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|
/* -*- mode: c++; tab-width: 4; indent-tabs-mode: nil; c-basic-offset: 4 -*- */
/*
Copyright (C) 2007 Cristina Duminuco
Copyright (C) 2007 Giorgio Facchinetti
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.
*/
#include "toplevelfixture.hpp"
#include "utilities.hpp"
#include <ql/indexes/ibor/euribor.hpp>
#include <ql/utilities/dataformatters.hpp>
#include <ql/cashflows/digitalcoupon.hpp>
#include <ql/cashflows/capflooredcoupon.hpp>
#include <ql/settings.hpp>
#include <ql/quotes/simplequote.hpp>
#include <ql/termstructures/volatility/optionlet/constantoptionletvol.hpp>
#include <ql/pricingengines/blackformula.hpp>
#include <ql/time/daycounters/actual360.hpp>
#include <ql/pricingengines/vanilla/analyticeuropeanengine.hpp>
#include <ql/processes/blackscholesprocess.hpp>
#include <ql/math/distributions/normaldistribution.hpp>
using namespace QuantLib;
using namespace boost::unit_test_framework;
BOOST_FIXTURE_TEST_SUITE(QuantLibTests, TopLevelFixture)
BOOST_AUTO_TEST_SUITE(DigitalCouponTests)
struct CommonVars {
// global data
Date today, settlement;
Real nominal;
Calendar calendar;
ext::shared_ptr<IborIndex> index;
Natural fixingDays;
RelinkableHandle<YieldTermStructure> termStructure;
Real optionTolerance;
Real blackTolerance;
// setup
CommonVars() {
fixingDays = 2;
nominal = 1000000.0;
index = ext::shared_ptr<IborIndex>(new Euribor6M(termStructure));
calendar = index->fixingCalendar();
today = calendar.adjust(Settings::instance().evaluationDate());
Settings::instance().evaluationDate() = today;
settlement = calendar.advance(today,fixingDays,Days);
termStructure.linkTo(flatRate(settlement,0.05,Actual365Fixed()));
optionTolerance = 1.e-04;
blackTolerance = 1e-10;
}
};
BOOST_AUTO_TEST_CASE(testAssetOrNothing) {
BOOST_TEST_MESSAGE("Testing European asset-or-nothing digital coupon...");
/* Call Payoff = (aL+b)Heaviside(aL+b-X) = a Max[L-X'] + (b+aX')Heaviside(L-X')
Value Call = aF N(d1') + bN(d2')
Put Payoff = (aL+b)Heaviside(X-aL-b) = -a Max[X-L'] + (b+aX')Heaviside(X'-L)
Value Put = aF N(-d1') + bN(-d2')
where:
d1' = ln(F/X')/stdDev + 0.5*stdDev;
*/
CommonVars vars;
Volatility vols[] = { 0.05, 0.15, 0.30 };
Rate strikes[] = { 0.01, 0.02, 0.03, 0.04, 0.05, 0.06, 0.07 };
Real gearings[] = { 1.0, 2.8 };
Rate spreads[] = { 0.0, 0.005 };
Real gap = 1e-7; /* low, in order to compare digital option value
with black formula result */
ext::shared_ptr<DigitalReplication>
replication(new DigitalReplication(Replication::Central, gap));
for (Real capletVol : vols) {
RelinkableHandle<OptionletVolatilityStructure> vol;
vol.linkTo(ext::shared_ptr<OptionletVolatilityStructure>(new ConstantOptionletVolatility(
vars.today, vars.calendar, Following, capletVol, Actual360())));
for (Real strike : strikes) {
for (Size k = 9; k < 10; k++) {
Date startDate = vars.calendar.advance(vars.settlement,(k+1)*Years);
Date endDate = vars.calendar.advance(vars.settlement,(k+2)*Years);
Rate nullstrike = Null<Rate>();
Date paymentDate = endDate;
for (Size h=0; h<std::size(gearings); h++) {
Real gearing = gearings[h];
Rate spread = spreads[h];
ext::shared_ptr<FloatingRateCoupon> underlying(new
IborCoupon(paymentDate, vars.nominal,
startDate, endDate,
vars.fixingDays, vars.index,
gearing, spread));
// Floating Rate Coupon - Call Digital option
DigitalCoupon digitalCappedCoupon(underlying,
strike, Position::Short, false, nullstrike,
nullstrike, Position::Short, false, nullstrike,
replication);
ext::shared_ptr<IborCouponPricer> pricer(new
BlackIborCouponPricer(vol));
digitalCappedCoupon.setPricer(pricer);
// Check digital option price vs N(d1) price
Time accrualPeriod = underlying->accrualPeriod();
Real discount = vars.termStructure->discount(endDate);
Date exerciseDate = underlying->fixingDate();
Rate forward = underlying->rate();
Rate effFwd = (forward-spread)/gearing;
Rate effStrike = (strike-spread)/gearing;
Real stdDev = std::sqrt(vol->blackVariance(exerciseDate, effStrike));
CumulativeNormalDistribution phi;
Real d1 = std::log(effFwd/effStrike)/stdDev + 0.5*stdDev;
Real d2 = d1 - stdDev;
Real N_d1 = phi(d1);
Real N_d2 = phi(d2);
Real nd1Price = (gearing * effFwd * N_d1 + spread * N_d2)
* vars.nominal * accrualPeriod * discount;
Real optionPrice = digitalCappedCoupon.callOptionRate() *
vars.nominal * accrualPeriod * discount;
Real error = std::abs(nd1Price - optionPrice);
if (error>vars.optionTolerance)
BOOST_ERROR("\nDigital Call Option:" <<
"\nVolatility = " << io::rate(capletVol) <<
"\nStrike = " << io::rate(strike) <<
"\nExercise = " << k+1 << " years" <<
"\nOption price by replication = " << optionPrice <<
"\nOption price by Cox-Rubinstein formula = " << nd1Price <<
"\nError " << error);
// Check digital option price vs N(d1) price using Vanilla Option class
if (spread==0.0) {
ext::shared_ptr<Exercise>
exercise(new EuropeanExercise(exerciseDate));
Real discountAtFixing = vars.termStructure->discount(exerciseDate);
ext::shared_ptr<SimpleQuote>
fwd(new SimpleQuote(effFwd*discountAtFixing));
ext::shared_ptr<SimpleQuote> qRate(new SimpleQuote(0.0));
ext::shared_ptr<YieldTermStructure>
