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|
/* -*- mode: c++; tab-width: 4; indent-tabs-mode: nil; c-basic-offset: 4 -*- */
/*
Copyright (C) 2003, 2004, 2005, 2006 Ferdinando Ametrano
Copyright (C) 2006 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
<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/pricingengines/blackcalculator.hpp>
#include <ql/math/distributions/normaldistribution.hpp>
#include <ql/math/comparison.hpp>
namespace QuantLib {
class BlackCalculator::Calculator : public AcyclicVisitor,
public Visitor<Payoff>,
public Visitor<PlainVanillaPayoff>,
public Visitor<CashOrNothingPayoff>,
public Visitor<AssetOrNothingPayoff>,
public Visitor<GapPayoff> {
private:
BlackCalculator& black_;
public:
explicit Calculator(BlackCalculator& black) : black_(black) {}
void visit(Payoff&) override;
void visit(PlainVanillaPayoff&) override;
void visit(CashOrNothingPayoff&) override;
void visit(AssetOrNothingPayoff&) override;
void visit(GapPayoff&) override;
};
BlackCalculator::BlackCalculator(const ext::shared_ptr<StrikedTypePayoff>& p,
Real forward,
Real stdDev,
Real discount)
: strike_(p->strike()), forward_(forward), stdDev_(stdDev),
discount_(discount), variance_(stdDev*stdDev) {
initialize(p);
}
BlackCalculator::BlackCalculator(Option::Type optionType,
Real strike,
Real forward,
Real stdDev,
Real discount)
: strike_(strike), forward_(forward), stdDev_(stdDev),
discount_(discount), variance_(stdDev*stdDev) {
initialize(ext::shared_ptr<StrikedTypePayoff>(new
PlainVanillaPayoff(optionType, strike)));
}
void BlackCalculator::initialize(const ext::shared_ptr<StrikedTypePayoff>& p) {
QL_REQUIRE(strike_>=0.0,
"strike (" << strike_ << ") must be non-negative");
QL_REQUIRE(forward_>0.0,
"forward (" << forward_ << ") must be positive");
//QL_REQUIRE(displacement_>=0.0,
// "displacement (" << displacement_ << ") must be non-negative");
QL_REQUIRE(stdDev_>=0.0,
"stdDev (" << stdDev_ << ") must be non-negative");
QL_REQUIRE(discount_>0.0,
"discount (" << discount_ << ") must be positive");
if (stdDev_>=QL_EPSILON) {
if (close(strike_, 0.0)) {
d1_ = QL_MAX_REAL;
d2_ = QL_MAX_REAL;
cum_d1_ = 1.0;
cum_d2_ = 1.0;
n_d1_ = 0.0;
n_d2_ = 0.0;
} else {
d1_ = std::log(forward_/strike_)/stdDev_ + 0.5*stdDev_;
d2_ = d1_-stdDev_;
CumulativeNormalDistribution f;
cum_d1_ = f(d1_);
cum_d2_ = f(d2_);
n_d1_ = f.derivative(d1_);
n_d2_ = f.derivative(d2_);
}
} else {
if (close(forward_, strike_)) {
d1_ = 0;
d2_ = 0;
cum_d1_ = 0.5;
cum_d2_ = 0.5;
n_d1_ = M_SQRT_2 * M_1_SQRTPI;
n_d2_ = M_SQRT_2 * M_1_SQRTPI;
} else if (forward_>strike_) {
d1_ = QL_MAX_REAL;
d2_ = QL_MAX_REAL;
cum_d1_ = 1.0;
cum_d2_ = 1.0;
n_d1_ = 0.0;
n_d2_ = 0.0;
} else {
d1_ = QL_MIN_REAL;
d2_ = QL_MIN_REAL;
cum_d1_ = 0.0;
cum_d2_ = 0.0;
n_d1_ = 0.0;
n_d2_ = 0.0;
}
}
x_ = strike_;
DxDstrike_ = 1.0;
// the following one will probably disappear as soon as
// super-share will be properly handled
DxDs_ = 0.0;
// this part is always executed.
