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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
Copyright (C) 2023 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
<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 <ql/any.hpp>
#include <ql/exercise.hpp>
#include <ql/math/functional.hpp>
#include <ql/math/comparison.hpp>
#include <ql/math/distributions/normaldistribution.hpp>
#include <ql/pricingengines/blackcalculator.hpp>
#include <ql/pricingengines/vanilla/bjerksundstenslandengine.hpp>
#include <utility>
#include <cmath>
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) * (cumNormalDist(d)
- std::pow((I / S), kappa) *
cumNormalDist(d - 2.0 * std::log(I/S) / std::sqrt(variance)));
}
Real phi_S(Real S, Real gamma, Real H, Real I, Real rT, Real bT, Real v) {
const Real lsh = std::log(S/H);
const Real lis = std::log(I/S);
const Real sv = std::sqrt(v);
return std::exp(bT*gamma - rT + ((-1 +gamma)*gamma*v)/2.)*((-(std::pow(I/S,2*(gamma + bT/v))/(std::exp(squared(2*bT - v + 2*gamma*v + 4*lis + 2*lsh)/(8.*v))*I))- 1/(std::exp(squared(2*bT - v + 2*gamma*v + 2*lsh)/(8.*v))*S))/(M_SQRT2*M_SQRTPI*sv) +(std::pow(I/S,2*(gamma + bT/v))*(2*bT + (-1 + 2*gamma)*v)*std::erfc((2*bT- v + 2*gamma*v + 4*lis + 2*lsh)/(2.*M_SQRT2*sv)))/(2.*I*v));
}
Real phi_SS(Real S, Real gamma, Real H, Real I, Real rT, Real bT, Real v) {
const Real lsh = std::log(S/H);
const Real lis = std::log(I/S);
const Real sv = std::sqrt(v);
const Real ex = std::exp(squared(2*bT - v + 2*gamma*v + 4*lis + 2*lsh)/(8.*v));
const Real ey = std::exp(squared(2*bT + (-1 + 2*gamma)*v + 2*lsh)/(8.*v));
return (std::exp(bT*gamma - rT + ((-1 +gamma)*gamma*v)/2.)*((M_SQRT2*I*v*sv)/ey +(2*M_SQRT2*std::pow(I/S,2*(gamma + bT/v))*S*sv*(2*bT +(-1 + 2*gamma)*v))/ex -2*std::sqrt(M_PI)*std::pow(I/S,2*(gamma + bT/v))*S*(bT +gamma*v)*(2*bT + (-1 + 2*gamma)*v)*std::erfc((2*bT - v + 2*gamma*v +4*lis + 2*lsh)/(2.*M_SQRT2*sv)) +(M_SQRT2*I*sv*(bT + (-0.5 + gamma)*v +lsh))/ey - (std::pow(I/S,2*(gamma + bT/v))*S*sv*(2*bT - 3*v + 2*gamma*v + 4*lis +2*lsh))/(M_SQRT2*ex)))/(2.*I*M_SQRTPI*squared(S*v));
}
Real phi_gamma(Real S, Real gamma, Real H, Real I, Real rT, Real bT, Real v) {
const Real lsh = std::log(S/H);
const Real lis = std::log(I/S);
const Real sv = std::sqrt(v);
return std::exp(bT*gamma - rT + ((-1 + gamma)*gamma*v)/2)*(((-std::exp(-squared(2*bT - v + 2*gamma*v +2*lsh)/(8*v)) + std::pow(I/S,-1 + 2*gamma +(2*bT)/v)/std::exp(squared(2*bT - v + 2*gamma*v + 4*lis +2*lsh)/(8*v)))*sv)/(M_SQRT2*M_SQRTPI) + ((2*bT+ (-1 + 2*gamma)*v)*std::erfc((2*bT + (-1 + 2*gamma)*v +2*lsh)/(2.*M_SQRT2*sv)))/4. -(std::pow(I/S,-1 + 2*gamma + (2*bT)/v)*std::erfc((2*bT - v + 2*gamma*v + 4*lis +2*lsh)/(2.*M_SQRT2*sv))*(2*bT + (-1 + 2*gamma)*v + 4*lis))/4.);
}
Real phi_H(Real S, Real gamma, Real H, Real I, Real rT, Real bT, Real v) {
const Real lsh = std::log(S/H);