qTS = flatRate(vars.today, qRate, Actual360());
ext::shared_ptr<SimpleQuote> vol(new SimpleQuote(0.0));
ext::shared_ptr<BlackVolTermStructure>
volTS = flatVol(vars.today, capletVol, Actual360());
ext::shared_ptr<StrikedTypePayoff>
callPayoff(new AssetOrNothingPayoff(Option::Call,effStrike));
ext::shared_ptr<BlackScholesMertonProcess> stochProcess(new
BlackScholesMertonProcess(Handle<Quote>(fwd),
Handle<YieldTermStructure>(qTS),
Handle<YieldTermStructure>(vars.termStructure),
Handle<BlackVolTermStructure>(volTS)));
ext::shared_ptr<PricingEngine>
engine(new AnalyticEuropeanEngine(stochProcess));
VanillaOption callOpt(callPayoff, exercise);
callOpt.setPricingEngine(engine);
Real callVO = vars.nominal * gearing
* accrualPeriod * callOpt.NPV()
* discount / discountAtFixing
* forward / effFwd;
error = std::abs(nd1Price - callVO);
if (error>vars.blackTolerance)
BOOST_ERROR("\nDigital Call Option:" <<
"\nVolatility = " << io::rate(capletVol) <<
"\nStrike = " << io::rate(strike) <<
"\nExercise = " << k+1 << " years" <<
"\nOption price by Black asset-ot-nothing payoff = " << callVO <<
"\nOption price by Cox-Rubinstein = " << nd1Price <<
"\nError " << error );
}
// Floating Rate Coupon + Put Digital option
DigitalCoupon digitalFlooredCoupon(underlying,
nullstrike, Position::Long, false, nullstrike,
strike, Position::Long, false, nullstrike,
replication);
digitalFlooredCoupon.setPricer(pricer);
// Check digital option price vs N(d1) price
N_d1 = phi(-d1);
N_d2 = phi(-d2);
nd1Price = (gearing * effFwd * N_d1 + spread * N_d2)
* vars.nominal * accrualPeriod * discount;
optionPrice = digitalFlooredCoupon.putOptionRate() *
vars.nominal * accrualPeriod * discount;
error = std::abs(nd1Price - optionPrice);
if (error>vars.optionTolerance)
BOOST_ERROR("\nDigital Put Option:" <<
"\nVolatility = " << io::rate(capletVol) <<
"\nStrike = " << io::rate(strike) <<
"\nExercise = " << k+1 << " years" <<
"\nOption price by replication = " << optionPrice <<
"\nOption price by Cox-Rubinstein = " << nd1Price <<
"\nError " << error );
// Check digital option price vs N(d1) price using Vanilla Option class
if (spread==0.0) {
ext::shared_ptr<Exercise>
exercise(new EuropeanExercise(exerciseDate));
Real discountAtFixing = vars.termStructure->discount(exerciseDate);
ext::shared_ptr<SimpleQuote>
fwd(new SimpleQuote(effFwd*discountAtFixing));
ext::shared_ptr<SimpleQuote> qRate(new SimpleQuote(0.0));
ext::shared_ptr<YieldTermStructure>
qTS = flatRate(vars.today, qRate, Actual360());
ext::shared_ptr<SimpleQuote> vol(new SimpleQuote(0.0));
ext::shared_ptr<BlackVolTermStructure>
volTS = flatVol(vars.today, capletVol, Actual360());
ext::shared_ptr<BlackScholesMertonProcess> stochProcess(new
BlackScholesMertonProcess(Handle<Quote>(fwd),
Handle<YieldTermStructure>(qTS),
Handle<YieldTermStructure>(vars.termStructure),
Handle<BlackVolTermStructure>(volTS)));
ext::shared_ptr<StrikedTypePayoff>
putPayoff(new AssetOrNothingPayoff(Option::Put, effStrike));
ext::shared_ptr<PricingEngine> engine(new AnalyticEuropeanEngine(stochProcess));
VanillaOption putOpt(putPayoff, exercise);
putOpt.setPricingEngine(engine);
Real putVO = vars.nominal * gearing
* accrualPeriod * putOpt.NPV()
* discount / discountAtFixing
* forward / effFwd;
error = std::abs(nd1Price - putVO);
if (error>vars.blackTolerance)
BOOST_ERROR("\nDigital Put Option:" <<
"\nVolatility = " << io::rate(capletVol) <<
"\nStrike = " << io::rate(strike) <<
"\nExercise = " << k+1 << " years" <<
"\nOption price by Black asset-ot-nothing payoff = " << putVO <<
"\nOption price by Cox-Rubinstein = " << nd1Price <<
"\nError " << error );
}
}
}
}
}
}
BOOST_AUTO_TEST_CASE(testAssetOrNothingDeepInTheMoney) {
BOOST_TEST_MESSAGE("Testing European deep in-the-money asset-or-nothing "
"digital coupon...");
CommonVars vars;
Real gearing = 1.0;
Real spread = 0.0;
Volatility capletVolatility = 0.0001;
RelinkableHandle<OptionletVolatilityStructure> volatility;
volatility.linkTo(ext::shared_ptr<OptionletVolatilityStructure>(new
ConstantOptionletVolatility(vars.today, vars.calendar, Following,
capletVolatility, Actual360())));
for (Size k = 0; k<10; k++) { // Loop on start and end dates
Date startDate = vars.calendar.advance(vars.settlement,(k+1)*Years);
Date endDate = vars.calendar.advance(vars.settlement,(k+2)*Years);
Rate nullstrike = Null<Rate>();
Date paymentDate = endDate;
ext::shared_ptr<FloatingRateCoupon> underlying(new
IborCoupon(paymentDate, vars.nominal,
startDate, endDate,
vars.fixingDays, vars.index,
gearing, spread));
// Floating Rate Coupon - Deep-in-the-money Call Digital option
Rate strike = 0.001;
DigitalCoupon digitalCappedCoupon(underlying,
strike, Position::Short, false, nullstrike,
nullstrike, Position::Short, false, nullstrike);
ext::shared_ptr<IborCouponPricer> pricer(new
BlackIborCouponPricer(volatility));
digitalCappedCoupon.setPricer(pricer);
// Check price vs its target price
Time accrualPeriod = underlying->accrualPeriod();
Real discount = vars.termStructure->discount(endDate);
Real targetOptionPrice = underlying->price(vars.termStructure);
Real targetPrice = 0.0;
