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hestonmodel.cpp
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hestonmodel.cpp
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/* -*- mode: c++; tab-width: 4; indent-tabs-mode: nil; c-basic-offset: 4 -*- */
/*
Copyright (C) 2005, 2007, 2009, 2010, 2012, 2014, 2017 Klaus Spanderen
This file is part of QuantLib, a free-software/open-source library
for financial quantitative analysts and developers - http://quantlib.org/
QuantLib is free software: you can redistribute it and/or modify it
under the terms of the QuantLib license. You should have received a
copy of the license along with this program; if not, please email
<quantlib-dev@lists.sf.net>. The license is also available online at
<http://quantlib.org/license.shtml>.
This program is distributed in the hope that it will be useful, but WITHOUT
ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or FITNESS
FOR A PARTICULAR PURPOSE. See the license for more details.
*/
#include "hestonmodel.hpp"
#include "utilities.hpp"
#include <ql/experimental/exoticoptions/analyticpdfhestonengine.hpp>
#include <ql/instruments/dividendbarrieroption.hpp>
#include <ql/instruments/dividendvanillaoption.hpp>
#include <ql/math/integrals/gausslobattointegral.hpp>
#include <ql/math/optimization/differentialevolution.hpp>
#include <ql/math/optimization/levenbergmarquardt.hpp>
#include <ql/math/randomnumbers/rngtraits.hpp>
#include <ql/methods/finitedifferences/operators/numericaldifferentiation.hpp>
#include <ql/methods/montecarlo/pathgenerator.hpp>
#include <ql/models/equity/hestonmodel.hpp>
#include <ql/models/equity/hestonmodelhelper.hpp>
#include <ql/models/equity/piecewisetimedependenthestonmodel.hpp>
#include <ql/pricingengines/barrier/fdblackscholesbarrierengine.hpp>
#include <ql/pricingengines/barrier/fdhestonbarrierengine.hpp>
#include <ql/pricingengines/blackformula.hpp>
#include <ql/pricingengines/vanilla/analyticdividendeuropeanengine.hpp>
#include <ql/pricingengines/vanilla/analytichestonengine.hpp>
#include <ql/pricingengines/vanilla/analyticptdhestonengine.hpp>
#include <ql/pricingengines/vanilla/coshestonengine.hpp>
#include <ql/pricingengines/vanilla/exponentialfittinghestonengine.hpp>
#include <ql/pricingengines/vanilla/fdblackscholesvanillaengine.hpp>
#include <ql/pricingengines/vanilla/fdhestonvanillaengine.hpp>
#include <ql/pricingengines/vanilla/hestonexpansionengine.hpp>
#include <ql/pricingengines/vanilla/mceuropeanhestonengine.hpp>
#include <ql/processes/hestonprocess.hpp>
#include <ql/quotes/simplequote.hpp>
#include <ql/termstructures/yield/flatforward.hpp>
#include <ql/termstructures/yield/zerocurve.hpp>
#include <ql/time/calendars/nullcalendar.hpp>
#include <ql/time/calendars/target.hpp>
#include <ql/time/daycounters/actual360.hpp>
#include <ql/time/daycounters/actual365fixed.hpp>
#include <ql/time/daycounters/actualactual.hpp>
#include <ql/time/period.hpp>
#include <cmath>
#include <utility>
using namespace QuantLib;
using namespace boost::unit_test_framework;
namespace {
struct CalibrationMarketData {
Handle<Quote> s0;
Handle<YieldTermStructure> riskFreeTS, dividendYield;
std::vector<ext::shared_ptr<CalibrationHelper> > options;
};
CalibrationMarketData getDAXCalibrationMarketData() {
/* this example is taken from A. Sepp
Pricing European-Style Options under Jump Diffusion Processes
with Stochstic Volatility: Applications of Fourier Transform
http://math.ut.ee/~spartak/papers/stochjumpvols.pdf
*/
Date settlementDate(Settings::instance().evaluationDate());
DayCounter dayCounter = Actual365Fixed();
Calendar calendar = TARGET();
Integer t[] = { 13, 41, 75, 165, 256, 345, 524, 703 };
Rate r[] = { 0.0357,0.0349,0.0341,0.0355,0.0359,0.0368,0.0386,0.0401 };
std::vector<Date> dates;
std::vector<Rate> rates;
dates.push_back(settlementDate);
rates.push_back(0.0357);
Size i;