return (std::exp(bT*gamma - rT + ((-1 + gamma)*gamma*v)/2.)*(I/std::exp(squared(2*bT - v + 2*gamma*v + 2*lsh)/(8.*v))- (std::pow(I/S,2*(gamma + bT/v))*S)/std::exp(squared(2*bT - v + 2*gamma*v + 4*std::log(I/S) + 2*lsh)/(8.*v))))/(H*I*std::sqrt(2*M_PI)*std::sqrt(v));
}
Real phi_I(Real S, Real gamma, Real H, Real I, Real rT, Real bT, Real v) {
const Real lsh = std::log(S/H);
const Real lis = std::log(I/S);
const Real sv = std::sqrt(v);
return (std::exp(bT*gamma - rT + ((-1 + gamma)*gamma*v)/2.)*std::pow(I/S,2*(gamma + bT/v))*S*((2*std::sqrt(2/M_PI))/(std::exp(squared(2*bT - v + 2*gamma*v + 4*lis + 2*lsh)/(8.*v))*sv) + (1 - 2*gamma - (2*bT)/v)*std::erfc((2*bT - v + 2*gamma*v + 4*lis +2*lsh)/(2.*M_SQRT2*sv))))/(2.*I*I);
}
Real phi_rt(Real S, Real gamma, Real H, Real I, Real rT, Real bT, Real v) {
const Real lsh = std::log(S/H);
return (std::exp(bT*gamma - rT + ((-1 + gamma)*gamma*v)/2.)*(-(I*std::erfc((2*bT- v + 2*gamma*v + 2*lsh)/(2.*std::sqrt(2*v)))) +std::pow(I/S,2*(gamma + bT/v))*S*std::erfc((2*bT - v + 2*gamma*v + 4*std::log(I/S) +2*lsh)/(2.*std::sqrt(2*v)))))/(2.*I);
}
Real phi_bt(Real S, Real gamma, Real H, Real I, Real rT, Real bT, Real v) {
const Real lsh = std::log(S/H);
const Real lis = std::log(I/S);
const Real sv = std::sqrt(v);
return (std::exp(bT*gamma - rT + ((-1 + gamma)*gamma*v)/2.)*(M_SQRT2*(-(I/std::exp(squared(2*bT - v +2*gamma*v + 2*lsh)/(8.*v))) + (std::pow(I/S,2*(gamma +bT/v))*S)/std::exp(squared(2*bT - v + 2*gamma*v + 4*lis +2*lsh)/(8.*v)))*sv + gamma*I*std::sqrt(M_PI)*v*std::erfc((2*bT - v + 2*gamma*v +2*lsh)/(2.*M_SQRT2*sv)) - M_SQRTPI*std::pow(I/S,2*(gamma + bT/v))*S*std::erfc((2*bT - v +2*gamma*v + 4*lis + 2*lsh)/(2.*M_SQRT2*sv))*(gamma*v +2*lis)))/(2.*I*std::sqrt(M_PI)*v);
}
Real phi_v(Real S, Real gamma, Real H, Real I, Real rT, Real bT, Real v) {
const Real lsh = std::log(S/H);
const Real lis = std::log(I/S);
const Real sv = std::sqrt(v);
const Real er = std::erfc((2*bT - v + 2*gamma*v + 4*lis + 2*lsh)/(2.*M_SQRT2*sv));
return (std::exp(bT*gamma - rT + ((-1 + gamma)*gamma*v)/2.)*(((-1 +gamma)*gamma*(I*std::erfc((2*bT - v + 2*gamma*v + 2*lsh)/(2.*M_SQRT2*sv)) -std::pow(I/S,2*(gamma + bT/v))*S*er))/(2.*I) +(2*bT*std::pow(I/S,-1 + 2*gamma + (2*bT)/v)*er*lis)/(v*v)+ (2*bT + v - 2*gamma*v + 2*lsh)/(2.*std::exp(std::pow(2*bT + (-1 + 2*gamma)*v +2*lsh,2)/(8.*v))*M_SQRT2*M_SQRTPI*v*sv) -(std::pow(I/S,-1 + 2*gamma + (2*bT)/v)*(2*bT + v - 2*gamma*v +4*lis + 2*lsh))/(2.*std::exp(squared(2*bT - v + 2*gamma*v + 4*lis + 2*lsh)/(8.*v))*M_SQRT2*M_SQRTPI*v*sv)))/2.;
}
}
BjerksundStenslandApproximationEngine::BjerksundStenslandApproximationEngine(
ext::shared_ptr<GeneralizedBlackScholesProcess> process)
: process_(std::move(process)) {
registerWith(process_);
}
OneAssetOption::results
BjerksundStenslandApproximationEngine::europeanCallResults(
Real S, Real X, Real rfD, Real dD, Real variance) const {
OneAssetOption::results results;
const Real forwardPrice = S * dD/rfD;
const BlackCalculator black(
Option::Call, X, forwardPrice, std::sqrt(variance), rfD);
results.value = black.value();