Real digitalPrice = digitalCappedCoupon.price(vars.termStructure);
Real error = std::fabs(targetPrice - digitalPrice);
Real tolerance = 1e-08;
if (error>tolerance)
BOOST_ERROR("\nFloating Coupon - Digital Call Option:" <<
"\nVolatility = " << io::rate(capletVolatility) <<
"\nStrike = " << io::rate(strike) <<
"\nExercise = " << k+1 << " years" <<
"\nCoupon Price = " << digitalPrice <<
"\nTarget price = " << targetPrice <<
"\nError = " << error );
// Check digital option price
Real replicationOptionPrice = digitalCappedCoupon.callOptionRate() *
vars.nominal * accrualPeriod * discount;
error = std::abs(targetOptionPrice - replicationOptionPrice);
Real optionTolerance = 1e-08;
if (error>optionTolerance)
BOOST_ERROR("\nDigital Call Option:" <<
"\nVolatility = " << io::rate(capletVolatility) <<
"\nStrike = " << io::rate(strike) <<
"\nExercise = " << k+1 << " years" <<
"\nPrice by replication = " << replicationOptionPrice <<
"\nTarget price = " << targetOptionPrice <<
"\nError = " << error);
// Floating Rate Coupon + Deep-in-the-money Put Digital option
strike = 0.99;
DigitalCoupon digitalFlooredCoupon(underlying,
nullstrike, Position::Long, false, nullstrike,
strike, Position::Long, false, nullstrike);
digitalFlooredCoupon.setPricer(pricer);
// Check price vs its target price
targetOptionPrice = underlying->price(vars.termStructure);
targetPrice = underlying->price(vars.termStructure) + targetOptionPrice ;
digitalPrice = digitalFlooredCoupon.price(vars.termStructure);
error = std::fabs(targetPrice - digitalPrice);
tolerance = 2.5e-06;
if (error>tolerance)
BOOST_ERROR("\nFloating Coupon + Digital Put Option:" <<
"\nVolatility = " << io::rate(capletVolatility) <<
"\nStrike = " << io::rate(strike) <<
"\nExercise = " << k+1 << " years" <<
"\nDigital coupon price = " << digitalPrice <<
"\nTarget price = " << targetPrice <<
"\nError " << error);
// Check digital option
replicationOptionPrice = digitalFlooredCoupon.putOptionRate() *
vars.nominal * accrualPeriod * discount;
error = std::abs(targetOptionPrice - replicationOptionPrice);
optionTolerance = 2.5e-06;
if (error>optionTolerance)
BOOST_ERROR("\nDigital Put Option:" <<
"\nVolatility = " << io::rate(capletVolatility) <<
"\nStrike = " << io::rate(strike) <<
"\nExercise = " << k+1 << " years" <<
"\nPrice by replication = " << replicationOptionPrice <<
"\nTarget price = " << targetOptionPrice <<
"\nError " << error);
}
}
BOOST_AUTO_TEST_CASE(testAssetOrNothingDeepOutTheMoney) {
BOOST_TEST_MESSAGE("Testing European deep out-the-money asset-or-nothing "
"digital coupon...");
CommonVars vars;
Real gearing = 1.0;
Real spread = 0.0;
Volatility capletVolatility = 0.0001;
RelinkableHandle<OptionletVolatilityStructure> volatility;
volatility.linkTo(ext::shared_ptr<OptionletVolatilityStructure>(new
ConstantOptionletVolatility(vars.today, vars.calendar, Following,
capletVolatility, Actual360())));
for (Size k = 0; k<10; k++) { // loop on start and end dates
Date startDate = vars.calendar.advance(vars.settlement,(k+1)*Years);
Date endDate = vars.calendar.advance(vars.settlement,(k+2)*Years);
Rate nullstrike = Null<Rate>();
Date paymentDate = endDate;
ext::shared_ptr<FloatingRateCoupon> underlying(new
IborCoupon(paymentDate, vars.nominal,
startDate, endDate,
vars.fixingDays, vars.index,
gearing, spread));
// Floating Rate Coupon - Deep-out-of-the-money Call Digital option
Rate strike = 0.99;
DigitalCoupon digitalCappedCoupon(underlying,
strike, Position::Short, false, nullstrike,
nullstrike, Position::Long, false, nullstrike);
ext::shared_ptr<IborCouponPricer> pricer(new BlackIborCouponPricer(volatility));
digitalCappedCoupon.setPricer(pricer);
// Check price vs its target
Time accrualPeriod = underlying->accrualPeriod();
Real discount = vars.termStructure->discount(endDate);
Real targetPrice = underlying->price(vars.termStructure);
Real digitalPrice = digitalCappedCoupon.price(vars.termStructure);
Real error = std::fabs(targetPrice - digitalPrice);
Real tolerance = 1e-10;
if (error>tolerance)
BOOST_ERROR("\nFloating Coupon - Digital Call Option :" <<
"\nVolatility = " << io::rate(capletVolatility) <<
"\nStrike = " << io::rate(strike) <<
"\nExercise = " << k+1 << " years" <<
"\nCoupon price = " << digitalPrice <<
"\nTarget price = " << targetPrice <<
"\nError = " << error );
// Check digital option price
Real targetOptionPrice = 0.;
Real replicationOptionPrice = digitalCappedCoupon.callOptionRate() *
vars.nominal * accrualPeriod * discount;
error = std::abs(targetOptionPrice - replicationOptionPrice);
Real optionTolerance = 1e-08;
if (error>optionTolerance)
BOOST_ERROR("\nDigital Call Option:" <<
"\nVolatility = " << io::rate(capletVolatility) <<
"\nStrike = " << io::rate(strike) <<
"\nExercise = " << k+1 << " years" <<
"\nPrice by replication = " << replicationOptionPrice <<
"\nTarget price = " << targetOptionPrice <<
"\nError = " << error );
// Floating Rate Coupon - Deep-out-of-the-money Put Digital option
strike = 0.01;
DigitalCoupon digitalFlooredCoupon(underlying,
nullstrike, Position::Long, false, nullstrike,
strike, Position::Long, false, nullstrike);