// in case of plain-vanilla payoffs, it is also the only part
// which is executed.
switch (p->optionType()) {
case Option::Call:
alpha_ = cum_d1_;// N(d1)
DalphaDd1_ = n_d1_;// n(d1)
beta_ = -cum_d2_;// -N(d2)
DbetaDd2_ = - n_d2_;// -n(d2)
break;
case Option::Put:
alpha_ = -1.0+cum_d1_;// -N(-d1)
DalphaDd1_ = n_d1_;// n( d1)
beta_ = 1.0-cum_d2_;// N(-d2)
DbetaDd2_ = - n_d2_;// -n( d2)
break;
default:
QL_FAIL("invalid option type");
}
// now dispatch on type.
Calculator calc(*this);
p->accept(calc);
}
void BlackCalculator::Calculator::visit(Payoff& p) {
QL_FAIL("unsupported payoff type: " << p.name());
}
void BlackCalculator::Calculator::visit(PlainVanillaPayoff&) {}
void BlackCalculator::Calculator::visit(CashOrNothingPayoff& payoff) {
black_.alpha_ = black_.DalphaDd1_ = 0.0;
black_.x_ = payoff.cashPayoff();
black_.DxDstrike_ = 0.0;
switch (payoff.optionType()) {
case Option::Call:
black_.beta_ = black_.cum_d2_;
black_.DbetaDd2_ = black_.n_d2_;
break;
case Option::Put:
black_.beta_ = 1.0-black_.cum_d2_;
black_.DbetaDd2_ = -black_.n_d2_;
break;
default:
QL_FAIL("invalid option type");
}
}
void BlackCalculator::Calculator::visit(AssetOrNothingPayoff& payoff) {
black_.beta_ = black_.DbetaDd2_ = 0.0;
switch (payoff.optionType()) {
case Option::Call:
black_.alpha_ = black_.cum_d1_;
black_.DalphaDd1_ = black_.n_d1_;
break;
case Option::Put:
black_.alpha_ = 1.0-black_.cum_d1_;
black_.DalphaDd1_ = -black_.n_d1_;
break;
default:
QL_FAIL("invalid option type");
}
}
void BlackCalculator::Calculator::visit(GapPayoff& payoff) {
black_.x_ = payoff.secondStrike();
black_.DxDstrike_ = 0.0;
}
Real BlackCalculator::value() const {
Real result = discount_ * (forward_ * alpha_ + x_ * beta_);
return result;
}
Real BlackCalculator::delta(Real spot) const {
QL_REQUIRE(spot > 0.0, "positive spot value required: " <<
spot << " not allowed");
// Handle zero volatility case
if (stdDev_ <= QL_EPSILON) {
// For zero volatility, delta is:
// ITM Call: 1.0, OTM Call: 0.0, ATM Call: 0.5
// ITM Put: -1.0, OTM Put: 0.0, ATM Put: -0.5
Real DforwardDs = forward_ / spot;
if (close(forward_, strike_)) {
// ATM case
if (alpha_ >= 0) { // Call
return discount_ * 0.5 * DforwardDs;
} else { // Put
return discount_ * (-0.5) * DforwardDs;
}
} else if (forward_ > strike_) {
// ITM Call, OTM Put
if (alpha_ >= 0) { // Call
return discount_ * 1.0 * DforwardDs;
} else { // Put
return 0.0;
}
} else {
// OTM Call, ITM Put
if (alpha_ >= 0) { // Call
return 0.0;
} else { // Put
return discount_ * (-1.0) * DforwardDs;
}
}
}
Real DforwardDs = forward_ / spot;
Real temp = stdDev_*spot;
Real DalphaDs = DalphaDd1_/temp;
Real DbetaDs = DbetaDd2_/temp;
Real temp2 = DalphaDs * forward_ + alpha_ * DforwardDs
+DbetaDs * x_ + beta_ * DxDs_;
return discount_ * temp2;
}
Real BlackCalculator::deltaForward() const {
// Handle zero volatility case
if (stdDev_ <= QL_EPSILON) {