for (i = 0; i < 8; ++i) {
dates.push_back(settlementDate + t[i]);
rates.push_back(r[i]);
}
// FLOATING_POINT_EXCEPTION
Handle<YieldTermStructure> riskFreeTS(
ext::make_shared<ZeroCurve>(dates, rates, dayCounter));
Handle<YieldTermStructure> dividendYield(
flatRate(settlementDate, 0.0, dayCounter));
Volatility v[] =
{ 0.6625,0.4875,0.4204,0.3667,0.3431,0.3267,0.3121,0.3121,
0.6007,0.4543,0.3967,0.3511,0.3279,0.3154,0.2984,0.2921,
0.5084,0.4221,0.3718,0.3327,0.3155,0.3027,0.2919,0.2889,
0.4541,0.3869,0.3492,0.3149,0.2963,0.2926,0.2819,0.2800,
0.4060,0.3607,0.3330,0.2999,0.2887,0.2811,0.2751,0.2775,
0.3726,0.3396,0.3108,0.2781,0.2788,0.2722,0.2661,0.2686,
0.3550,0.3277,0.3012,0.2781,0.2781,0.2661,0.2661,0.2681,
0.3428,0.3209,0.2958,0.2740,0.2688,0.2627,0.2580,0.2620,
0.3302,0.3062,0.2799,0.2631,0.2573,0.2533,0.2504,0.2544,
0.3343,0.2959,0.2705,0.2540,0.2504,0.2464,0.2448,0.2462,
0.3460,0.2845,0.2624,0.2463,0.2425,0.2385,0.2373,0.2422,
0.3857,0.2860,0.2578,0.2399,0.2357,0.2327,0.2312,0.2351,
0.3976,0.2860,0.2607,0.2356,0.2297,0.2268,0.2241,0.2320 };
Handle<Quote> s0(ext::make_shared<SimpleQuote>(4468.17));
Real strike[] = { 3400,3600,3800,4000,4200,4400,
4500,4600,4800,5000,5200,5400,5600 };
std::vector<ext::shared_ptr<CalibrationHelper> > options;
for (Size s = 0; s < 13; ++s) {
for (Size m = 0; m < 8; ++m) {
Handle<Quote> vol(ext::make_shared<SimpleQuote>(v[s*8+m]));
Period maturity((int)((t[m]+3)/7.), Weeks); // round to weeks
options.push_back(ext::make_shared<HestonModelHelper>(maturity, calendar,
s0, strike[s], vol,
riskFreeTS, dividendYield,
BlackCalibrationHelper::ImpliedVolError));
}
}
CalibrationMarketData marketData = { s0, riskFreeTS, dividendYield, options };
return marketData;
}
}
void HestonModelTest::testBlackCalibration() {
BOOST_TEST_MESSAGE(
"Testing Heston model calibration using a flat volatility surface...");
SavedSettings backup;
/* calibrate a Heston model to a constant volatility surface without
smile. expected result is a vanishing volatility of the volatility.
In addition theta and v0 should be equal to the constant variance */
Date today = Date::todaysDate();
Settings::instance().evaluationDate() = today;
DayCounter dayCounter = Actual360();
Calendar calendar = NullCalendar();
Handle<YieldTermStructure> riskFreeTS(flatRate(0.04, dayCounter));
Handle<YieldTermStructure> dividendTS(flatRate(0.50, dayCounter));
std::vector<Period> optionMaturities = {1 * Months, 2 * Months, 3 * Months, 6 * Months,
9 * Months, 1 * Years, 2 * Years};
std::vector<ext::shared_ptr<CalibrationHelper> > options;
Handle<Quote> s0(ext::make_shared<SimpleQuote>(1.0));
Handle<Quote> vol(ext::make_shared<SimpleQuote>(0.1));
Volatility volatility = vol->value();
for (auto& optionMaturitie : optionMaturities) {
for (Real moneyness = -1.0; moneyness < 2.0; moneyness += 1.0) {
// FLOATING_POINT_EXCEPTION
const Time tau = dayCounter.yearFraction(
riskFreeTS->referenceDate(),
calendar.advance(riskFreeTS->referenceDate(), optionMaturitie));
const Real fwdPrice =
s0->value() * dividendTS->discount(tau) / riskFreeTS->discount(tau);
const Real strikePrice = fwdPrice * std::exp(-moneyness * volatility * std::sqrt(tau));
options.push_back(ext::make_shared<HestonModelHelper>(
optionMaturitie, calendar, s0, strikePrice, vol, riskFreeTS, dividendTS));
}
}
for (Real sigma = 0.1; sigma < 0.7; sigma += 0.2) {
const Real v0=0.01;
const Real kappa=0.2;
const Real theta=0.02;
const Real rho=-0.75;
ext::shared_ptr<HestonProcess> process(
ext::make_shared<HestonProcess>(riskFreeTS, dividendTS,
s0, v0, kappa, theta, sigma, rho));
ext::shared_ptr<HestonModel> model(ext::make_shared<HestonModel>(process));
ext::shared_ptr<PricingEngine> engine(
ext::make_shared<AnalyticHestonEngine>(model, 96));
for (auto& option : options)
ext::dynamic_pointer_cast<BlackCalibrationHelper>(option)->setPricingEngine(engine);
