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/* -*- mode: c++; tab-width: 4; indent-tabs-mode: nil; c-basic-offset: 4 -*- */
/*
Copyright (C) 2014 Bernd Lewerenz
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/exercise.hpp>
#include <ql/experimental/exoticoptions/continuousarithmeticasianvecerengine.hpp>
#include <ql/instruments/vanillaoption.hpp>
#include <ql/math/distributions/normaldistribution.hpp>
#include <ql/math/rounding.hpp>
#include <ql/methods/finitedifferences/dminus.hpp>
#include <ql/methods/finitedifferences/dplus.hpp>
#include <ql/methods/finitedifferences/dplusdminus.hpp>
#include <ql/methods/finitedifferences/tridiagonaloperator.hpp>
#include <ql/pricingengines/blackcalculator.hpp>
#include <ql/pricingengines/vanilla/analyticeuropeanengine.hpp>
#include <utility>
namespace QuantLib {
ContinuousArithmeticAsianVecerEngine::ContinuousArithmeticAsianVecerEngine(
ext::shared_ptr<GeneralizedBlackScholesProcess> process,
Handle<Quote> currentAverage,
Date startDate,
Size timeSteps,
Size assetSteps,
Real z_min,
Real z_max)
: process_(std::move(process)), currentAverage_(std::move(currentAverage)),
startDate_(startDate), z_min_(z_min), z_max_(z_max), timeSteps_(timeSteps),
assetSteps_(assetSteps) {
registerWith(process_);
registerWith(currentAverage_);
}
void ContinuousArithmeticAsianVecerEngine::calculate() const {
Real expectedAverage;
QL_REQUIRE(arguments_.averageType == Average::Arithmetic,
"not an Arithmetic average option");
QL_REQUIRE(arguments_.exercise->type() == Exercise::European,
"not an European Option");
DayCounter rfdc = process_->riskFreeRate()->dayCounter();
DayCounter divdc = process_->dividendYield()->dayCounter();
DayCounter voldc = process_->blackVolatility()->dayCounter();
Real S_0 = process_->stateVariable()->value();
// payoff
ext::shared_ptr<StrikedTypePayoff> payoff =
ext::dynamic_pointer_cast<StrikedTypePayoff>(arguments_.payoff);
QL_REQUIRE(payoff, "non-plain payoff given");
// original time to maturity
Date maturity = arguments_.exercise->lastDate();
Real X = payoff->strike();
QL_REQUIRE(z_min_<=0 && z_max_>=0,
"strike (0 for vecer fixed strike asian) not on Grid");
Volatility sigma =
process_->blackVolatility()->blackVol(maturity, X);
Rate r = process_->riskFreeRate()->
zeroRate(maturity, rfdc, Continuous, NoFrequency);
Rate q = process_->dividendYield()->
zeroRate(maturity, divdc, Continuous, NoFrequency);
Date today(Settings::instance().evaluationDate());
QL_REQUIRE(startDate_>=today,
"Seasoned Asian not yet implemented");
// Expiry in Years
Time T = rfdc.yearFraction(today,
arguments_.exercise->lastDate());
Time T1 = rfdc.yearFraction(today,
startDate_ ); // Average Begin
Time T2 = T; // Average End (In this version only Maturity...)
