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/* -*- mode: c++; tab-width: 4; indent-tabs-mode: nil; c-basic-offset: 4 -*- */
/*
Copyright (C) 2004 Neil Firth
Copyright (C) 2007 StatPro Italia srl
Copyright (C) 2013 Fabien Le Floc'h
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/juquadraticengine.hpp>
#include <ql/pricingengines/vanilla/baroneadesiwhaleyengine.hpp>
#include <ql/pricingengines/blackcalculator.hpp>
#include <ql/pricingengines/blackformula.hpp>
#include <ql/math/distributions/normaldistribution.hpp>
#include <ql/exercise.hpp>
namespace QuantLib {
/* An Approximate Formula for Pricing American Options
Journal of Derivatives Winter 1999
Ju, N.
*/
JuQuadraticApproximationEngine::JuQuadraticApproximationEngine(
const boost::shared_ptr<GeneralizedBlackScholesProcess>& process)
: process_(process) {
registerWith(process_);
}
void JuQuadraticApproximationEngine::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<StrikedTypePayoff> payoff =
boost::dynamic_pointer_cast<StrikedTypePayoff>(arguments_.payoff);
QL_REQUIRE(payoff, "non-striked 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 forwardPrice = spot * dividendDiscount / riskFreeDiscount;
BlackCalculator black(payoff, forwardPrice,
std::sqrt(variance), riskFreeDiscount);
if (dividendDiscount>=1.0 && payoff->optionType()==Option::Call) {
// early exercise never optimal
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
CumulativeNormalDistribution cumNormalDist;
NormalDistribution normalDist;
Real tolerance = 1e-6;
Real Sk = BaroneAdesiWhaleyApproximationEngine::criticalPrice(
payoff, riskFreeDiscount, dividendDiscount, variance,
tolerance);
Real forwardSk = Sk * dividendDiscount / riskFreeDiscount;
Real alpha = -2.0*std::log(riskFreeDiscount)/(variance);
Real beta = 2.0*std::log(dividendDiscount/riskFreeDiscount)/
(variance);
Real h = 1 - riskFreeDiscount;
Real phi;
switch (payoff->optionType()) {
case Option::Call:
phi = 1;
break;
case Option::Put:
phi = -1;
break;
default:
QL_FAIL("unknown option type");
}
//it can throw: to be fixed
// FLOATING_POINT_EXCEPTION
Real temp_root = std::sqrt ((beta-1)*(beta-1) + (4*alpha)/h);
Real lambda = (-(beta-1) + phi * temp_root) / 2;
Real lambda_prime = - phi * alpha / (h*h * temp_root);
Real black_Sk = blackFormula(payoff->optionType(), payoff->strike(),
forwardSk, std::sqrt(variance)) * riskFreeDiscount;
Real hA = phi * (Sk - payoff->strike()) - black_Sk;
Real d1_Sk = (std::log(forwardSk/payoff->strike()) + 0.5*variance)
/std::sqrt(variance);
Real d2_Sk = d1_Sk - std::sqrt(variance);
Real part1 = forwardSk * normalDist(d1_Sk) /
(alpha * std::sqrt(variance));
Real part2 = - phi * forwardSk * cumNormalDist(phi * d1_Sk) *
std::log(dividendDiscount) / std::log(riskFreeDiscount);
Real part3 = + phi * payoff->strike() * cumNormalDist(phi * d2_Sk);
Real V_E_h = part1 + part2 + part3;
Real b = (1-h) * alpha * lambda_prime / (2*(2*lambda + beta - 1));
Real c = - ((1 - h) * alpha / (2 * lambda + beta - 1)) *
(V_E_h / (hA) + 1 / h + lambda_prime / (2*lambda + beta - 1));
Real temp_spot_ratio = std::log(spot / Sk);
Real chi = temp_spot_ratio * (b * temp_spot_ratio + c);
if (phi*(Sk-spot) > 0) {
results_.value = black.value() +
hA * std::pow((spot/Sk), lambda) / (1 - chi);
Real temp_chi_prime = (2 * b / spot) * std::log(spot/Sk);
Real chi_prime = temp_chi_prime + c / spot;
Real chi_double_prime = 2*b/(spot*spot)
- temp_chi_prime / spot - c / (spot*spot);
Real d1_S = (std::log(forwardPrice/payoff->strike()) + 0.5*variance)
/ std::sqrt(variance);
//There is a typo in the original paper from Ju-Zhong
//the first term is the Black-Scholes delta/gamma.
results_.delta = phi * dividendDiscount * cumNormalDist (phi * d1_S)
+ (lambda / (spot * (1 - chi)) + chi_prime / ((1 - chi)*(1 - chi))) *
(phi * (Sk - payoff->strike()) - black_Sk) * std::pow((spot/Sk), lambda);
results_.gamma = dividendDiscount * normalDist (phi*d1_S)
/ (spot * std::sqrt(variance))
+ (2 * lambda * chi_prime / (spot * (1 - chi) * (1 - chi))
+ 2 * chi_prime * chi_prime / ((1 - chi) * (1 - chi) * (1 - chi))
+ chi_double_prime / ((1 - chi) * (1 - chi))
+ lambda * (lambda - 1) / (spot * spot * (1 - chi)))
* (phi * (Sk - payoff->strike()) - black_Sk)
* std::pow((spot/Sk), lambda);
} else {
results_.value = phi * (spot - payoff->strike());
results_.delta = phi;
results_.gamma = 0;
}
} // end of "early exercise can be optimal"
}
}
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