LevenbergMarquardt om(1e-8, 1e-8, 1e-8);
model->calibrate(options, om, EndCriteria(400, 40, 1.0e-8,
1.0e-8, 1.0e-8));
Real tolerance = 3.0e-3;
if (model->sigma() > tolerance) {
BOOST_ERROR("Failed to reproduce expected sigma"
<< "\n calculated: " << model->sigma()
<< "\n expected: " << 0.0
<< "\n tolerance: " << tolerance);
}
if (std::fabs(model->kappa()
*(model->theta()-volatility*volatility)) > tolerance) {
BOOST_ERROR("Failed to reproduce expected theta"
<< "\n calculated: " << model->theta()
<< "\n expected: " << volatility*volatility);
}
if (std::fabs(model->v0()-volatility*volatility) > tolerance) {
BOOST_ERROR("Failed to reproduce expected v0"
<< "\n calculated: " << model->v0()
<< "\n expected: " << volatility*volatility);
}
}
}
void HestonModelTest::testDAXCalibration() {
BOOST_TEST_MESSAGE(
"Testing Heston model calibration using DAX volatility data...");
SavedSettings backup;
Date settlementDate(5, July, 2002);
Settings::instance().evaluationDate() = settlementDate;
CalibrationMarketData marketData = getDAXCalibrationMarketData();
const Handle<YieldTermStructure> riskFreeTS = marketData.riskFreeTS;
const Handle<YieldTermStructure> dividendTS = marketData.dividendYield;
const Handle<Quote> s0 = marketData.s0;
const std::vector<ext::shared_ptr<CalibrationHelper> >& options = marketData.options;
const Real v0=0.1;
const Real kappa=1.0;
const Real theta=0.1;
const Real sigma=0.5;
const Real rho=-0.5;
const ext::shared_ptr<HestonProcess> process(
ext::make_shared<HestonProcess>(
riskFreeTS, dividendTS, s0, v0, kappa, theta, sigma, rho));
const ext::shared_ptr<HestonModel> model(
ext::make_shared<HestonModel>(process));
const ext::shared_ptr<PricingEngine> engines[] = {
ext::make_shared<AnalyticHestonEngine>(model, 64),
ext::make_shared<COSHestonEngine>(model, 12, 75),
ext::make_shared<ExponentialFittingHestonEngine>(model)
};
const Array params = model->params();
for (const auto& engine : engines) {
model->setParams(params);
for (const auto& option : options)
ext::dynamic_pointer_cast<BlackCalibrationHelper>(option)->setPricingEngine(engine);
LevenbergMarquardt om(1e-8, 1e-8, 1e-8);
model->calibrate(options, om,
EndCriteria(400, 40, 1.0e-8, 1.0e-8, 1.0e-8));
Real sse = 0;
for (Size i = 0; i < 13*8; ++i) {
const Real diff = options[i]->calibrationError()*100.0;
sse += diff*diff;
}
Real expected = 177.2; //see article by A. Sepp.
if (std::fabs(sse - expected) > 1.0) {
BOOST_FAIL("Failed to reproduce calibration error"
<< "\n calculated: " << sse
<< "\n expected: " << expected);
}
}
}
void HestonModelTest::testAnalyticVsBlack() {
BOOST_TEST_MESSAGE("Testing analytic Heston engine against Black formula...");
SavedSettings backup;
Date settlementDate = Date::todaysDate();
Settings::instance().evaluationDate() = settlementDate;
DayCounter dayCounter = ActualActual(ActualActual::ISDA);
Date exerciseDate = settlementDate + 6*Months;
ext::shared_ptr<StrikedTypePayoff> payoff(
ext::make_shared<PlainVanillaPayoff>(Option::Put, 30));
ext::shared_ptr<Exercise> exercise(
ext::make_shared<EuropeanExercise>(exerciseDate));
Handle<YieldTermStructure> riskFreeTS(flatRate(0.1, dayCounter));
Handle<YieldTermStructure> dividendTS(flatRate(0.04, dayCounter));
Handle<Quote> s0(ext::make_shared<SimpleQuote>(32.0));
const Real v0=0.05;
const Real kappa=5.0;
const Real theta=0.05;
const Real sigma=1.0e-4;
const Real rho=0.0;
ext::shared_ptr<HestonProcess> process(ext::make_shared<HestonProcess>(
riskFreeTS, dividendTS, s0, v0, kappa, theta, sigma, rho));
VanillaOption option(payoff, exercise);
// FLOATING_POINT_EXCEPTION
ext::shared_ptr<PricingEngine> engine(
ext::make_shared<AnalyticHestonEngine>(
ext::make_shared<HestonModel>(process), 144));
option.setPricingEngine(engine);
Real calculated = option.NPV();
Real yearFraction = dayCounter.yearFraction(settlementDate, exerciseDate);
Real forwardPrice = 32*std::exp((0.1-0.04)*yearFraction);