results.delta = black.delta(S);
results.gamma = black.gamma(S);
const DayCounter rfdc = process_->riskFreeRate()->dayCounter();
const DayCounter divdc = process_->dividendYield()->dayCounter();
const 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(S, t);
results.thetaPerDay = black.thetaPerDay(S, t);
results.strikeSensitivity = black.strikeSensitivity();
results.additionalResults["strikeGamma"] = Real(results.gamma*squared(S/X));
results.additionalResults["exerciseType"] = std::string("European");
return results;
}
OneAssetOption::results
BjerksundStenslandApproximationEngine::immediateExercise(Real S, Real X) const {
OneAssetOption::results results;
results.value = std::max(0.0, S - X);
results.delta = (S >= X)? 1.0 : 0.0;
results.gamma = 0.0;
results.rho = 0.0;
results.dividendRho = 0.0;
results.vega = 0.0;
results.theta = 0.0;
results.thetaPerDay = 0.0;
results.strikeSensitivity = -results.delta;
results.additionalResults["strikeGamma"] = Real(0.0);
results.additionalResults["exerciseType"] = std::string("Immediate");
return results;
}
OneAssetOption::results
BjerksundStenslandApproximationEngine::americanCallApproximation(
Real S, Real X, Real rfD, Real dD, Real variance) const {
const OneAssetOption::results europeanResults
= europeanCallResults(S, X, rfD, dD, variance);
OneAssetOption::results results;
const Real bT = std::log(dD/rfD);
const Real rT = std::log(1.0/rfD);
const Real beta = (0.5 - bT/variance) +
std::sqrt(squared(bT/variance - 0.5) + 2.0 * rT/variance);
const Real BInfinity = beta / (beta - 1.0) * X;
const Real B0 = (bT == rT) ? X : std::max(X, rT / (rT - bT) * X);
const Real ht = -(bT + 2.0*std::sqrt(variance)) * B0 / (BInfinity - B0);
const Real I = B0 + (BInfinity - B0) * (1 - std::exp(ht));
const Real fwd = S * dD/rfD;
const Real q = std::log(I/fwd)/std::sqrt(variance);
if (S >= I) {
results = immediateExercise(S, X);
}
else if (q > 12.5) {
// We have a run away exercise boundary. It is numerically
// more accurate to use the Greeks of the European engine.
results = europeanResults;
}
else {
const Real phi_S_beta_I_I_rT_bT_v
= phi(S, beta, I, I, rT, bT, variance);
const Real phi_S_1_I_I_rT_bT_v
= phi(S, 1.0, I, I, rT, bT, variance);
const Real phi_S_1_X_I_rT_bT_V
= phi(S, 1.0, X, I, rT, bT, variance);
results.value = (I - X) * std::pow(S/I, beta)
*(1 - phi_S_beta_I_I_rT_bT_v)
+ S * phi_S_1_I_I_rT_bT_v
- S * phi_S_1_X_I_rT_bT_V
- X * phi(S, 0.0, I, I, rT, bT, variance)
+ X * phi(S, 0.0, X, I, rT, bT, variance);
const Real phi_S_S_beta_I_I_rT_bT_v
= phi_S(S, beta, I, I, rT, bT, variance);
const Real phi_S_S_1_I_I_rT_bT_v
= phi_S(S, 1.0, I, I, rT, bT, variance);
const Real phi_S_S_1_X_I_rT_bT_v
= phi_S(S, 1.0, X, I, rT, bT, variance);
results.delta = (I - X) * std::pow(S/I, beta-1)*beta/I
* (1 - phi_S_beta_I_I_rT_bT_v)
- (I - X) * std::pow(S/I, beta)
* phi_S_S_beta_I_I_rT_bT_v
+ phi_S_1_I_I_rT_bT_v
+ S*phi_S_S_1_I_I_rT_bT_v
- phi_S_1_X_I_rT_bT_V
- S*phi_S_S_1_X_I_rT_bT_v