if ((T2 - T1) < 0.001) {
// its a vanilla option. Use vanilla engine
VanillaOption europeanOption(payoff, arguments_.exercise);
europeanOption.setPricingEngine(
ext::make_shared<AnalyticEuropeanEngine>(process_));
results_.value = europeanOption.NPV();
} else {
Real Theta = 0.5; // Mixed Scheme: 0.5 = Crank Nicolson
Real Z_0 = cont_strategy(0,T1,T2,q,r) - std::exp(-r*T) * X /S_0;
QL_REQUIRE(Z_0>=z_min_ && Z_0<=z_max_,
"spot not on grid");
Real h = (z_max_ - z_min_) / assetSteps_; // Space step size
Real k = T / timeSteps_; // Time Step size
Real sigma2 = sigma * sigma, vecerTerm;
Array SVec(assetSteps_+1),u_initial(assetSteps_+1),
u(assetSteps_+1),rhs(assetSteps_+1);
for (Natural i= 0; i<= SVec.size()-1;i++) {
SVec[i] = z_min_ + i * h; // Value of Underlying on the grid
}
// Begin gamma construction
TridiagonalOperator gammaOp = DPlusDMinus(assetSteps_+1,h);
Array upperD = gammaOp.upperDiagonal();
Array lowerD = gammaOp.lowerDiagonal();
Array Dia = gammaOp.diagonal();
// Construct Vecer operator
TridiagonalOperator explicit_part(gammaOp.size());
TridiagonalOperator implicit_part(gammaOp.size());
for (Natural i= 0; i<= SVec.size()-1;i++) {
u_initial[i] = std::max<Real>(SVec[i] , 0.0); // Call Payoff
}
u = u_initial;
// Start Time Loop
for (Natural j = 1; j<=timeSteps_;j++) {
if (Theta != 1.0) { // Explicit Part
for (Natural i = 1; i<= SVec.size()-2;i++) {
vecerTerm = SVec[i] - std::exp(-q * (T-(j-1)*k))
* cont_strategy(T-(j-1)*k,T1,T2,q,r);
gammaOp.setMidRow(i,
0.5 * sigma2 * vecerTerm * vecerTerm * lowerD[i-1],
0.5 * sigma2 * vecerTerm * vecerTerm * Dia[i],
0.5 * sigma2 * vecerTerm * vecerTerm * upperD[i]);
}
explicit_part = TridiagonalOperator::identity(gammaOp.size()) +
(1 - Theta) * k * gammaOp;
explicit_part.setFirstRow(1.0,0.0); // Apply before applying
explicit_part.setLastRow(-1.0,1.0); // Neumann BC
u = explicit_part.applyTo(u);
// Apply after applying (Neumann BC)
u[assetSteps_] = u[assetSteps_-1] + h;
u[0] = 0;
} // End Explicit Part
if (Theta != 0.0) { // Implicit Part
for (Natural i = 1; i<= SVec.size()-2;i++) {
vecerTerm = SVec[i] - std::exp(-q * (T-j*k)) *
cont_strategy(T-j*k,T1,T2,q,r);
gammaOp.setMidRow(i,
0.5 * sigma2 * vecerTerm * vecerTerm * lowerD[i-1],
0.5 * sigma2 * vecerTerm * vecerTerm * Dia[i],
0.5 * sigma2 * vecerTerm * vecerTerm * upperD[i]);
}
implicit_part = TridiagonalOperator::identity(gammaOp.size()) -
Theta * k * gammaOp;
// Apply before solving
implicit_part.setFirstRow(1.0,0.0);
implicit_part.setLastRow(-1.0,1.0);
rhs = u;
rhs[0] = 0; // Lower BC
rhs[assetSteps_] = h; // Upper BC (Neumann) Delta=1
u = implicit_part.solveFor(rhs);
} // End implicit Part
} // End Time Loop
DownRounding Rounding(0);
auto lowerI = Integer(Rounding((Z_0 - z_min_) / h));
// Interpolate solution
Real pv;
pv = u[lowerI] + (u[lowerI+1] - u[lowerI]) * (Z_0 - SVec[lowerI])/h;
results_.value = S_0 * pv;
if (payoff->optionType()==Option::Put) {
// Apply Call Put Parity for Asians
if (r == q) {
expectedAverage = S_0;
} else {
expectedAverage =
S_0 * (std::exp( (r-q) * T2) -
std::exp( (r-q) * T1)) / ((r-q) * (T2-T1));
}
Real asianForward = std::exp(-r * T2) * (expectedAverage - X);
results_.value = results_.value - asianForward;
}
}
}
// Replication of Average by holding this amount in Assets
Real ContinuousArithmeticAsianVecerEngine::cont_strategy(Time t,
Time T1,
Time T2,
Real v,
Real r) const {
Real const eps= 0.00001;
QL_REQUIRE(T1 <= T2, "Average Start must be before Average End");
if (std::fabs(t-T2) < eps) {
return 0.0;
} else {
if (t<T1) {
if (std::fabs(r-v) >= eps) {
return (std::exp(v * (t-T2)) *
(1 - std::exp((v-r) * (T2-T1) )) /
(( r - v) * (T2 - T1) ));
} else {
return std::exp(v*(t-T2));
} // end else v-r ==0
} else { // t<T1
if (std::fabs(r-v) >= eps) {
return std::exp(v * (t-T2)) *
(1 - std::exp( (v - r) * (T2-t) )) /
(( r - v) * (T2 - T1) );
} else {
return std::exp(v * (t-T2)) * (T2 - t) / (T2 - T1);
}
}
}
}
}
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