// For zero volatility, forward delta is:
// ITM Call: 1.0, OTM Call: 0.0, ATM Call: 0.5
// ITM Put: -1.0, OTM Put: 0.0, ATM Put: -0.5
if (close(forward_, strike_)) {
// ATM case
if (alpha_ >= 0) { // Call
return discount_ * 0.5;
} else { // Put
return discount_ * (-0.5);
}
} else if (forward_ > strike_) {
// ITM Call, OTM Put
if (alpha_ >= 0) { // Call
return discount_ * 1.0;
} else { // Put
return 0.0;
}
} else {
// OTM Call, ITM Put
if (alpha_ >= 0) { // Call
return 0.0;
} else { // Put
return discount_ * (-1.0);
}
}
}
Real temp = stdDev_*forward_;
Real DalphaDforward = DalphaDd1_/temp;
Real DbetaDforward = DbetaDd2_/temp;
Real temp2 = DalphaDforward * forward_ + alpha_
+DbetaDforward * x_; // DXDforward = 0.0
return discount_ * temp2;
}
Real BlackCalculator::elasticity(Real spot) const {
Real val = value();
Real del = delta(spot);
if (val>QL_EPSILON)
return del/val*spot;
else if (std::fabs(del)<QL_EPSILON)
return 0.0;
else if (del>0.0)
return QL_MAX_REAL;
else
return QL_MIN_REAL;
}
Real BlackCalculator::elasticityForward() const {
Real val = value();
Real del = deltaForward();
if (val>QL_EPSILON)
return del/val*forward_;
else if (std::fabs(del)<QL_EPSILON)
return 0.0;
else if (del>0.0)
return QL_MAX_REAL;
else
return QL_MIN_REAL;
}
Real BlackCalculator::gamma(Real spot) const {
QL_REQUIRE(spot > 0.0, "positive spot value required: " <<
spot << " not allowed");
// Handle zero volatility case
if (stdDev_ <= QL_EPSILON) {
// For zero volatility, gamma is always 0 (no convexity)
return 0.0;
}
Real DforwardDs = forward_ / spot;
Real temp = stdDev_*spot;
Real DalphaDs = DalphaDd1_/temp;
Real DbetaDs = DbetaDd2_/temp;
Real D2alphaDs2 = - DalphaDs/spot*(1+d1_/stdDev_);
Real D2betaDs2 = - DbetaDs /spot*(1+d2_/stdDev_);
Real temp2 = D2alphaDs2 * forward_ + 2.0 * DalphaDs * DforwardDs
+D2betaDs2 * x_ + 2.0 * DbetaDs * DxDs_;
return discount_ * temp2;
}
Real BlackCalculator::gammaForward() const {
// Handle zero volatility case
if (stdDev_ <= QL_EPSILON) {
// For zero volatility, gamma is always 0 (no convexity)
return 0.0;
}
Real temp = stdDev_*forward_;
Real DalphaDforward = DalphaDd1_/temp;
Real DbetaDforward = DbetaDd2_/temp;
Real D2alphaDforward2 = - DalphaDforward/forward_*(1+d1_/stdDev_);
Real D2betaDforward2 = - DbetaDforward /forward_*(1+d2_/stdDev_);
Real temp2 = D2alphaDforward2 * forward_ + 2.0 * DalphaDforward
+D2betaDforward2 * x_; // DXDforward = 0.0
return discount_ * temp2;
}
Real BlackCalculator::theta(Real spot,
Time maturity) const {
QL_REQUIRE(maturity>=0.0,
"maturity (" << maturity << ") must be non-negative");
if (close(maturity, 0.0)) return 0.0;
return -( std::log(discount_) * value()
+std::log(forward_/spot) * spot * delta(spot)
+0.5*variance_ * spot * spot * gamma(spot))/maturity;
}
Real BlackCalculator::vega(Time maturity) const {
QL_REQUIRE(maturity>=0.0,
"negative maturity not allowed");
// Handle zero volatility case