- X*phi_S(S, 0.0, I, I, rT, bT, variance)
+ X*phi_S(S, 0.0, X, I, rT, bT, variance);
const Date refDate = process_->riskFreeRate()->referenceDate();
const Date exerciseDate = arguments_.exercise->lastDate();
const DayCounter qdc = process_->dividendYield()->dayCounter();
const Time tq = qdc.yearFraction(refDate, exerciseDate);
const Real betaDq = tq*(1/variance
- 1/(2*std::sqrt(squared(bT/variance - 0.5) + 2.0 * rT/variance))
* 2*(bT/variance-0.5)/variance);
const Real BInfinityDq = -X/squared(beta-1.0)*betaDq;
const Real B0Dq = (dD <= rfD) ? Real(0.0)
: Real(X*std::log(rfD)/squared(std::log(dD))*tq);
const Real htDq = tq * B0 / (BInfinity - B0)
- (bT + 2.0*std::sqrt(variance))
*(B0Dq*(BInfinity - B0) - B0*(BInfinityDq - B0Dq))
/squared(BInfinity - B0);
const Real IDq = B0Dq + (BInfinityDq - B0Dq) * (1 - std::exp(ht))
- (BInfinity - B0) * std::exp(ht)*htDq;
const Real phi_H_S_beta_I_I_rT_bT_v
= phi_H(S, beta, I, I, rT, bT, variance);
const Real phi_I_S_beta_I_I_rT_bT_v
= phi_I(S, beta, I, I, rT, bT, variance);
const Real phi_gamma_S_beta_I_I_rT_bT_v
= phi_gamma(S, beta, I, I, rT, bT, variance);
const Real phi_bt_S_beta_I_I_rT_bT_v
= phi_bt(S, beta, I, I, rT, bT, variance);
const Real phi_H_S_1_I_I_rT_bT_v
= phi_H(S, 1.0, I, I, rT, bT, variance);
const Real phi_I_S_1_I_I_rT_bT_v
= phi_I(S, 1.0, I, I, rT, bT, variance);
const Real phi_bt_S_1_I_I_rT_bT_v
= phi_bt(S, 1.0, I, I, rT, bT, variance);
const Real phi_I_S_1_X_I_rT_bT_v
= phi_I(S, 1.0, X, I, rT, bT, variance);
const Real phi_bt_S_1_X_I_rT_bT_v
= phi_bt(S, 1.0, X, I, rT, bT, variance);
const Real phi_H_S_0_I_I_rT_bT_v
= phi_H(S, 0.0, I, I, rT, bT, variance);
const Real phi_I_S_0_I_I_rT_bT_v
= phi_I(S, 0.0, I, I, rT, bT, variance);
const Real phi_bt_S_0_I_I_rT_bT_v
= phi_bt(S, 0.0, I, I, rT, bT, variance);
const Real phi_I_S_0_X_I_rT_bT_v
= phi_I(S, 0.0, X, I, rT, bT, variance);
const Real phi_bt_S_0_X_I_rT_bT_v
= phi_bt(S, 0.0, X, I, rT, bT, variance);
results.dividendRho =
(IDq*std::pow(S/I, beta)
+ (I-X)*std::pow(S/I, beta)*(betaDq*std::log(S/I) - beta*1/I*IDq))
* (1 - phi_S_beta_I_I_rT_bT_v)
- (I - X) * std::pow(S/I, beta)
*( phi_H_S_beta_I_I_rT_bT_v*IDq
+phi_I_S_beta_I_I_rT_bT_v*IDq
+phi_gamma_S_beta_I_I_rT_bT_v*betaDq
-phi_bt_S_beta_I_I_rT_bT_v*tq)
+ S*( phi_H_S_1_I_I_rT_bT_v*IDq
+ phi_I_S_1_I_I_rT_bT_v*IDq
- phi_bt_S_1_I_I_rT_bT_v*tq)
- S*( phi_I_S_1_X_I_rT_bT_v*IDq
- phi_bt_S_1_X_I_rT_bT_v*tq)
- X*( phi_H_S_0_I_I_rT_bT_v*IDq
+ phi_I_S_0_I_I_rT_bT_v*IDq
- phi_bt_S_0_I_I_rT_bT_v*tq)
+ X*( phi_I_S_0_X_I_rT_bT_v*IDq
- phi_bt_S_0_X_I_rT_bT_v*tq);
const DayCounter rdc = process_->riskFreeRate()->dayCounter();
const Time tr = rdc.yearFraction(refDate, exerciseDate);
const Real betaDr = tr*(-1/variance
+ 1/(2*std::sqrt(squared(bT/variance - 0.5) + 2.0 * rT/variance))
* 2*((bT/variance-0.5)/variance + 1/variance));
const Real BInfinityDr = -X/squared(beta-1.0)*betaDr;
const Real B0Dr = (dD <= rfD) ? Real(0) : Real(-X*tr/std::log(dD));
const Real htDr = -tr * B0 / (BInfinity - B0)