Real expected = blackFormula(payoff->optionType(), payoff->strike(),
forwardPrice, std::sqrt(0.05*yearFraction)) *
std::exp(-0.1*yearFraction);
Real error = std::fabs(calculated - expected);
Real tolerance = 2.0e-7;
if (error > tolerance) {
BOOST_FAIL("failed to reproduce Black price with AnalyticHestonEngine"
<< "\n calculated: " << calculated
<< "\n expected: " << expected
<< "\n error: " << std::scientific << error);
}
engine =
ext::make_shared<FdHestonVanillaEngine>(
ext::make_shared<HestonModel>(process),
200,200,100);
option.setPricingEngine(engine);
calculated = option.NPV();
error = std::fabs(calculated - expected);
tolerance = 1.0e-3;
if (error > tolerance) {
BOOST_FAIL("failed to reproduce Black price with FdHestonVanillaEngine"
<< "\n calculated: " << calculated
<< "\n expected: " << expected
<< "\n error: " << std::scientific << error);
}
}
void HestonModelTest::testAnalyticVsCached() {
BOOST_TEST_MESSAGE("Testing analytic Heston engine against cached values...");
SavedSettings backup;
Date settlementDate(27, December, 2004);
Settings::instance().evaluationDate() = settlementDate;
DayCounter dayCounter = ActualActual(ActualActual::ISDA);
Date exerciseDate(28, March, 2005);
ext::shared_ptr<StrikedTypePayoff> payoff(
ext::make_shared<PlainVanillaPayoff>(Option::Call, 1.05));
ext::shared_ptr<Exercise> exercise(
ext::make_shared<EuropeanExercise>(exerciseDate));
Handle<YieldTermStructure> riskFreeTS(flatRate(0.0225, dayCounter));
Handle<YieldTermStructure> dividendTS(flatRate(0.02, dayCounter));
Handle<Quote> s0(ext::make_shared<SimpleQuote>(1.0));
const Real v0 = 0.1;
const Real kappa = 3.16;
const Real theta = 0.09;
const Real sigma = 0.4;
const Real rho = -0.2;
ext::shared_ptr<HestonProcess> process(ext::make_shared<HestonProcess>(
riskFreeTS, dividendTS, s0, v0, kappa, theta, sigma, rho));
VanillaOption option(payoff, exercise);
ext::shared_ptr<AnalyticHestonEngine> engine(
ext::make_shared<AnalyticHestonEngine>(
ext::make_shared<HestonModel>(process), 64));
option.setPricingEngine(engine);
Real expected1 = 0.0404774515;
Real calculated1 = option.NPV();
Real tolerance = 1.0e-8;
if (std::fabs(calculated1 - expected1) > tolerance) {
BOOST_ERROR("Failed to reproduce cached analytic price"
<< "\n calculated: " << calculated1
<< "\n expected: " << expected1);
}
// reference values from www.wilmott.com, technical forum
// search for "Heston or VG price check"
Real K[] = {0.9,1.0,1.1};
Real expected2[] = { 0.1330371,0.0641016, 0.0270645 };
Real calculated2[6];
Size i;
for (i = 0; i < 6; ++i) {
Date exerciseDate(8+i/3, September, 2005);
ext::shared_ptr<StrikedTypePayoff> payoff(
ext::make_shared<PlainVanillaPayoff>(Option::Call, K[i%3]));
ext::shared_ptr<Exercise> exercise(
ext::make_shared<EuropeanExercise>(exerciseDate));
Handle<YieldTermStructure> riskFreeTS(flatRate(0.05, dayCounter));
Handle<YieldTermStructure> dividendTS(flatRate(0.02, dayCounter));
Real s = riskFreeTS->discount(0.7)/dividendTS->discount(0.7);
Handle<Quote> s0(ext::make_shared<SimpleQuote>(s));
ext::shared_ptr<HestonProcess> process(
ext::make_shared<HestonProcess>(
riskFreeTS, dividendTS, s0, 0.09, 1.2, 0.08, 1.8, -0.45));
VanillaOption option(payoff, exercise);
ext::shared_ptr<PricingEngine> engine(
ext::make_shared<AnalyticHestonEngine>(
ext::make_shared<HestonModel>(process)));
option.setPricingEngine(engine);
calculated2[i] = option.NPV();
}
// we are after the value for T=0.7
Time t1 = dayCounter.yearFraction(settlementDate, Date(8, September,2005));
Time t2 = dayCounter.yearFraction(settlementDate, Date(9, September,2005));
for (i = 0; i < 3; ++i) {
const Real interpolated =
calculated2[i]+(calculated2[i+3]-calculated2[i])/(t2-t1)*(0.7-t1);
if (std::fabs(interpolated - expected2[i]) > 100*tolerance) {
BOOST_ERROR("Failed to reproduce cached analytic prices:"
<< "\n calculated: " << interpolated