if (stdDev_ <= QL_EPSILON) {
return 0.0;
}
Real temp = std::log(strike_/forward_)/variance_;
// actually DalphaDsigma / SQRT(T)
Real DalphaDsigma = DalphaDd1_*(temp+0.5);
Real DbetaDsigma = DbetaDd2_ *(temp-0.5);
Real temp2 = DalphaDsigma * forward_ + DbetaDsigma * x_;
return discount_ * std::sqrt(maturity) * temp2;
}
Real BlackCalculator::rho(Time maturity) const {
QL_REQUIRE(maturity>=0.0,
"negative maturity not allowed");
// Handle zero volatility case
if (stdDev_ <= QL_EPSILON) {
// For zero volatility, rho = T * (delta_forward * forward - value/discount)
Real deltaFwd = deltaForward();
return maturity * (deltaFwd * forward_ - value());
}
// actually DalphaDr / T
Real DalphaDr = DalphaDd1_/stdDev_;
Real DbetaDr = DbetaDd2_/stdDev_;
Real temp = DalphaDr * forward_ + alpha_ * forward_ + DbetaDr * x_;
return maturity * (discount_ * temp - value());
}
Real BlackCalculator::dividendRho(Time maturity) const {
QL_REQUIRE(maturity>=0.0,
"negative maturity not allowed");
// Handle zero volatility case
if (stdDev_ <= QL_EPSILON) {
// For zero volatility, dividend rho = -T * discount * delta_forward * forward
Real deltaFwd = deltaForward() / discount_; // Remove discount to get pure delta
return -maturity * discount_ * deltaFwd * forward_;
}
// actually DalphaDq / T
Real DalphaDq = -DalphaDd1_/stdDev_;
Real DbetaDq = -DbetaDd2_/stdDev_;
Real temp = DalphaDq * forward_ - alpha_ * forward_ + DbetaDq * x_;
return maturity * discount_ * temp;
}
Real BlackCalculator::strikeSensitivity() const {
// Handle zero volatility case
if (stdDev_ <= QL_EPSILON) {
// For zero volatility, strike sensitivity is:
// Call: -N(d2) where d2 -> 1 (ITM), 0 (OTM), 0.5 (ATM)
// Put: N(-d2) = 1 - N(d2)
if (close(forward_, strike_)) {
// ATM case
if (alpha_ >= 0) { // Call
return -discount_ * 0.5;
} else { // Put
return discount_ * 0.5;
}
} else if (forward_ > strike_) {
// ITM Call, OTM Put
if (alpha_ >= 0) { // Call
return -discount_ * 1.0;
} else { // Put
return discount_ * 0.0;
}
} else {
// OTM Call, ITM Put
if (alpha_ >= 0) { // Call
return -discount_ * 0.0;
} else { // Put
return discount_ * 1.0;
}
}
}
Real temp = stdDev_*strike_;
Real DalphaDstrike = -DalphaDd1_/temp;
Real DbetaDstrike = -DbetaDd2_/temp;
Real temp2 =
DalphaDstrike * forward_ + DbetaDstrike * x_ + beta_ * DxDstrike_;
return discount_ * temp2;
}
Real BlackCalculator::strikeGamma() const {
// Handle zero volatility case
if (stdDev_ <= QL_EPSILON) {
// For zero volatility, strike gamma is 0 (no convexity)
return 0.0;
}
Real temp = stdDev_*strike_;
Real DalphaDstrike = -DalphaDd1_/temp;
Real DbetaDstrike = -DbetaDd2_/temp;
Real D2alphaD2strike = -DalphaDstrike/strike_*(1-d1_/stdDev_);
Real D2betaD2strike = -DbetaDstrike /strike_*(1-d2_/stdDev_);
Real temp2 =
D2alphaD2strike * forward_ + D2betaD2strike * x_
+ 2.0*DbetaDstrike *DxDstrike_;
return discount_ * temp2;
}
}
|