digitalFlooredCoupon.setPricer(pricer);
// Check price vs its target
targetPrice = underlying->price(vars.termStructure);
digitalPrice = digitalFlooredCoupon.price(vars.termStructure);
tolerance = 1e-08;
error = std::fabs(targetPrice - digitalPrice);
if (error>tolerance)
BOOST_ERROR("\nFloating Coupon + Digital Put Coupon:" <<
"\nVolatility = " << io::rate(capletVolatility) <<
"\nStrike = " << io::rate(strike) <<
"\nExercise = " << k+1 << " years" <<
"\nCoupon price = " << digitalPrice <<
"\nTarget price = " << targetPrice <<
"\nError = " << error );
// Check digital option
targetOptionPrice = 0.0;
replicationOptionPrice = digitalFlooredCoupon.putOptionRate() *
vars.nominal * accrualPeriod * discount;
error = std::abs(targetOptionPrice - replicationOptionPrice);
if (error>optionTolerance)
BOOST_ERROR("\nDigital Put Coupon:" <<
"\nVolatility = " << io::rate(capletVolatility) <<
"\nStrike = " << io::rate(strike) <<
"\nExercise = " << k+1 << " years" <<
"\nPrice by replication = " << replicationOptionPrice <<
"\nTarget price = " << targetOptionPrice <<
"\nError = " << error );
}
}
BOOST_AUTO_TEST_CASE(testCashOrNothing) {
BOOST_TEST_MESSAGE("Testing European cash-or-nothing digital coupon...");
/* Call Payoff = R Heaviside(aL+b-X)
Value Call = R N(d2')
Put Payoff = R Heaviside(X-aL-b)
Value Put = R N(-d2')
where:
d2' = ln(F/X')/stdDev - 0.5*stdDev;
*/
CommonVars vars;
Volatility vols[] = { 0.05, 0.15, 0.30 };
Rate strikes[] = { 0.01, 0.02, 0.03, 0.04, 0.05, 0.06, 0.07 };
Real gearing = 3.0;
Real spread = -0.0002;
Real gap = 1e-08; /* very low, in order to compare digital option value
with black formula result */
ext::shared_ptr<DigitalReplication> replication(new
DigitalReplication(Replication::Central, gap));
for (Real capletVol : vols) {
RelinkableHandle<OptionletVolatilityStructure> vol;
vol.linkTo(ext::shared_ptr<OptionletVolatilityStructure>(new ConstantOptionletVolatility(
vars.today, vars.calendar, Following, capletVol, Actual360())));
for (Real strike : strikes) {
for (Size k = 0; k < 10; k++) {
Date startDate = vars.calendar.advance(vars.settlement,(k+1)*Years);
Date endDate = vars.calendar.advance(vars.settlement,(k+2)*Years);
Rate nullstrike = Null<Rate>();
Rate cashRate = 0.01;
Date paymentDate = endDate;
ext::shared_ptr<FloatingRateCoupon> underlying(new
IborCoupon(paymentDate, vars.nominal,
startDate, endDate,
vars.fixingDays, vars.index,
gearing, spread));
// Floating Rate Coupon - Call Digital option
DigitalCoupon digitalCappedCoupon(underlying,
strike, Position::Short, false, cashRate,
nullstrike, Position::Short, false, nullstrike,
replication);
ext::shared_ptr<IborCouponPricer> pricer(new BlackIborCouponPricer(vol));
digitalCappedCoupon.setPricer(pricer);
// Check digital option price vs N(d2) price
Date exerciseDate = underlying->fixingDate();
Rate forward = underlying->rate();
Rate effFwd = (forward-spread)/gearing;
Rate effStrike = (strike-spread)/gearing;
Time accrualPeriod = underlying->accrualPeriod();
Real discount = vars.termStructure->discount(endDate);
Real stdDev = std::sqrt(vol->blackVariance(exerciseDate, effStrike));
Real ITM = blackFormulaCashItmProbability(Option::Call, effStrike,
effFwd, stdDev);
Real nd2Price = ITM * vars.nominal * accrualPeriod * discount * cashRate;
Real optionPrice = digitalCappedCoupon.callOptionRate() *
vars.nominal * accrualPeriod * discount;
Real error = std::abs(nd2Price - optionPrice);
if (error>vars.optionTolerance)
BOOST_ERROR("\nDigital Call Option:" <<
"\nVolatility = " << io::rate(capletVol) <<
"\nStrike = " << io::rate(strike) <<
"\nExercise = " << k+1 << " years" <<
"\nPrice by replication = " << optionPrice <<
"\nPrice by Reiner-Rubinstein = " << nd2Price <<
"\nError = " << error );
// Check digital option price vs N(d2) price using Vanilla Option class
ext::shared_ptr<Exercise> exercise(new EuropeanExercise(exerciseDate));
Real discountAtFixing = vars.termStructure->discount(exerciseDate);
ext::shared_ptr<SimpleQuote> fwd(new SimpleQuote(effFwd*discountAtFixing));
ext::shared_ptr<SimpleQuote> qRate(new SimpleQuote(0.0));
ext::shared_ptr<YieldTermStructure> qTS = flatRate(vars.today, qRate, Actual360());
ext::shared_ptr<SimpleQuote> vol(new SimpleQuote(0.0));
ext::shared_ptr<BlackVolTermStructure> volTS = flatVol(vars.today, capletVol,
Actual360());
ext::shared_ptr<StrikedTypePayoff> callPayoff(new CashOrNothingPayoff(
Option::Call, effStrike, cashRate));
ext::shared_ptr<BlackScholesMertonProcess> stochProcess(new
BlackScholesMertonProcess(Handle<Quote>(fwd),
Handle<YieldTermStructure>(qTS),
Handle<YieldTermStructure>(vars.termStructure),
Handle<BlackVolTermStructure>(volTS)));
ext::shared_ptr<PricingEngine> engine(new AnalyticEuropeanEngine(stochProcess));
VanillaOption callOpt(callPayoff, exercise);
callOpt.setPricingEngine(engine);
Real callVO = vars.nominal * accrualPeriod * callOpt.NPV()
* discount / discountAtFixing;
error = std::abs(nd2Price - callVO);
if (error>vars.blackTolerance)
BOOST_ERROR("\nDigital Call Option:" <<
"\nVolatility = " << io::rate(capletVol) <<
"\nStrike = " << io::rate(strike) <<
"\nExercise = " << k+1 << " years" <<
