QuantLib: Replication.cpp

A free/open-source library for quantitative finance Reference manual - version 1.20

Replication.cpp

This example uses the CompositeInstrument class to statically replicate a down-and-out barrier options.

/* -*- mode: c++; tab-width: 4; indent-tabs-mode: nil; c-basic-offset: 4 -*- */
/* This example showcases the CompositeInstrument class. Such class
is used to build a static replication of a down-and-out barrier
option, as outlined in Section 10.2 of Mark Joshi's "The Concepts
and Practice of Mathematical Finance" to which we refer the
reader.
*/
#include <ql/qldefines.hpp>
#ifdef BOOST_MSVC
# include <ql/auto_link.hpp>
#endif
#include <ql/instruments/compositeinstrument.hpp>
#include <ql/instruments/barrieroption.hpp>
#include <ql/instruments/europeanoption.hpp>
#include <ql/pricingengines/barrier/analyticbarrierengine.hpp>
#include <ql/pricingengines/vanilla/analyticeuropeanengine.hpp>
#include <ql/exercise.hpp>
#include <ql/termstructures/yield/flatforward.hpp>
#include <ql/termstructures/volatility/equityfx/blackconstantvol.hpp>
#include <ql/quotes/simplequote.hpp>
#include <ql/time/calendars/nullcalendar.hpp>
#include <iostream>
#include <iomanip>
using namespace QuantLib;
#if defined(QL_ENABLE_SESSIONS)
namespace QuantLib {
ThreadKey sessionId() { return 0; }
}
#endif
int main(int, char* []) {
try {
std::cout << std::endl;
Date today(29, May, 2006);
Settings::instance().evaluationDate() = today;
// the option to replicate
Barrier::Type barrierType = Barrier::DownOut;
Real barrier = 70.0;
Real rebate = 0.0;
Option::Type type = Option::Put;
Real underlyingValue = 100.0;
ext::shared_ptr<SimpleQuote> underlying(
new SimpleQuote(underlyingValue));
Real strike = 100.0;
ext::shared_ptr<SimpleQuote> riskFreeRate(new SimpleQuote(0.04));
ext::shared_ptr<SimpleQuote> volatility(new SimpleQuote(0.20));
Date maturity = today + 1*Years;
std::cout << std::endl ;
// write column headings
Size widths[] = { 45, 15, 15 };
Size totalWidth = widths[0]+widths[1]+widths[2];
std::string rule(totalWidth, '-'), dblrule(totalWidth, '=');
std::cout << dblrule << std::endl;
std::cout << "Initial market conditions" << std::endl;
std::cout << dblrule << std::endl;
std::cout << std::setw(widths[0]) << std::left << "Option"
<< std::setw(widths[1]) << std::left << "NPV"
<< std::setw(widths[2]) << std::left << "Error"
<< std::endl;
std::cout << rule << std::endl;
// bootstrap the yield/vol curves
DayCounter dayCounter = Actual365Fixed();
Handle<Quote> h1(riskFreeRate);
Handle<Quote> h2(volatility);
Handle<YieldTermStructure> flatRate(
ext::shared_ptr<YieldTermStructure>(
new FlatForward(0, NullCalendar(),
h1, dayCounter)));
Handle<BlackVolTermStructure> flatVol(
ext::shared_ptr<BlackVolTermStructure>(
new BlackConstantVol(0, NullCalendar(),
h2, dayCounter)));
// instantiate the option
ext::shared_ptr<Exercise> exercise(
new EuropeanExercise(maturity));
ext::shared_ptr<StrikedTypePayoff> payoff(
new PlainVanillaPayoff(type, strike));
ext::shared_ptr<BlackScholesProcess> bsProcess(
new BlackScholesProcess(Handle<Quote>(underlying),
flatRate, flatVol));
ext::shared_ptr<PricingEngine> barrierEngine(
new AnalyticBarrierEngine(bsProcess));
ext::shared_ptr<PricingEngine> europeanEngine(
new AnalyticEuropeanEngine(bsProcess));
BarrierOption referenceOption(barrierType, barrier, rebate,
payoff, exercise);
referenceOption.setPricingEngine(barrierEngine);
Real referenceValue = referenceOption.NPV();
std::cout << std::setw(widths[0]) << std::left
<< "Original barrier option"
<< std::fixed
<< std::setw(widths[1]) << std::left << referenceValue
<< std::setw(widths[2]) << std::left << "N/A"
<< std::endl;
// Replicating portfolios
CompositeInstrument portfolio1, portfolio2, portfolio3;
// Final payoff first (the same for all portfolios):
// as shown in Joshi, a put struck at K...