<< "\n expected: " << expected2[i] );
}
}
}
void HestonModelTest::testMcVsCached() {
BOOST_TEST_MESSAGE(
"Testing Monte Carlo Heston engine against cached values...");
SavedSettings backup;
Date settlementDate(27, December, 2004);
Settings::instance().evaluationDate() = settlementDate;
DayCounter dayCounter = ActualActual(ActualActual::ISDA);
Date exerciseDate(28, March, 2005);
ext::shared_ptr<StrikedTypePayoff> payoff(
ext::make_shared<PlainVanillaPayoff>(Option::Put, 1.05));
ext::shared_ptr<Exercise> exercise(
ext::make_shared<EuropeanExercise>(exerciseDate));
Handle<YieldTermStructure> riskFreeTS(flatRate(0.7, dayCounter));
Handle<YieldTermStructure> dividendTS(flatRate(0.4, dayCounter));
Handle<Quote> s0(ext::make_shared<SimpleQuote>(1.05));
ext::shared_ptr<HestonProcess> process(
ext::make_shared<HestonProcess>(
riskFreeTS, dividendTS, s0, 0.3, 1.16, 0.2, 0.8, 0.8,
HestonProcess::QuadraticExponentialMartingale));
VanillaOption option(payoff, exercise);
ext::shared_ptr<PricingEngine> engine;
engine = MakeMCEuropeanHestonEngine<PseudoRandom>(process)
.withStepsPerYear(11)
.withAntitheticVariate()
.withSamples(50000)
.withSeed(1234);
option.setPricingEngine(engine);
Real expected = 0.0632851308977151;
Real calculated = option.NPV();
Real errorEstimate = option.errorEstimate();
Real tolerance = 7.5e-4;
if (std::fabs(calculated - expected) > 2.34*errorEstimate) {
BOOST_ERROR("Failed to reproduce cached price"
<< "\n calculated: " << calculated
<< "\n expected: " << expected
<< " +/- " << errorEstimate);
}
if (errorEstimate > tolerance) {
BOOST_ERROR("failed to reproduce error estimate"
<< "\n calculated: " << errorEstimate
<< "\n expected: " << tolerance);
}
}
void HestonModelTest::testFdBarrierVsCached() {
BOOST_TEST_MESSAGE("Testing FD barrier Heston engine against cached values...");
SavedSettings backup;
DayCounter dc = Actual360();
Date today = Date::todaysDate();
Handle<Quote> s0(ext::make_shared<SimpleQuote>(100.0));
Handle<YieldTermStructure> rTS(flatRate(today, 0.08, dc));
Handle<YieldTermStructure> qTS(flatRate(today, 0.04, dc));
Date exDate = today + 180;
ext::shared_ptr<Exercise> exercise(
ext::make_shared<EuropeanExercise>(exDate));
ext::shared_ptr<StrikedTypePayoff> payoff(
ext::make_shared<PlainVanillaPayoff>(Option::Call, 90.0));
ext::shared_ptr<HestonProcess> process(
ext::make_shared<HestonProcess>(
rTS, qTS, s0, 0.25*0.25, 1.0, 0.25*0.25, 0.001, 0.0));
ext::shared_ptr<PricingEngine> engine;
engine = ext::make_shared<FdHestonBarrierEngine>(
ext::make_shared<HestonModel>(process),
200,400,100);
BarrierOption option(Barrier::DownOut, 95.0, 3.0, payoff, exercise);
option.setPricingEngine(engine);
Real calculated = option.NPV();
Real expected = 9.0246;
Real error = std::fabs(calculated-expected);
if (error > 1.0e-3) {
BOOST_FAIL("failed to reproduce cached price with FD Barrier engine"
<< "\n calculated: " << calculated
<< "\n expected: " << expected
<< "\n error: " << std::scientific << error);
}
option = BarrierOption(Barrier::DownIn, 95.0, 3.0, payoff, exercise);
option.setPricingEngine(engine);
calculated = option.NPV();
expected = 7.7627;
error = std::fabs(calculated-expected);
if (error > 1.0e-3) {
BOOST_FAIL("failed to reproduce cached price with FD Barrier engine"
<< "\n calculated: " << calculated
<< "\n expected: " << expected
<< "\n error: " << std::scientific << error);
}
}
void HestonModelTest::testFdVanillaVsCached() {
BOOST_TEST_MESSAGE("Testing FD vanilla Heston engine against cached values...");
SavedSettings backup;
Date settlementDate(27, December, 2004);
Settings::instance().evaluationDate() = settlementDate;
DayCounter dayCounter = ActualActual(ActualActual::ISDA);
Date exerciseDate(28, March, 2005);
ext::shared_ptr<StrikedTypePayoff> payoff(
ext::make_shared<PlainVanillaPayoff>(Option::Put, 1.05));
ext::shared_ptr<Exercise> exercise(