- (bT + 2.0*std::sqrt(variance))
*(B0Dr*(BInfinity - B0) - B0*(BInfinityDr - B0Dr))
/squared(BInfinity - B0);
const Real IDr = B0Dr + (BInfinityDr - B0Dr) * (1 - std::exp(ht))
- (BInfinity - B0) * std::exp(ht)*htDr;
results.rho =
(IDr*std::pow(S/I, beta)
+ (I-X)*std::pow(S/I, beta)*(betaDr*std::log(S/I) - beta/I*IDr))
* (1 - phi_S_beta_I_I_rT_bT_v)
- (I - X) * std::pow(S/I, beta)
*( phi_H_S_beta_I_I_rT_bT_v*IDr
+ phi_I_S_beta_I_I_rT_bT_v*IDr
+ phi_gamma_S_beta_I_I_rT_bT_v*betaDr
+ phi_rt(S, beta, I, I, rT, bT, variance)*tr
+ phi_bt_S_beta_I_I_rT_bT_v*tr)
+ S*( phi_H_S_1_I_I_rT_bT_v*IDr
+ phi_I_S_1_I_I_rT_bT_v*IDr
+ phi_rt(S, 1.0, I, I, rT, bT, variance)*tr
+ phi_bt_S_1_I_I_rT_bT_v*tr)
- S*( phi_I_S_1_X_I_rT_bT_v*IDr
+ phi_rt(S, 1.0, X, I, rT, bT, variance)*tr
+ phi_bt_S_1_X_I_rT_bT_v*tr)
- X*( phi_H_S_0_I_I_rT_bT_v*IDr
+ phi_I_S_0_I_I_rT_bT_v*IDr
+ phi_rt(S, 0.0, I, I, rT, bT, variance)*tr
+ phi_bt_S_0_I_I_rT_bT_v*tr)
+ X*( phi_I_S_0_X_I_rT_bT_v*IDr
+ phi_rt(S, 0.0, X, I, rT, bT, variance)*tr
+ phi_bt_S_0_X_I_rT_bT_v*tr);
const Real beta = (0.5 - bT/variance) +
std::sqrt(squared(bT/variance - 0.5) + 2.0 * rT/variance);
const DayCounter vdc = process_->blackVolatility()->dayCounter();
const Time tv = vdc.yearFraction(refDate, exerciseDate);
const Real varianceDv = 2*std::sqrt(variance*tv);
const Real betaDv = bT/squared(variance)*varianceDv +
- 1/(2*std::sqrt(squared(bT/variance - 0.5) + 2.0 * rT/variance))
*( 2*(bT/variance - 0.5)*bT*varianceDv/squared(variance)
+2*rT/squared(variance)*varianceDv );
const Real BInfinityDv = -X/squared(beta-1.0)*betaDv;
const Real htDv = -1/std::sqrt(variance)*varianceDv*B0/(BInfinity-B0)
+ (bT + 2*std::sqrt(variance))*B0/squared(BInfinity-B0)*BInfinityDv;
const Real IDv = BInfinityDv*(1-std::exp(ht))
- (BInfinity-B0)*std::exp(ht)*htDv;
results.vega =
(IDv*std::pow(S/I, beta)
+ (I-X)*std::pow(S/I, beta)*(betaDv*std::log(S/I) - beta/I*IDv))
* (1 - phi_S_beta_I_I_rT_bT_v)
- (I - X) * std::pow(S/I, beta)
*( phi_H_S_beta_I_I_rT_bT_v*IDv
+ phi_I_S_beta_I_I_rT_bT_v*IDv
+ phi_gamma_S_beta_I_I_rT_bT_v*betaDv
+ phi_v(S, beta, I, I, rT, bT, variance)*varianceDv)
+ S*( phi_H_S_1_I_I_rT_bT_v*IDv
+ phi_I_S_1_I_I_rT_bT_v*IDv
+ phi_v(S, 1.0, I, I, rT, bT, variance)*varianceDv)
- S*( phi_I_S_1_X_I_rT_bT_v*IDv
+ phi_v(S, 1.0, X, I, rT, bT, variance)*varianceDv)
- X*( phi_H_S_0_I_I_rT_bT_v*IDv
+ phi_I_S_0_I_I_rT_bT_v*IDv
+ phi_v(S, 0.0, I, I, rT, bT, variance)*varianceDv)
+ X*( phi_I_S_0_X_I_rT_bT_v*IDv
+ phi_v(S, 0.0, X, I, rT, bT, variance)*varianceDv);
results.gamma =
(I - X) * std::pow(S/I, beta-2)*beta*(beta-1)/squared(I)
* (1 - phi_S_beta_I_I_rT_bT_v)
- 2*(I - X) * std::pow(S/I, beta-1)*beta/I
*phi_S_S_beta_I_I_rT_bT_v
- (I - X) * std::pow(S/I, beta)
* phi_SS(S, beta, I, I, rT, bT, variance)
+ 2*phi_S_S_1_I_I_rT_bT_v
+ S*phi_SS(S, 1.0, I, I, rT, bT, variance)
- 2*phi_S_S_1_X_I_rT_bT_v
- S*phi_SS(S, 1.0, X, I, rT, bT, variance)
- X*phi_SS(S, 0.0, I, I, rT, bT, variance)