"\nOption price by Black asset-ot-nothing payoff = " << callVO <<
"\nOption price by Reiner-Rubinstein = " << nd2Price <<
"\nError " << error );
// Floating Rate Coupon + Put Digital option
DigitalCoupon digitalFlooredCoupon(underlying,
nullstrike, Position::Long, false, nullstrike,
strike, Position::Long, false, cashRate,
replication);
digitalFlooredCoupon.setPricer(pricer);
// Check digital option price vs N(d2) price
ITM = blackFormulaCashItmProbability(Option::Put,
effStrike,
effFwd,
stdDev);
nd2Price = ITM * vars.nominal * accrualPeriod * discount * cashRate;
optionPrice = digitalFlooredCoupon.putOptionRate() *
vars.nominal * accrualPeriod * discount;
error = std::abs(nd2Price - optionPrice);
if (error>vars.optionTolerance)
BOOST_ERROR("\nPut Digital Option:" <<
"\nVolatility = " << io::rate(capletVol) <<
"\nStrike = " << io::rate(strike) <<
"\nExercise = " << k+1 << " years" <<
"\nPrice by replication = " << optionPrice <<
"\nPrice by Reiner-Rubinstein = " << nd2Price <<
"\nError = " << error );
// Check digital option price vs N(d2) price using Vanilla Option class
ext::shared_ptr<StrikedTypePayoff> putPayoff(new
CashOrNothingPayoff(Option::Put, effStrike, cashRate));
VanillaOption putOpt(putPayoff, exercise);
putOpt.setPricingEngine(engine);
Real putVO = vars.nominal * accrualPeriod * putOpt.NPV()
* discount / discountAtFixing;
error = std::abs(nd2Price - putVO);
if (error>vars.blackTolerance)
BOOST_ERROR("\nDigital Put Option:" <<
"\nVolatility = " << io::rate(capletVol) <<
"\nStrike = " << io::rate(strike) <<
"\nExercise = " << k+1 << " years" <<
"\nOption price by Black asset-ot-nothing payoff = " << putVO <<
"\nOption price by Reiner-Rubinstein = " << nd2Price <<
"\nError " << error );
}
}
}
}
BOOST_AUTO_TEST_CASE(testCashOrNothingDeepInTheMoney) {
BOOST_TEST_MESSAGE("Testing European deep in-the-money cash-or-nothing "
"digital coupon...");
CommonVars vars;
Real gearing = 1.0;
Real spread = 0.0;
Volatility capletVolatility = 0.0001;
RelinkableHandle<OptionletVolatilityStructure> volatility;
volatility.linkTo(ext::shared_ptr<OptionletVolatilityStructure>(new
ConstantOptionletVolatility(vars.today, vars.calendar, Following,
capletVolatility, Actual360())));
for (Size k = 0; k<10; k++) { // Loop on start and end dates
Date startDate = vars.calendar.advance(vars.settlement,(k+1)*Years);
Date endDate = vars.calendar.advance(vars.settlement,(k+2)*Years);
Rate nullstrike = Null<Rate>();
Rate cashRate = 0.01;
Date paymentDate = endDate;
ext::shared_ptr<FloatingRateCoupon> underlying(new
IborCoupon(paymentDate, vars.nominal,
startDate, endDate,
vars.fixingDays, vars.index,
gearing, spread));
// Floating Rate Coupon - Deep-in-the-money Call Digital option
Rate strike = 0.001;
DigitalCoupon digitalCappedCoupon(underlying,
strike, Position::Short, false, cashRate,
nullstrike, Position::Short, false, nullstrike);
ext::shared_ptr<IborCouponPricer> pricer(new
BlackIborCouponPricer(volatility));
digitalCappedCoupon.setPricer(pricer);
// Check price vs its target
Time accrualPeriod = underlying->accrualPeriod();
Real discount = vars.termStructure->discount(endDate);
Real targetOptionPrice = cashRate * vars.nominal * accrualPeriod * discount;
Real targetPrice = underlying->price(vars.termStructure) - targetOptionPrice;
Real digitalPrice = digitalCappedCoupon.price(vars.termStructure);
Real error = std::fabs(targetPrice - digitalPrice);
Real tolerance = 1e-07;
if (error>tolerance)
BOOST_ERROR("\nFloating Coupon - Digital Call Coupon:" <<
"\nVolatility = " << io::rate(capletVolatility) <<
"\nStrike = " << io::rate(strike) <<
"\nExercise = " << k+1 << " years" <<
"\nCoupon price = " << digitalPrice <<
"\nTarget price = " << targetPrice <<
"\nError " << error );
// Check digital option price
Real replicationOptionPrice = digitalCappedCoupon.callOptionRate() *
vars.nominal * accrualPeriod * discount;
error = std::abs(targetOptionPrice - replicationOptionPrice);
Real optionTolerance = 1e-07;
if (error>optionTolerance)
BOOST_ERROR("\nDigital Call Option:" <<
"\nVolatility = " << io::rate(capletVolatility) <<
"\nStrike = " << io::rate(strike) <<
"\nExercise = " << k+1 << " years" <<
"\nPrice by replication = " << replicationOptionPrice <<
"\nTarget price = " << targetOptionPrice <<
"\nError = " << error);
// Floating Rate Coupon + Deep-in-the-money Put Digital option
strike = 0.99;
DigitalCoupon digitalFlooredCoupon(underlying,
nullstrike, Position::Long, false, nullstrike,
strike, Position::Long, false, cashRate);
digitalFlooredCoupon.setPricer(pricer);
// Check price vs its target
targetPrice = underlying->price(vars.termStructure) + targetOptionPrice;
digitalPrice = digitalFlooredCoupon.price(vars.termStructure);
error = std::fabs(targetPrice - digitalPrice);
if (error>tolerance)
BOOST_ERROR("\nFloating Coupon + Digital Put Option:" <<
"\nVolatility = " << io::rate(capletVolatility) <<
"\nStrike = " << io::rate(strike) <<
"\nExercise = " << k+1 << " years" <<
"\nCoupon price = " << digitalPrice <<
"\nTarget price = " << targetPrice <<
"\nError = " << error );
// Check digital option