ext::make_shared<EuropeanExercise>(exerciseDate));
Handle<YieldTermStructure> riskFreeTS(flatRate(0.7, dayCounter));
Handle<YieldTermStructure> dividendTS(flatRate(0.4, dayCounter));
Handle<Quote> s0(ext::make_shared<SimpleQuote>(1.05));
VanillaOption option(payoff, exercise);
ext::shared_ptr<HestonProcess> process(
ext::make_shared<HestonProcess>(
riskFreeTS, dividendTS, s0, 0.3, 1.16, 0.2, 0.8, 0.8));
option.setPricingEngine(
MakeFdHestonVanillaEngine(ext::make_shared<HestonModel>(process))
.withTGrid(100)
.withXGrid(200)
.withVGrid(100)
);
Real expected = 0.06325;
Real calculated = option.NPV();
Real error = std::fabs(calculated - expected);
Real tolerance = 1.0e-4;
if (error > tolerance) {
BOOST_FAIL("failed to reproduce cached price with FD engine"
<< "\n calculated: " << calculated
<< "\n expected: " << expected
<< "\n error: " << std::scientific << error);
}
BOOST_TEST_MESSAGE("Testing FD vanilla Heston engine for discrete dividends...");
payoff = ext::make_shared<PlainVanillaPayoff>(Option::Call, 95.0);
s0 = Handle<Quote>(ext::make_shared<SimpleQuote>(100.0));
riskFreeTS = Handle<YieldTermStructure>(flatRate(0.05, dayCounter));
dividendTS = Handle<YieldTermStructure>(flatRate(0.0, dayCounter));
exerciseDate = Date(28, March, 2006);
exercise = ext::make_shared<EuropeanExercise>(exerciseDate);
std::vector<Date> dividendDates;
std::vector<Real> dividends;
for (Date d = settlementDate + 3*Months;
d < exercise->lastDate();
d += 6*Months) {
dividendDates.push_back(d);
dividends.push_back(1.0);
}
DividendVanillaOption divOption(payoff, exercise,
dividendDates, dividends);
process = ext::make_shared<HestonProcess>(
riskFreeTS, dividendTS, s0, 0.04, 1.0, 0.04, 0.001, 0.0);
divOption.setPricingEngine(
MakeFdHestonVanillaEngine(ext::make_shared<HestonModel>(process))
.withTGrid(200)
.withXGrid(400)
.withVGrid(100)
);
calculated = divOption.NPV();
// Value calculated with an independent FD framework, validated with
// an independent MC framework
expected = 12.946;
error = std::fabs(calculated - expected);
tolerance = 5.0e-3;
if (error > tolerance) {
BOOST_FAIL("failed to reproduce discrete dividend price with FD engine"
<< "\n calculated: " << calculated
<< "\n expected: " << expected
<< "\n error: " << std::scientific << error);
}
BOOST_TEST_MESSAGE("Testing FD vanilla Heston engine for american exercise...");
dividendTS = Handle<YieldTermStructure>(flatRate(0.03, dayCounter));
process = ext::make_shared<HestonProcess>(
riskFreeTS, dividendTS, s0, 0.04, 1.0, 0.04, 0.001, 0.0);
payoff = ext::make_shared<PlainVanillaPayoff>(Option::Put, 95.0);
exercise = ext::make_shared<AmericanExercise>(
settlementDate, exerciseDate);
option = VanillaOption(payoff, exercise);
option.setPricingEngine(
MakeFdHestonVanillaEngine(ext::make_shared<HestonModel>(process))
.withTGrid(200)
.withXGrid(400)
.withVGrid(100)
);
calculated = option.NPV();
Handle<BlackVolTermStructure> volTS(flatVol(settlementDate, 0.2,
dayCounter));
ext::shared_ptr<BlackScholesMertonProcess> ref_process(
ext::make_shared<BlackScholesMertonProcess>(s0, dividendTS, riskFreeTS, volTS));
ext::shared_ptr<PricingEngine> ref_engine(
ext::make_shared<FdBlackScholesVanillaEngine>(ref_process, 200, 400));
option.setPricingEngine(ref_engine);
expected = option.NPV();
error = std::fabs(calculated - expected);
tolerance = 1.0e-3;
if (error > tolerance) {
BOOST_FAIL("failed to reproduce american option price with FD engine"
<< "\n calculated: " << calculated
<< "\n expected: " << expected
<< "\n error: " << std::scientific << error);
}
}
namespace {
struct HestonProcessDiscretizationDesc {
HestonProcess::Discretization discretization;
Size nSteps;
std::string name;
};
}
void HestonModelTest::testKahlJaeckelCase() {
BOOST_TEST_MESSAGE(
"Testing MC and FD Heston engines for the Kahl-Jaeckel example...");
/* Example taken from Wilmott mag (Sept. 2005).
