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/* -*- mode: c++; tab-width: 4; indent-tabs-mode: nil; c-basic-offset: 4 -*- */
/*
Copyright (C) 2003 Ferdinando Ametrano
Copyright (C) 2007 StatPro Italia srl
This file is part of QuantLib, a free-software/open-source library
for financial quantitative analysts and developers - http://quantlib.org/
QuantLib is free software: you can redistribute it and/or modify it
under the terms of the QuantLib license. You should have received a
copy of the license along with this program; if not, please email
<quantlib-dev@lists.sf.net>. The license is also available online at
<http://quantlib.org/license.shtml>.
This program is distributed in the hope that it will be useful, but WITHOUT
ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or FITNESS
FOR A PARTICULAR PURPOSE. See the license for more details.
*/
#include <ql/pricingengines/vanilla/bjerksundstenslandengine.hpp>
#include <ql/pricingengines/blackcalculator.hpp>
#include <ql/math/distributions/normaldistribution.hpp>
#include <ql/exercise.hpp>
namespace QuantLib {
namespace {
CumulativeNormalDistribution cumNormalDist;
Real phi(Real S, Real gamma, Real H, Real I,
Real rT, Real bT, Real variance) {
Real lambda = (-rT + gamma * bT + 0.5 * gamma * (gamma - 1.0)
* variance);
Real d = -(std::log(S / H) + (bT + (gamma - 0.5) * variance) )
/ std::sqrt(variance);
Real kappa = 2.0 * bT / variance + (2.0 * gamma - 1.0);
return std::exp(lambda) * std::pow(S, gamma) * (cumNormalDist(d)
- std::pow((I / S), kappa) *
cumNormalDist(d - 2.0 * std::log(I/S) / std::sqrt(variance)));
}
Real americanCallApproximation(Real S, Real X,
Real rfD, Real dD, Real variance) {
Real bT = std::log(dD/rfD);
Real rT = std::log(1.0/rfD);
Real beta = (0.5 - bT/variance) +
std::sqrt(std::pow((bT/variance - 0.5), Real(2.0))
+ 2.0 * rT/variance);
Real BInfinity = beta / (beta - 1.0) * X;
// Real B0 = std::max(X, std::log(rfD) / std::log(dD) * X);
Real B0 = std::max(X, rT / (rT - bT) * X);
Real ht = -(bT + 2.0*std::sqrt(variance)) * B0 / (BInfinity - B0);
// investigate what happen to I for dD->0.0
Real I = B0 + (BInfinity - B0) * (1 - std::exp(ht));
QL_REQUIRE(I >= X,
"Bjerksund-Stensland approximation not applicable "
"to this set of parameters");
if (S >= I) {
return S - X;
} else {
// investigate what happen to alpha for dD->0.0
Real alpha = (I - X) * std::pow(I, (-beta));
return alpha * std::pow(S, beta)
- alpha * phi(S, beta, I, I, rT, bT, variance)
+ phi(S, 1.0, I, I, rT, bT, variance)
- phi(S, 1.0, X, I, rT, bT, variance)
- X * phi(S, 0.0, I, I, rT, bT, variance)
+ X * phi(S, 0.0, X, I, rT, bT, variance);
}
}
}
BjerksundStenslandApproximationEngine::
BjerksundStenslandApproximationEngine(
const boost::shared_ptr<GeneralizedBlackScholesProcess>& process)
: process_(process) {
registerWith(process_);
}
void BjerksundStenslandApproximationEngine::calculate() const {
QL_REQUIRE(arguments_.exercise->type() == Exercise::American,
"not an American Option");
boost::shared_ptr<AmericanExercise> ex =
boost::dynamic_pointer_cast<AmericanExercise>(arguments_.exercise);
QL_REQUIRE(ex, "non-American exercise given");
QL_REQUIRE(!ex->payoffAtExpiry(),
"payoff at expiry not handled");
boost::shared_ptr<PlainVanillaPayoff> payoff =
boost::dynamic_pointer_cast<PlainVanillaPayoff>(arguments_.payoff);
QL_REQUIRE(payoff, "non-plain payoff given");
Real variance =
process_->blackVolatility()->blackVariance(ex->lastDate(),
payoff->strike());
DiscountFactor dividendDiscount =
process_->dividendYield()->discount(ex->lastDate());
DiscountFactor riskFreeDiscount =
process_->riskFreeRate()->discount(ex->lastDate());
Real spot = process_->stateVariable()->value();
QL_REQUIRE(spot > 0.0, "negative or null underlying given");
Real strike = payoff->strike();
if (payoff->optionType()==Option::Put) {
// use put-call simmetry
std::swap(spot, strike);
std::swap(riskFreeDiscount, dividendDiscount);
payoff = boost::shared_ptr<PlainVanillaPayoff>(
new PlainVanillaPayoff(Option::Call, strike));
}
if (dividendDiscount>=1.0) {
// early exercise is never optimal - use Black formula
Real forwardPrice = spot * dividendDiscount / riskFreeDiscount;
BlackCalculator black(payoff, forwardPrice, std::sqrt(variance),
riskFreeDiscount);
results_.value = black.value();
results_.delta = black.delta(spot);
results_.deltaForward = black.deltaForward();
results_.elasticity = black.elasticity(spot);
results_.gamma = black.gamma(spot);
DayCounter rfdc = process_->riskFreeRate()->dayCounter();
DayCounter divdc = process_->dividendYield()->dayCounter();
DayCounter voldc = process_->blackVolatility()->dayCounter();
Time t =
rfdc.yearFraction(process_->riskFreeRate()->referenceDate(),
arguments_.exercise->lastDate());
results_.rho = black.rho(t);
t = divdc.yearFraction(process_->dividendYield()->referenceDate(),
arguments_.exercise->lastDate());
results_.dividendRho = black.dividendRho(t);
t = voldc.yearFraction(process_->blackVolatility()->referenceDate(),
arguments_.exercise->lastDate());
results_.vega = black.vega(t);
results_.theta = black.theta(spot, t);
results_.thetaPerDay = black.thetaPerDay(spot, t);
results_.strikeSensitivity = black.strikeSensitivity();
results_.itmCashProbability = black.itmCashProbability();
} else {
// early exercise can be optimal - use approximation
results_.value = americanCallApproximation(spot,
strike,
riskFreeDiscount,
dividendDiscount,
variance);
}
}
}
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