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/* -*- mode: c++; tab-width: 4; indent-tabs-mode: nil; c-basic-offset: 4 -*- */
/*
Copyright (C) 2013, 2015 Peter Caspers
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/termstructures/volatility/kahalesmilesection.hpp>
using std::sqrt;
namespace QuantLib {
KahaleSmileSection::KahaleSmileSection(const ext::shared_ptr<SmileSection>& source,
const Real atm,
const bool interpolate,
const bool exponentialExtrapolation,
const bool deleteArbitragePoints,
const std::vector<Real>& moneynessGrid,
const Real gap,
const int forcedLeftIndex,
const int forcedRightIndex)
: SmileSection(*source), source_(source), moneynessGrid_(moneynessGrid), gap_(gap),
interpolate_(interpolate), exponentialExtrapolation_(exponentialExtrapolation),
forcedLeftIndex_(forcedLeftIndex), forcedRightIndex_(forcedRightIndex) {
// only shifted lognormal smile sections are supported
QL_REQUIRE(source->volatilityType() == ShiftedLognormal,
"KahaleSmileSection only supports shifted lognormal source sections");
ssutils_ = ext::make_shared<SmileSectionUtils>(
*source, moneynessGrid, atm, deleteArbitragePoints);
moneynessGrid_ = ssutils_->moneyGrid();
k_ = ssutils_->strikeGrid();
c_ = ssutils_->callPrices();
f_ = ssutils_->atmLevel();
// for shifted smile sections we shift the forward and the strikes
// and do as if we were in a lognormal setting
for (Real& i : k_) {
i += source_->shift();
}
f_ += source_->shift();
compute();
}
void KahaleSmileSection::compute() {
std::pair<Size, Size> afIdx = ssutils_->arbitragefreeIndices();
leftIndex_ = afIdx.first;
rightIndex_ = afIdx.second;
cFunctions_ = std::vector<ext::shared_ptr<cFunction> >(
rightIndex_ - leftIndex_ + 2);
// extrapolation in the leftmost interval
Brent brent;
bool success;
Real secl = 0.0;
do {
success = true;
try {
Real k1 = k_[leftIndex_];
Real c1 = c_[leftIndex_];
Real c0 = c_[0];
secl = (c_[leftIndex_] - c_[0]) / (k_[leftIndex_] - k_[0]);
Real sec = (c_[leftIndex_ + 1] - c_[leftIndex_]) /
(k_[leftIndex_ + 1] - k_[leftIndex_]);
Real c1p;
if (interpolate_)
c1p = (secl + sec) / 2;
else {
c1p = -source_->digitalOptionPrice(k1 - source_->shift() + gap_ / 2.0, Option::Call, 1.0, gap_);
QL_REQUIRE(secl < c1p && c1p <= 0.0, "dummy");
// can not extrapolate so throw exception which is caught
// below
}
sHelper1 sh1(k1, c0, c1, c1p);
Real s = brent.solve(sh1, QL_KAHALE_ACC, 0.20, 0.00,
QL_KAHALE_SMAX); // numerical parameters
// hardcoded here
sh1(s);
ext::shared_ptr<cFunction> cFct1(
new cFunction(sh1.f_, s, 0.0, sh1.b_));
cFunctions_[0] = cFct1;
// sanity check - in rare cases we can get digitials
// which are not monotonic or greater than 1.0
// due to numerical effects. Move to the next index in
// these cases.