"Not-so-complex logarithms in the Heston model",
Example was also discussed within the Wilmott thread
"QuantLib code is very high quatlity"
*/
SavedSettings backup;
Date settlementDate(30, March, 2007);
Settings::instance().evaluationDate() = settlementDate;
DayCounter dayCounter = ActualActual(ActualActual::ISDA);
Date exerciseDate(30, March, 2017);
const ext::shared_ptr<StrikedTypePayoff> payoff(
ext::make_shared<PlainVanillaPayoff>(Option::Call, 200));
const ext::shared_ptr<Exercise> exercise(
ext::make_shared<EuropeanExercise>(exerciseDate));
VanillaOption option(payoff, exercise);
Handle<YieldTermStructure> riskFreeTS(flatRate(0.0, dayCounter));
Handle<YieldTermStructure> dividendTS(flatRate(0.0, dayCounter));
Handle<Quote> s0(ext::make_shared<SimpleQuote>(100));
const Real v0 = 0.16;
const Real theta = v0;
const Real kappa = 1.0;
const Real sigma = 2.0;
const Real rho =-0.8;
const HestonProcessDiscretizationDesc descriptions[] = {
{ HestonProcess::NonCentralChiSquareVariance, 10,
"NonCentralChiSquareVariance" },
{ HestonProcess::QuadraticExponentialMartingale, 100,
"QuadraticExponentialMartingale" },
};
const Real tolerance = 0.2;
const Real expected = 4.95212;
for (const auto& description : descriptions) {
const ext::shared_ptr<HestonProcess> process(ext::make_shared<HestonProcess>(
riskFreeTS, dividendTS, s0, v0, kappa, theta, sigma, rho, description.discretization));
const ext::shared_ptr<PricingEngine> engine =
MakeMCEuropeanHestonEngine<PseudoRandom>(process)
.withSteps(description.nSteps)
.withAntitheticVariate()
.withAbsoluteTolerance(tolerance)
.withSeed(1234);
option.setPricingEngine(engine);
const Real calculated = option.NPV();
const Real errorEstimate = option.errorEstimate();
if (std::fabs(calculated - expected) > 2.34*errorEstimate) {
BOOST_ERROR("Failed to reproduce cached price with MC engine"
<< "\n discretization: " << description.name
<< "\n expected: " << expected
<< "\n calculated: " << calculated << " +/- " << errorEstimate);
}
if (errorEstimate > tolerance) {
BOOST_ERROR("failed to reproduce error estimate with MC engine"
<< "\n discretization: " << description.name << "\n calculated : "
<< errorEstimate << "\n expected : " << tolerance);
}
}
option.setPricingEngine(
MakeMCEuropeanHestonEngine<LowDiscrepancy>(
ext::make_shared<HestonProcess>(
riskFreeTS, dividendTS, s0, v0, kappa, theta, sigma, rho,
HestonProcess::BroadieKayaExactSchemeLaguerre))
.withSteps(1)
.withSamples(1023));
Real calculated = option.NPV();
if (std::fabs(calculated - expected) > 0.5*tolerance) {
BOOST_ERROR("Failed to reproduce cached price with MC engine"
<< "\n discretization: BroadieKayaExactSchemeLobatto"
<< "\n calculated: " << calculated
<< "\n expected: " << expected
<< "\n tolerance: " << tolerance);
}
const ext::shared_ptr<HestonModel> hestonModel(
ext::make_shared<HestonModel>(
ext::make_shared<HestonProcess>(
riskFreeTS, dividendTS, s0, v0,
kappa, theta, sigma, rho)));
option.setPricingEngine(
ext::make_shared<FdHestonVanillaEngine>(hestonModel, 200, 401, 101));
calculated = option.NPV();
Real error = std::fabs(calculated - expected);
if (error > 5.0e-2) {
BOOST_FAIL("failed to reproduce cached price with FD engine"
<< "\n calculated: " << calculated
<< "\n expected: " << expected
<< "\n error: " << std::scientific << error);
}
option.setPricingEngine(
ext::make_shared<AnalyticHestonEngine>(hestonModel, 1e-6, 1000));
calculated = option.NPV();
error = std::fabs(calculated - expected);
if (error > 0.00002) {
BOOST_FAIL("failed to reproduce cached price with "
"GaussLobatto engine"
<< "\n calculated: " << calculated
<< "\n expected: " << expected
<< "\n error: " << std::scientific << error);
}
option.setPricingEngine(
ext::make_shared<COSHestonEngine>(hestonModel, 16, 400));
calculated = option.NPV();
error = std::fabs(calculated - expected);
if (error > 0.00002) {
BOOST_FAIL("failed to reproduce cached price with "
"Cosine engine"
<< "\n calculated: " << calculated
<< "\n expected: " << expected
<< "\n error: " << std::scientific << error);
}
option.setPricingEngine(
ext::make_shared<ExponentialFittingHestonEngine>(hestonModel));
calculated = option.NPV();
error = std::fabs(calculated - expected);
if (error > 0.00002) {
BOOST_FAIL("failed to reproduce cached price with "
"exponential fitting Heston engine"
