapIndex1()->exogenousDiscount()
? index_->swapIndex1()->discountingTermStructure()
: index_->swapIndex1()->forwardingTermStructure();
discount_ = paymentDate_ > couponDiscountCurve_->referenceDate()
? couponDiscountCurve_->discount(paymentDate_)
: 1.0;
spreadLegValue_ = spread_ * coupon_->accrualPeriod() * discount_;
gearing1_ = index_->gearing1();
gearing2_ = index_->gearing2();
QL_REQUIRE(gearing1_ > 0.0 && gearing2_ < 0.0,
"gearing1 (" << gearing1_
<< ") should be positive while gearing2 ("
<< gearing2_ << ") should be negative");
c1_ = ext::make_shared<CmsCoupon>(
coupon_->date(), coupon_->nominal(), coupon_->accrualStartDate(),
coupon_->accrualEndDate(), coupon_->fixingDays(),
index_->swapIndex1(), 1.0, 0.0, coupon_->referencePeriodStart(),
coupon_->referencePeriodEnd(), coupon_->dayCounter(),
coupon_->isInArrears());
c2_ = ext::make_shared<CmsCoupon>(
coupon_->date(), coupon_->nominal(), coupon_->accrualStartDate(),
coupon_->accrualEndDate(), coupon_->fixingDays(),
index_->swapIndex2(), 1.0, 0.0, coupon_->referencePeriodStart(),
coupon_->referencePeriodEnd(), coupon_->dayCounter(),
coupon_->isInArrears());
c1_->setPricer(cmsPricer_);
c2_->setPricer(cmsPricer_);
if (fixingDate_ > today_) {
fixingTime_ = cmsPricer_->swaptionVolatility()->timeFromReference(
fixingDate_);
swapRate1_ = c1_->indexFixing();
swapRate2_ = c2_->indexFixing();
adjustedRate1_ = c1_->adjustedFixing();
adjustedRate2_ = c2_->adjustedFixing();
ext::shared_ptr<SwaptionVolatilityStructure> swvol =
*cmsPricer_->swaptionVolatility();
ext::shared_ptr<SwaptionVolatilityCube> swcub =
ext::dynamic_pointer_cast<SwaptionVolatilityCube>(swvol);
if(inheritedVolatilityType_ && volType_ == ShiftedLognormal) {
shift1_ =
swvol->shift(fixingDate_, index_->swapIndex1()->tenor());
shift2_ =
swvol->shift(fixingDate_, index_->swapIndex2()->tenor());
}
if (swcub == nullptr) {
// not a cube, just an atm surface given, so we can
// not easily convert volatilities and just forbid it
QL_REQUIRE(inheritedVolatilityType_,
"if only an atm surface is given, the volatility "
"type must be inherited");
vol1_ = swvol->volatility(
fixingDate_, index_->swapIndex1()->tenor(), swapRate1_);
vol2_ = swvol->volatility(
fixingDate_, index_->swapIndex2()->tenor(), swapRate2_);
} else {
vol1_ = swcub->smileSection(fixingDate_,
index_->swapIndex1()->tenor())
->volatility(swapRate1_, volType_, shift1_);
vol2_ = swcub->smileSection(fixingDate_,
index_->swapIndex2()->tenor())
->volatility(swapRate2_, volType_, shift2_);
}
if(volType_ == ShiftedLognormal) {
mu1_ = 1.0 / fixingTime_ * std::log((adjustedRate1_ + shift1_) /
(swapRate1_ + shift1_));
mu2_ = 1.0 / fixingTime_ * std::log((adjustedRate2_ + shift2_) /
(swapRate2_ + shift2_));
}
// for the normal volatility case we do not need the drifts
// but rather use adjusted rates directly in the integrand
rho_ = std::max(std::min(correlation()->value(), 0.9999),
-0.9999); // avoid division by zero in integrand
} else {
// fixing is in the past or today
adjustedRate1_ = c1_->indexFixing();
adjustedRate2_ = c2_->indexFixing();
}
}
Real LognormalCmsSpreadPricer::optionletPrice(Option::Type optionType,
Real strike) const {
// this method is only called for future fixings
optionType_ = optionType;
phi_ = optionType == Option::Call ? 1.0 : -1.0;
Real res = 0.0;
if (volType_ == ShiftedLognormal) {
// (shifted) lognormal volatility
if (strike >= 0.0) {
a_ = gearing1_;
b_ = gearing2_;
s1_ = swapRate1_ + shift1_;
s2_ = swapRate2_ + shift2_;
m1_ = mu1_;
m2_ = mu2_;
v1_ = vol1_;
v2_ = vol2_;
k_ = strike + gearing1_ * shift1_ + gearing2_ * shift2_;
} else {
a_ = -gearing2_;
b_ = -gearing1_;
s1_ = swapRate2_ + shift1_;
s2_ = swapRate1_ + shift2_;
m1_ = mu2_;
m2_ = mu1_;
v1_ = vol2_;
v2_ = vol1_;
k_ = -strike - gearing1_ * shift1_ - gearing2_ * shift2_;
res += phi_ * (gearing1_ * adjustedRate1_ +
gearing2_ * adjustedRate2_ - strike);
}
res +=
1.0 / M_SQRTPI * (*integrator_)(integrand_f(this));
} else {
// normal volatility
Real forward = gearing1_ * adjustedRate1_ +
gearing2_ * adjustedRate2_;
Real stddev =
std::sqrt(fixingTime_ *
(gearing1_ * gearing1_ * vol1_ * vol1_ +
gearing2_ * gearing2_ * vol2_ * vol2_ +
2.0 * gearing1_ * gearing2_ * rho_ * vol1_ * vol2_));
res =
bachelierBlackFormula(optionType_, strike, forward, stddev, 1.0);
}
return res * discount_ * coupon_->accrualPeriod();
}
Rate LognormalCmsSpreadPricer::swapletRate() const {
return swapletPrice() / (coupon_->accrualPeriod() * discount_);
}
Real LognormalCmsSpreadPricer::capletPrice(Rate effectiveCap) const {
// caplet is equivalent to call option on fixing
if (fixingDate_ <= today_) {
// the fixing is determined
const Rate Rs = std::max(
coupon_->index()->fixing(fixingDate_) - effectiveCap, 0.);
Rate price = gearing_ * Rs * coupon_->accrualPeriod() * discount_;
return price;
} else {
Real capletPrice = optionletPrice(Option::Call, effectiveCap);
return gearing_ * capletPrice;
}
}
Rate LognormalCmsSpreadPricer::capletRate(Rate effectiveCap) const {
return capletPrice(effectiveCap) /
(coupon_->accrualPeriod() * discount_);
}
Real LognormalCmsSpreadPricer::floorletPrice(Rate effectiveFloor) const {
// floorlet is equivalent to put option on fixing
if (fixingDate_ <= today_) {
// the fixing is determined
const Rate Rs = std::max(
effectiveFloor - coupon_->index()->fixing(fixingDate_), 0.);
Rate price = gearing_ * Rs * coupon_->accrualPeriod() * discount_;
return price;
} else {
Real floorletPrice = optionletPrice(Option::Put, effectiveFloor);
return gearing_ * floorletPrice;
}
}
Rate LognormalCmsSpreadPricer::floorletRate(Rate effectiveFloor) const {
return floorletPrice(effectiveFloor) /
(coupon_->accrualPeriod() * discount_);
}
Real LognormalCmsSpreadPricer::swapletPrice() const {
return gearing_ * coupon_->accrualPeriod() * discount_ *
(gearing1_ * adjustedRate1_ + gearing2_ * adjustedRate2_) +
spreadLegValue_;
}
}
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