+ X*phi_SS(S, 0.0, X, I, rT, bT, variance);
const Volatility vol = std::sqrt(variance/tv);
const Date tomorrow = refDate + Period(1, Days);
const Time dtq = qdc.yearFraction(refDate, exerciseDate)
- qdc.yearFraction(tomorrow, exerciseDate);
const Time dtr = rdc.yearFraction(refDate, exerciseDate)
- rdc.yearFraction(tomorrow, exerciseDate);
const Time dtv = vdc.yearFraction(refDate, exerciseDate)
- vdc.yearFraction(tomorrow, exerciseDate);
results.thetaPerDay = -(0.5*results.vega*vol/tv*dtv
+ results.rho*rT/(tr*tr)*dtr + results.dividendRho*(rT-bT)/(tq*tq)*dtq);
results.theta = 365*results.thetaPerDay;
results.strikeSensitivity = results.value/X - S/X*results.delta;
results.additionalResults["strikeGamma"] = Real(results.gamma*squared(S/X));
results.additionalResults["exerciseType"] = std::string("American");
}
// check if European engine gives higher NPV
if (results.value < europeanResults.value) {
results = europeanResults;
}
return results;
}
void BjerksundStenslandApproximationEngine::calculate() const {
QL_REQUIRE(arguments_.exercise->type() == Exercise::American,
"not an American Option");
ext::shared_ptr<AmericanExercise> ex =
ext::dynamic_pointer_cast<AmericanExercise>(arguments_.exercise);
QL_REQUIRE(ex, "non-American exercise given");
QL_REQUIRE(!ex->payoffAtExpiry(),
"payoff at expiry not handled");
ext::shared_ptr<PlainVanillaPayoff> payoff =
ext::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 symmetry
std::swap(spot, strike);
std::swap(riskFreeDiscount, dividendDiscount);
payoff = ext::make_shared<PlainVanillaPayoff>(
Option::Call, strike);
}
if (dividendDiscount > 1.0 && riskFreeDiscount > dividendDiscount)
QL_FAIL("double-boundary case r<q<0 for a call given");
if (dividendDiscount >= 1.0 && dividendDiscount >= riskFreeDiscount) {
results_ = europeanCallResults(
spot, strike, riskFreeDiscount, dividendDiscount, variance);
} else {
// early exercise can be optimal - use approximation
results_ = americanCallApproximation(
spot, strike, riskFreeDiscount, dividendDiscount, variance);
}
// check if immediate exercise gives higher NPV
if (results_.value < (spot - strike)*(1+10*QL_EPSILON) ) {
results_ = immediateExercise(spot, strike);
}
if (ext::dynamic_pointer_cast<PlainVanillaPayoff>(arguments_.payoff)
->optionType() == Option::Put) {
std::swap(results_.delta, results_.strikeSensitivity);
Real tmp = results_.gamma;
results_.gamma =
ext::any_cast<Real>(results_.additionalResults["strikeGamma"]);
results_.additionalResults["strikeGamma"] = tmp;
std::swap(results_.rho, results_.dividendRho);
Time tr = process_->riskFreeRate()->dayCounter().yearFraction(
process_->riskFreeRate()->referenceDate(),
arguments_.exercise->lastDate());
Time tq = process_->dividendYield()->dayCounter().yearFraction(
process_->dividendYield()->referenceDate(),
arguments_.exercise->lastDate());
results_.rho *= tr/tq;
results_.dividendRho *= tq/tr;
}
}
}
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