<< "\n calculated: " << calculated
<< "\n expected: " << expected
<< "\n error: " << std::scientific << error);
}
}
namespace {
struct HestonParameter {
Real v0, kappa, theta, sigma, rho; };
}
void HestonModelTest::testDifferentIntegrals() {
BOOST_TEST_MESSAGE(
"Testing different numerical Heston integration algorithms...");
SavedSettings backup;
const Date settlementDate(27, December, 2004);
Settings::instance().evaluationDate() = settlementDate;
const DayCounter dayCounter = ActualActual(ActualActual::ISDA);
Handle<YieldTermStructure> riskFreeTS(flatRate(0.05, dayCounter));
Handle<YieldTermStructure> dividendTS(flatRate(0.03, dayCounter));
const Real strikes[] = { 0.5, 0.7, 1.0, 1.25, 1.5, 2.0 };
const Integer maturities[] = { 1, 2, 3, 12, 60, 120, 360};
const Option::Type types[] ={ Option::Put, Option::Call };
const HestonParameter equityfx = { 0.07, 2.0, 0.04, 0.55, -0.8 };
const HestonParameter highCorr = { 0.07, 1.0, 0.04, 0.55, 0.995 };
const HestonParameter lowVolOfVol = { 0.07, 1.0, 0.04, 0.025, -0.75 };
const HestonParameter highVolOfVol = { 0.07, 1.0, 0.04, 5.0, -0.75 };
const HestonParameter kappaEqSigRho = { 0.07, 0.4, 0.04, 0.5, 0.8 };
std::vector<HestonParameter> params = {
equityfx,
highCorr,
lowVolOfVol,
highVolOfVol,
kappaEqSigRho
};
const Real tol[] = { 1e-3, 1e-3, 0.2, 0.01, 1e-3 };
for (std::vector<HestonParameter>::const_iterator iter = params.begin();
iter != params.end(); ++iter) {
Handle<Quote> s0(ext::make_shared<SimpleQuote>(1.0));
ext::shared_ptr<HestonProcess> process(
ext::make_shared<HestonProcess>(
riskFreeTS, dividendTS,
s0, iter->v0, iter->kappa,
iter->theta, iter->sigma, iter->rho));
ext::shared_ptr<HestonModel> model(
ext::make_shared<HestonModel>(process));
ext::shared_ptr<AnalyticHestonEngine> lobattoEngine(
ext::make_shared<AnalyticHestonEngine>(model, 1e-10,
1000000));
ext::shared_ptr<AnalyticHestonEngine> laguerreEngine(
ext::make_shared<AnalyticHestonEngine>(model, 128));
ext::shared_ptr<AnalyticHestonEngine> legendreEngine(
ext::make_shared<AnalyticHestonEngine>(
model, AnalyticHestonEngine::Gatheral,
AnalyticHestonEngine::Integration::gaussLegendre(512)));
ext::shared_ptr<AnalyticHestonEngine> chebyshevEngine(
ext::make_shared<AnalyticHestonEngine>(
model, AnalyticHestonEngine::Gatheral,
AnalyticHestonEngine::Integration::gaussChebyshev(512)));
ext::shared_ptr<AnalyticHestonEngine> chebyshev2ndEngine(
ext::make_shared<AnalyticHestonEngine>(
model, AnalyticHestonEngine::Gatheral,
AnalyticHestonEngine::Integration::gaussChebyshev2nd(512)));
Real maxLegendreDiff = 0.0;
Real maxChebyshevDiff = 0.0;
Real maxChebyshev2ndDiff= 0.0;
Real maxLaguerreDiff = 0.0;
for (int maturitie : maturities) {
ext::shared_ptr<Exercise> exercise(
ext::make_shared<EuropeanExercise>(settlementDate + Period(maturitie, Months)));
for (double strike : strikes) {
for (auto type : types) {
ext::shared_ptr<StrikedTypePayoff> payoff(
ext::make_shared<PlainVanillaPayoff>(type, strike));
VanillaOption option(payoff, exercise);
option.setPricingEngine(lobattoEngine);
const Real lobattoNPV = option.NPV();
option.setPricingEngine(laguerreEngine);
const Real laguerre = option.NPV();
option.setPricingEngine(legendreEngine);
const Real legendre = option.NPV();
option.setPricingEngine(chebyshevEngine);
const Real chebyshev = option.NPV();
option.setPricingEngine(chebyshev2ndEngine);
const Real chebyshev2nd = option.NPV();
maxLaguerreDiff
= std::max(maxLaguerreDiff,
std::fabs(lobattoNPV-laguerre));
maxLegendreDiff
= std::max(maxLegendreDiff,
std::fabs(lobattoNPV-legendre));
maxChebyshevDiff
= std::max(maxChebyshevDiff,
std::fabs(lobattoNPV-chebyshev));
maxChebyshev2ndDiff
= std::max(maxChebyshev2ndDiff,
std::fabs(lobattoNPV-chebyshev2nd));
}
}
}
const Real maxDiff = std::max(std::max(
std::max(maxLaguerreDiff,maxLegendreDiff),
maxChebyshevDiff), maxChebyshev2ndDiff);
const Real tr = tol[iter - params.begin()];
if (maxDiff > tr) {
BOOST_ERROR("Failed to reproduce Heston pricing values "
"within given tolerance"
<< "\n maxDifference: " << maxDiff
<< "\n tolerance: " << tr);
}
}
}
void HestonModelTest::testMultipleStrikesEngine() {
BOOST_TEST_MESSAGE("Testing multiple-strikes FD Heston engine...");
SavedSettings backup;
Date settlementDate(27, December, 2004);
Settings::instance().evaluationDate() = settlementDate;