replicationOptionPrice = digitalFlooredCoupon.putOptionRate() *
vars.nominal * accrualPeriod * discount;
error = std::abs(targetOptionPrice - replicationOptionPrice);
if (error>optionTolerance)
BOOST_ERROR("\nDigital Put Coupon:" <<
"\nVolatility = " << io::rate(capletVolatility) <<
"\nStrike = " << io::rate(strike) <<
"\nExercise = " << k+1 << " years" <<
"\nPrice by replication = " << replicationOptionPrice <<
"\nTarget price = " << targetOptionPrice <<
"\nError = " << error );
}
}
BOOST_AUTO_TEST_CASE(testCashOrNothingDeepOutTheMoney) {
BOOST_TEST_MESSAGE("Testing European deep out-the-money cash-or-nothing "
"digital coupon...");
CommonVars vars;
Real gearing = 1.0;
Real spread = 0.0;
Volatility capletVolatility = 0.0001;
RelinkableHandle<OptionletVolatilityStructure> volatility;
volatility.linkTo(ext::shared_ptr<OptionletVolatilityStructure>(new
ConstantOptionletVolatility(vars.today, vars.calendar, Following,
capletVolatility, Actual360())));
for (Size k = 0; k<10; k++) { // loop on start and end dates
Date startDate = vars.calendar.advance(vars.settlement,(k+1)*Years);
Date endDate = vars.calendar.advance(vars.settlement,(k+2)*Years);
Rate nullstrike = Null<Rate>();
Rate cashRate = 0.01;
Date paymentDate = endDate;
ext::shared_ptr<FloatingRateCoupon> underlying(new
IborCoupon(paymentDate, vars.nominal,
startDate, endDate,
vars.fixingDays, vars.index,
gearing, spread));
// Deep out-of-the-money Capped Digital Coupon
Rate strike = 0.99;
DigitalCoupon digitalCappedCoupon(underlying,
strike, Position::Short, false, cashRate,
nullstrike, Position::Short, false, nullstrike);
ext::shared_ptr<IborCouponPricer> pricer(new BlackIborCouponPricer(volatility));
digitalCappedCoupon.setPricer(pricer);
// Check price vs its target
Time accrualPeriod = underlying->accrualPeriod();
Real discount = vars.termStructure->discount(endDate);
Real targetPrice = underlying->price(vars.termStructure);
Real digitalPrice = digitalCappedCoupon.price(vars.termStructure);
Real error = std::fabs(targetPrice - digitalPrice);
Real tolerance = 1e-10;
if (error>tolerance)
BOOST_ERROR("\nFloating Coupon + Digital Call Option:" <<
"\nVolatility = " << io::rate(capletVolatility) <<
"\nStrike = " << io::rate(strike) <<
"\nExercise = " << k+1 << " years" <<
"\nCoupon price = " << digitalPrice <<
"\nTarget price = " << targetPrice <<
"\nError = " << error );
// Check digital option price
Real targetOptionPrice = 0.;
Real replicationOptionPrice = digitalCappedCoupon.callOptionRate() *
vars.nominal * accrualPeriod * discount;
error = std::abs(targetOptionPrice - replicationOptionPrice);
Real optionTolerance = 1e-10;
if (error>optionTolerance)
BOOST_ERROR("\nDigital Call Option:" <<
"\nVolatility = " << io::rate(capletVolatility) <<
"\nStrike = " << io::rate(strike) <<
"\nExercise = " << k+1 << " years" <<
"\nPrice by replication = " << replicationOptionPrice <<
"\nTarget price = " << targetOptionPrice <<
"\nError = " << error );
// Deep out-of-the-money Floored Digital Coupon
strike = 0.01;
DigitalCoupon digitalFlooredCoupon(underlying,
nullstrike, Position::Long, false, nullstrike,
strike, Position::Long, false, cashRate);
digitalFlooredCoupon.setPricer(pricer);
// Check price vs its target
targetPrice = underlying->price(vars.termStructure);
digitalPrice = digitalFlooredCoupon.price(vars.termStructure);
tolerance = 1e-09;
error = std::fabs(targetPrice - digitalPrice);
if (error>tolerance)
BOOST_ERROR("\nDigital Floored Coupon:" <<
"\nVolatility = " << io::rate(capletVolatility) <<
"\nStrike = " << io::rate(strike) <<
"\nExercise = " << k+1 << " years" <<
"\nCoupon price = " << digitalPrice <<
"\nTarget price = " << targetPrice <<
"\nError = " << error );
// Check digital option
targetOptionPrice = 0.0;
replicationOptionPrice = digitalFlooredCoupon.putOptionRate() *
vars.nominal * accrualPeriod * discount;
error = std::abs(targetOptionPrice - replicationOptionPrice);
if (error>optionTolerance)
BOOST_ERROR("\nDigital Put Option:" <<
"\nVolatility = " << io::rate(capletVolatility) <<
"\nStrike = " << io::rate(strike) <<
"\nExercise = " << k+1 << " years" <<
"\nPrice by replication " << replicationOptionPrice <<
"\nTarget price " << targetOptionPrice <<
"\nError " << error );
}
}
BOOST_AUTO_TEST_CASE(testCallPutParity) {
BOOST_TEST_MESSAGE("Testing call/put parity for European digital coupon...");
CommonVars vars;
Volatility vols[] = { 0.05, 0.15, 0.30 };
Rate strikes[] = { 0.01, 0.02, 0.03, 0.04, 0.05, 0.06, 0.07 };
Real gearing = 1.0;
Real spread = 0.0;
for (Real capletVolatility : vols) {
RelinkableHandle<OptionletVolatilityStructure> volatility;
volatility.linkTo(
ext::shared_ptr<OptionletVolatilityStructure>(new ConstantOptionletVolatility(
vars.today, vars.calendar, Following, capletVolatility, Actual360())));
for (Real strike : strikes) {
for (Size k = 0; k < 10; k++) {
Date startDate = vars.calendar.advance(vars.settlement,(k+1)*Years);
Date endDate = vars.calendar.advance(vars.settlement,(k+2)*Years);
Rate nullstrike = Null<Rate>();
Date paymentDate = endDate;
ext::shared_ptr<FloatingRateCoupon> underlying(new
IborCoupon(paymentDate, vars.nominal,