Real dig = digitalOptionPrice((k1 - source_->shift()) / 2.0, Option::Call, 1.0, gap_);
QL_REQUIRE(dig >= -c1p && dig <= 1.0, "dummy");
if(static_cast<int>(leftIndex_) < forcedLeftIndex_) {
leftIndex_++;
success = false;
}
}
catch (...) {
leftIndex_++;
success = false;
}
} while (!success && leftIndex_ < rightIndex_);
QL_REQUIRE(
leftIndex_ < rightIndex_,
"can not extrapolate to left, right index of af region reached ("
<< rightIndex_ << ")");
// interpolation
Real cp0 = 0.0, cp1 = 0.0;
if (interpolate_) {
for (Size i = leftIndex_; i < rightIndex_; i++) {
Real k0 = k_[i];
Real k1 = k_[i + 1];
Real c0 = c_[i];
Real c1 = c_[i + 1];
Real sec = (c_[i + 1] - c_[i]) / (k_[i + 1] - k_[i]);
if (i == leftIndex_)
cp0 = leftIndex_ > 0 ? (secl + sec) / 2.0 : sec;
Real secr;
if (i == rightIndex_ - 1)
secr = 0.0;
else
secr = (c_[i + 2] - c_[i + 1]) / (k_[i + 2] - k_[i + 1]);
cp1 = (sec + secr) / 2.0;
aHelper ah(k0, k1, c0, c1, cp0, cp1);
Real a;
bool valid = false;
try {
a = brent.solve(
ah, QL_KAHALE_ACC, 0.5 * (cp1 + (1.0 + cp0)),
cp1 + QL_KAHALE_EPS, 1.0 + cp0 - QL_KAHALE_EPS);
// numerical parameters hardcoded here
valid = true;
}
catch (...) {
// delete the right point of the interval where we try to
// interpolate
moneynessGrid_.erase(moneynessGrid_.begin() + (i + 1));
k_.erase(k_.begin() + (i + 1));
c_.erase(c_.begin() + (i + 1));
cFunctions_.erase(cFunctions_.begin() + (i + 1));
rightIndex_--;
i--;
}
if (valid) {
ah(a);
ext::shared_ptr<cFunction> cFct(
new cFunction(ah.f_, ah.s_, a, ah.b_));
cFunctions_[leftIndex_ > 0 ? i - leftIndex_ + 1 : 0] = cFct;
cp0 = cp1;
}
}
}
// extrapolation of right wing
do {
success = true;
try {
Real k0 = k_[rightIndex_];
Real c0 = c_[rightIndex_];
Real cp0;
if (interpolate_)
cp0 = 0.5 * (c_[rightIndex_] - c_[rightIndex_ - 1]) /
(k_[rightIndex_] - k_[rightIndex_ - 1]);
else {
cp0 = -source_->digitalOptionPrice(
k0 - shift() - gap_ / 2.0, Option::Call, 1.0, gap_);
}
ext::shared_ptr<cFunction> cFct;
if (exponentialExtrapolation_) {
QL_REQUIRE(-cp0 / c0 > 0.0, "dummy"); // this is caught
// below
cFct = ext::make_shared<cFunction>(
-cp0 / c0, std::log(c0) - cp0 / c0 * k0);
} else {
sHelper sh(k0, c0, cp0);
Real s;
s = brent.solve(sh, QL_KAHALE_ACC, 0.20, 0.0,
QL_KAHALE_SMAX); // numerical parameters
// hardcoded here
sh(s);
cFct = ext::make_shared<cFunction>(
sh.f_, s, 0.0, 0.0);
}
cFunctions_[rightIndex_ - leftIndex_ + 1] = cFct;
}
catch (...) {
rightIndex_--;
success = false;
}
if(static_cast<int>(rightIndex_) > forcedRightIndex_) {
rightIndex_--;
success = false;
}
} while (!success && rightIndex_ > leftIndex_);
QL_REQUIRE(
leftIndex_ < rightIndex_,
"can not extrapolate to right, left index of af region reached ("
<< leftIndex_ << ")");
}
Real KahaleSmileSection::optionPrice(Rate strike, Option::Type type,
Real discount) const {
// option prices are directly available, so implement this function
// rather than use smileSection
// standard implementation
Real shifted_strike = std::max(strike + shift(), QL_KAHALE_EPS);
int i = index(shifted_strike);
if (interpolate_ ||
(i == 0 || i == (int)(rightIndex_ - leftIndex_ + 1)))
return discount *
(type == Option::Call
? (*cFunctions_[i])(shifted_strike)
: (*cFunctions_[i])(shifted_strike) + shifted_strike - f_);
else
return source_->optionPrice(strike, type, discount);
}
Real KahaleSmileSection::volatilityImpl(Rate strike) const {
Real shifted_strike = std::max(strike + shift(), QL_KAHALE_EPS);
int i = index(shifted_strike);
if (!interpolate_ &&
!(i == 0 || i == (int)(rightIndex_ - leftIndex_ + 1)))
return source_->volatility(strike);
Real c = (*cFunctions_[i])(shifted_strike);
Real vol = 0.0;
try {
Option::Type type = shifted_strike >= f_ ? Option::Call : Option::Put;
vol = blackFormulaImpliedStdDev(
type, shifted_strike, f_,
type == Option::Put ? strike - f_ + c : c) /
sqrt(exerciseTime());
}
catch (...) {
}
return vol;
}
Size KahaleSmileSection::index(Rate strike) const {
int i =
static_cast<int>(std::upper_bound(k_.begin(), k_.end(), strike) -
k_.begin()) -
static_cast<int>(leftIndex_);
return std::max(
std::min(i, static_cast<int>(rightIndex_ - leftIndex_ + 1)), 0);
}
}
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