ext::shared_ptr<Instrument> put1(
new EuropeanOption(payoff, exercise));
put1->setPricingEngine(europeanEngine);
portfolio1. add(put1);
portfolio2.add(put1);
portfolio3.add(put1);
// ...minus a digital put struck at B of notional K-B...
ext::shared_ptr<StrikedTypePayoff> digitalPayoff(
new CashOrNothingPayoff(Option::Put, barrier, 1.0));
ext::shared_ptr<Instrument> digitalPut(
new EuropeanOption(digitalPayoff, exercise));
digitalPut->setPricingEngine(europeanEngine);
portfolio1. subtract(digitalPut, strike-barrier);
portfolio2.subtract(digitalPut, strike-barrier);
portfolio3.subtract(digitalPut, strike-barrier);
// ...minus a put option struck at B.
ext::shared_ptr<StrikedTypePayoff> lowerPayoff(
new PlainVanillaPayoff(Option::Put, barrier));
ext::shared_ptr<Instrument> put2(
new EuropeanOption(lowerPayoff, exercise));
put2->setPricingEngine(europeanEngine);
portfolio1.subtract(put2);
portfolio2.subtract(put2);
portfolio3.subtract(put2);
// Now we use puts struck at B to kill the value of the
// portfolio on a number of points (B,t). For the first
// portfolio, we'll use 12 dates at one-month's distance.
Integer i;
for (i=12; i>=1; i--) {
// First, we instantiate the option...
Date innerMaturity = today + i*Months;
ext::shared_ptr<Exercise> innerExercise(
new EuropeanExercise(innerMaturity));
ext::shared_ptr<StrikedTypePayoff> innerPayoff(
new PlainVanillaPayoff(Option::Put, barrier));
ext::shared_ptr<Instrument> putn(
new EuropeanOption(innerPayoff, innerExercise));
putn->setPricingEngine(europeanEngine);
// ...second, we evaluate the current portfolio and the
// latest put at (B,t)...
Date killDate = today + (i-1)*Months;
Settings::instance().evaluationDate() = killDate;
underlying->setValue(barrier);
Real portfolioValue = portfolio1. NPV();
Real putValue = putn->NPV();
// ...finally, we estimate the notional that kills the
// portfolio value at that point...
Real notional = portfolioValue/putValue;
// ...and we subtract from the portfolio a put with such
// notional.
portfolio1.subtract(putn, notional);
}
// The portfolio being complete, we return to today's market...
Settings::instance().evaluationDate() = today;
underlying->setValue(underlyingValue);
// ...and output the value.
Real portfolioValue = portfolio1.NPV();
Real error = portfolioValue - referenceValue;
std::cout << std::setw(widths[0]) << std::left
<< "Replicating portfolio (12 dates)"
<< std::fixed
<< std::setw(widths[1]) << std::left << portfolioValue
<< std::setw(widths[2]) << std::left << error
<< std::endl;
// For the second portfolio, we'll use 26 dates at two-weeks'
// distance.
for (i=52; i>=2; i-=2) {
// Same as above.
Date innerMaturity = today + i*Weeks;
ext::shared_ptr<Exercise> innerExercise(
new EuropeanExercise(innerMaturity));
ext::shared_ptr<StrikedTypePayoff> innerPayoff(
new PlainVanillaPayoff(Option::Put, barrier));
ext::shared_ptr<Instrument> putn(
new EuropeanOption(innerPayoff, innerExercise));
putn->setPricingEngine(europeanEngine);
Date killDate = today + (i-2)*Weeks;
Settings::instance().evaluationDate() = killDate;
underlying->setValue(barrier);
Real portfolioValue = portfolio2.NPV();
Real putValue = putn->NPV();
Real notional = portfolioValue/putValue;
portfolio2.subtract(putn, notional);
}
Settings::instance().evaluationDate() = today;
underlying->setValue(underlyingValue);
portfolioValue = portfolio2.NPV();
error = portfolioValue - referenceValue;
std::cout << std::setw(widths[0]) << std::left
<< "Replicating portfolio (26 dates)"
<< std::fixed
<< std::setw(widths[1]) << std::left << portfolioValue
<< std::setw(widths[2]) << std::left << error
<< std::endl;
// For the third portfolio, we'll use 52 dates at one-week's
// distance.
for (i=52; i>=1; i--) {
// Same as above.