startDate, endDate,
vars.fixingDays, vars.index,
gearing, spread));
// Cash-or-Nothing
Rate cashRate = 0.01;
// Floating Rate Coupon + Call Digital option
DigitalCoupon cash_digitalCallCoupon(underlying,
strike, Position::Long, false, cashRate,
nullstrike, Position::Long, false, nullstrike);
ext::shared_ptr<IborCouponPricer> pricer(new
BlackIborCouponPricer(volatility));
cash_digitalCallCoupon.setPricer(pricer);
// Floating Rate Coupon - Put Digital option
DigitalCoupon cash_digitalPutCoupon(underlying,
nullstrike, Position::Long, false, nullstrike,
strike, Position::Short, false, cashRate);
cash_digitalPutCoupon.setPricer(pricer);
Real digitalPrice = cash_digitalCallCoupon.price(vars.termStructure) -
cash_digitalPutCoupon.price(vars.termStructure);
// Target price
Time accrualPeriod = underlying->accrualPeriod();
Real discount = vars.termStructure->discount(endDate);
Real targetPrice = vars.nominal * accrualPeriod * discount * cashRate;
Real error = std::fabs(targetPrice - digitalPrice);
Real tolerance = 1.e-08;
if (error>tolerance)
BOOST_ERROR("\nCash-or-nothing:" <<
"\nVolatility = " << io::rate(capletVolatility) <<
"\nStrike = " << io::rate(strike) <<
"\nExercise = " << k+1 << " years" <<
"\nPrice = " << digitalPrice <<
"\nTarget Price = " << targetPrice <<
"\nError = " << error );
// Asset-or-Nothing
// Floating Rate Coupon + Call Digital option
DigitalCoupon asset_digitalCallCoupon(underlying,
strike, Position::Long, false, nullstrike,
nullstrike, Position::Long, false, nullstrike);
asset_digitalCallCoupon.setPricer(pricer);
// Floating Rate Coupon - Put Digital option
DigitalCoupon asset_digitalPutCoupon(underlying,
nullstrike, Position::Long, false, nullstrike,
strike, Position::Short, false, nullstrike);
asset_digitalPutCoupon.setPricer(pricer);
digitalPrice = asset_digitalCallCoupon.price(vars.termStructure) -
asset_digitalPutCoupon.price(vars.termStructure);
// Target price
targetPrice = vars.nominal * accrualPeriod * discount * underlying->rate();
error = std::fabs(targetPrice - digitalPrice);
tolerance = 1.e-07;
if (error>tolerance)
BOOST_ERROR("\nAsset-or-nothing:" <<
"\nVolatility = " << io::rate(capletVolatility) <<
"\nStrike = " << io::rate(strike) <<
"\nExercise = " << k+1 << " years" <<
"\nPrice = " << digitalPrice <<
"\nTarget Price = " << targetPrice <<
"\nError = " << error );
}
}
}
}
BOOST_AUTO_TEST_CASE(testReplicationType) {
BOOST_TEST_MESSAGE("Testing replication type for European digital coupon...");
CommonVars vars;
Volatility vols[] = { 0.05, 0.15, 0.30 };
Rate strikes[] = { 0.01, 0.02, 0.03, 0.04, 0.05, 0.06, 0.07 };
Real gearing = 1.0;
Real spread = 0.0;
Real gap = 1e-04;
ext::shared_ptr<DigitalReplication> subReplication(new
DigitalReplication(Replication::Sub, gap));
ext::shared_ptr<DigitalReplication> centralReplication(new
DigitalReplication(Replication::Central, gap));
ext::shared_ptr<DigitalReplication> superReplication(new
DigitalReplication(Replication::Super, gap));
for (Real capletVolatility : vols) {
RelinkableHandle<OptionletVolatilityStructure> volatility;
volatility.linkTo(ext::shared_ptr<OptionletVolatilityStructure>(new
ConstantOptionletVolatility(vars.today, vars.calendar, Following,
capletVolatility, Actual360())));
for (Real strike : strikes) {
for (Size k = 0; k < 10; k++) {
Date startDate = vars.calendar.advance(vars.settlement,(k+1)*Years);
Date endDate = vars.calendar.advance(vars.settlement,(k+2)*Years);
Rate nullstrike = Null<Rate>();
Date paymentDate = endDate;
ext::shared_ptr<FloatingRateCoupon> underlying(new
IborCoupon(paymentDate, vars.nominal,
startDate, endDate,
vars.fixingDays, vars.index,
gearing, spread));
// Cash-or-Nothing
Rate cashRate = 0.005;
// Floating Rate Coupon + Call Digital option
DigitalCoupon sub_cash_longDigitalCallCoupon(underlying,
strike, Position::Long, false, cashRate,
nullstrike, Position::Long, false, nullstrike,
subReplication);
DigitalCoupon central_cash_longDigitalCallCoupon(underlying,
strike, Position::Long, false, cashRate,
nullstrike, Position::Long, false, nullstrike,
centralReplication);
DigitalCoupon over_cash_longDigitalCallCoupon(underlying,
strike, Position::Long, false, cashRate,
nullstrike, Position::Long, false, nullstrike,
superReplication);
ext::shared_ptr<IborCouponPricer> pricer(new
BlackIborCouponPricer(volatility));
sub_cash_longDigitalCallCoupon.setPricer(pricer);
central_cash_longDigitalCallCoupon.setPricer(pricer);
over_cash_longDigitalCallCoupon.setPricer(pricer);
Real sub_digitalPrice = sub_cash_longDigitalCallCoupon.price(vars.termStructure);
Real central_digitalPrice = central_cash_longDigitalCallCoupon.price(vars.termStructure);
Real over_digitalPrice = over_cash_longDigitalCallCoupon.price(vars.termStructure);
Real tolerance = 1.e-09;
if ( ( (sub_digitalPrice > central_digitalPrice) &&
std::abs(central_digitalPrice - sub_digitalPrice)>tolerance ) ||
( (central_digitalPrice>over_digitalPrice) &&
std::abs(central_digitalPrice - over_digitalPrice)>tolerance ) ) {
BOOST_ERROR("\nCash-or-nothing: Floating Rate Coupon + Call Digital option" <<
"\nVolatility = " << io::rate(capletVolatility) <<
"\nStrike = " << io::rate(strike) <<
"\nExercise = " << k+1 << " years" <<
std::setprecision(20) <<
"\nSub-Replication Price = " << sub_digitalPrice <<
"\nCentral-Replication Price = " << central_digitalPrice <<
"\nOver-Replication Price = " << over_digitalPrice);
}
// Floating Rate Coupon - Call Digital option
DigitalCoupon sub_cash_shortDigitalCallCoupon(underlying,
strike, Position::Short, false, cashRate,
nullstrike, Position::Long, false, nullstrike,
subReplication);
DigitalCoupon central_cash_shortDigitalCallCoupon(underlying,
strike, Position::Short, false, cashRate,
nullstrike, Position::Long, false, nullstrike,
centralReplication);
DigitalCoupon over_cash_shortDigitalCallCoupon(underlying,
strike, Position::Short, false, cashRate,
nullstrike, Position::Long, false, nullstrike,
superReplication);
sub_cash_shortDigitalCallCoupon.setPricer(pricer);
central_cash_shortDigitalCallCoupon.setPricer(pricer);
over_cash_shortDigitalCallCoupon.setPricer(pricer);
sub_digitalPrice = sub_cash_shortDigitalCallCoupon.price(vars.termStructure);
central_digitalPrice = central_cash_shortDigitalCallCoupon.price(vars.termStructure);
over_digitalPrice = over_cash_shortDigitalCallCoupon.price(vars.termStructure);
if ( ( (sub_digitalPrice > central_digitalPrice) &&
std::abs(central_digitalPrice - sub_digitalPrice)>tolerance ) ||
( (central_digitalPrice>over_digitalPrice) &&
std::abs(central_digitalPrice - over_digitalPrice)>tolerance ) )
BOOST_ERROR("\nCash-or-nothing: Floating Rate Coupon - Call Digital option" <<
"\nVolatility = " << io::rate(capletVolatility) <<
"\nStrike = " << io::rate(strike) <<
"\nExercise = " << k+1 << " years" <<
std::setprecision(20) <<
"\nSub-Replication Price = " << sub_digitalPrice <<
"\nCentral-Replication Price = " << central_digitalPrice <<
"\nOver-Replication Price = " << over_digitalPrice);
// Floating Rate Coupon + Put Digital option
DigitalCoupon sub_cash_longDigitalPutCoupon(underlying,
nullstrike, Position::Long, false, nullstrike,
strike, Position::Long, false, cashRate,
subReplication);
DigitalCoupon central_cash_longDigitalPutCoupon(underlying,
nullstrike, Position::Long, false, nullstrike,
strike, Position::Long, false, cashRate,
centralReplication);
DigitalCoupon over_cash_longDigitalPutCoupon(underlying,
nullstrike, Position::Long, false, nullstrike,
strike, Position::Long, false, cashRate,
superReplication);
sub_cash_longDigitalPutCoupon.setPricer(pricer);
central_cash_longDigitalPutCoupon.setPricer(pricer);
over_cash_longDigitalPutCoupon.setPricer(pricer);
sub_digitalPrice = sub_cash_longDigitalPutCoupon.price(vars.termStructure);
central_digitalPrice = central_cash_longDigitalPutCoupon.price(vars.termStructure);
over_digitalPrice = over_cash_longDigitalPutCoupon.price(vars.termStructure);
if ( ( (sub_digitalPrice > central_digitalPrice) &&
std::abs(central_digitalPrice - sub_digitalPrice)>tolerance ) ||
( (central_digitalPrice>over_digitalPrice) &&
std::abs(central_digitalPrice - over_digitalPrice)>tolerance ) )
BOOST_ERROR("\nCash-or-nothing: Floating Rate Coupon + Put Digital option" <<
"\nVolatility = " << io::rate(capletVolatility) <<
"\nStrike = " << io::rate(strike) <<
"\nExercise = " << k+1 << " years" <<
std::setprecision(20) <<
"\nSub-Replication Price = " << sub_digitalPrice <<
"\nCentral-Replication Price = " << central_digitalPrice <<
"\nOver-Replication Price = " << over_digitalPrice);
// Floating Rate Coupon - Put Digital option
DigitalCoupon sub_cash_shortDigitalPutCoupon(underlying,
nullstrike, Position::Long, false, nullstrike,
strike, Position::Short, false, cashRate,
subReplication);
DigitalCoupon central_cash_shortDigitalPutCoupon(underlying,
nullstrike, Position::Long, false, nullstrike,
strike, Position::Short, false, cashRate,
centralReplication);
DigitalCoupon over_cash_shortDigitalPutCoupon(underlying,
nullstrike, Position::Long, false, nullstrike,
strike, Position::Short, false, cashRate,
superReplication);
sub_cash_shortDigitalPutCoupon.setPricer(pricer);
central_cash_shortDigitalPutCoupon.setPricer(pricer);
over_cash_shortDigitalPutCoupon.setPricer(pricer);
sub_digitalPrice = sub_cash_shortDigitalPutCoupon.price(vars.termStructure);
central_digitalPrice = central_cash_shortDigitalPutCoupon.price(vars.termStructure);
over_digitalPrice = over_cash_shortDigitalPutCoupon.price(vars.termStructure);
if ( ( (sub_digitalPrice > central_digitalPrice) &&
std::abs(central_digitalPrice - sub_digitalPrice)>tolerance ) ||
( (central_digitalPrice>over_digitalPrice) &&
std::abs(central_digitalPrice - over_digitalPrice)>tolerance ) )
BOOST_ERROR("\nCash-or-nothing: Floating Rate Coupon + Call Digital option" <<
"\nVolatility = " << io::rate(capletVolatility) <<
"\nStrike = " << io::rate(strike) <<
"\nExercise = " << k+1 << " years" <<
std::setprecision(20) <<
"\nSub-Replication Price = " << sub_digitalPrice <<
"\nCentral-Replication Price = " << central_digitalPrice <<
"\nOver-Replication Price = " << over_digitalPrice);
}
}
}
}
BOOST_AUTO_TEST_SUITE_END()
BOOST_AUTO_TEST_SUITE_END()
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