Date innerMaturity = today + i*Weeks;
ext::shared_ptr<Exercise> innerExercise(
new EuropeanExercise(innerMaturity));
ext::shared_ptr<StrikedTypePayoff> innerPayoff(
new PlainVanillaPayoff(Option::Put, barrier));
ext::shared_ptr<Instrument> putn(
new EuropeanOption(innerPayoff, innerExercise));
putn->setPricingEngine(europeanEngine);
Date killDate = today + (i-1)*Weeks;
Settings::instance().evaluationDate() = killDate;
underlying->setValue(barrier);
Real portfolioValue = portfolio3.NPV();
Real putValue = putn->NPV();
Real notional = portfolioValue/putValue;
portfolio3.subtract(putn, notional);
}
Settings::instance().evaluationDate() = today;
underlying->setValue(underlyingValue);
portfolioValue = portfolio3.NPV();
error = portfolioValue - referenceValue;
std::cout << std::setw(widths[0]) << std::left
<< "Replicating portfolio (52 dates)"
<< std::fixed
<< std::setw(widths[1]) << std::left << portfolioValue
<< std::setw(widths[2]) << std::left << error
<< std::endl;
// Now we modify the market condition to see whether the
// replication holds. First, we change the underlying value so
// that the option is out of the money.
std::cout << dblrule << std::endl;
std::cout << "Modified market conditions: out of the money"
<< std::endl;
std::cout << dblrule << std::endl;
std::cout << std::setw(widths[0]) << std::left << "Option"
<< std::setw(widths[1]) << std::left << "NPV"
<< std::setw(widths[2]) << std::left << "Error"
<< std::endl;
std::cout << rule << std::endl;
underlying->setValue(110.0);
referenceValue = referenceOption.NPV();
std::cout << std::setw(widths[0]) << std::left
<< "Original barrier option"
<< std::fixed
<< std::setw(widths[1]) << std::left << referenceValue
<< std::setw(widths[2]) << std::left << "N/A"
<< std::endl;
portfolioValue = portfolio1.NPV();
error = portfolioValue - referenceValue;
std::cout << std::setw(widths[0]) << std::left
<< "Replicating portfolio (12 dates)"
<< std::fixed
<< std::setw(widths[1]) << std::left << portfolioValue
<< std::setw(widths[2]) << std::left << error
<< std::endl;
portfolioValue = portfolio2.NPV();
error = portfolioValue - referenceValue;
std::cout << std::setw(widths[0]) << std::left
<< "Replicating portfolio (26 dates)"
<< std::fixed
<< std::setw(widths[1]) << std::left << portfolioValue
<< std::setw(widths[2]) << std::left << error
<< std::endl;
portfolioValue = portfolio3.NPV();
error = portfolioValue - referenceValue;
std::cout << std::setw(widths[0]) << std::left
<< "Replicating portfolio (52 dates)"
<< std::fixed
<< std::setw(widths[1]) << std::left << portfolioValue
<< std::setw(widths[2]) << std::left << error
<< std::endl;
// Next, we change the underlying value so that the option is
// in the money.
std::cout << dblrule << std::endl;
std::cout << "Modified market conditions: in the money" << std::endl;
std::cout << dblrule << std::endl;
std::cout << std::setw(widths[0]) << std::left << "Option"
<< std::setw(widths[1]) << std::left << "NPV"
<< std::setw(widths[2]) << std::left << "Error"
<< std::endl;
std::cout << rule << std::endl;
underlying->setValue(90.0);
referenceValue = referenceOption.NPV();
std::cout << std::setw(widths[0]) << std::left
<< "Original barrier option"
<< std::fixed
<< std::setw(widths[1]) << std::left << referenceValue
<< std::setw(widths[2]) << std::left << "N/A"
<< std::endl;
portfolioValue = portfolio1.NPV();
error = portfolioValue - referenceValue;
std::cout << std::setw(widths[0]) << std::left
<< "Replicating portfolio (12 dates)"
<< std::fixed
<< std::setw(widths[1]) << std::left << portfolioValue
<< std::setw(widths[2]) << std::left << error
<< std::endl;
portfolioValue = portfolio2.NPV();
error = portfolioValue - referenceValue;
std::cout << std::setw(widths[0]) << std::left
<< "Replicating portfolio (26 dates)"
<< std::fixed
<< std::setw(widths[1]) << std::left << portfolioValue
<< std::setw(widths[2]) << std::left << error
<< std::endl;
portfolioValue = portfolio3.NPV();
error = portfolioValue - referenceValue;
std::cout << std::setw(widths[0]) << std::left
<< "Replicating portfolio (52 dates)"
<< std::fixed
<< std::setw(widths[1]) << std::left << portfolioValue
<< std::setw(widths[2]) << std::left << error
<< std::endl;
// Finally, a word of warning for those (shame on them) who
// run the example but do not read the code.
std::cout << dblrule << std::endl;
std::cout
<< std::endl
<< "The replication seems to be less robust when volatility and \n"
<< "risk-free rate are changed. Feel free to experiment with \n"
<< "the example and contribute a patch if you spot any errors."
<< std::endl;
return 0;
} catch (std::exception& e) {
std::cerr << e.what() << std::endl;
return 1;
} catch (...) {
std::cerr << "unknown error" << std::endl;
return 1;
}
}