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andreasenhugevolatilityinterpl.cpp
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andreasenhugevolatilityinterpl.cpp
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/* -*- mode: c++; tab-width: 4; indent-tabs-mode: nil; c-basic-offset: 4 -*- */
/*
Copyright (C) 2017, 2018 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
<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/exercise.hpp>
#include <ql/instruments/vanillaoption.hpp>
#include <ql/math/array.hpp>
#include <ql/math/comparison.hpp>
#include <ql/math/interpolations/backwardflatinterpolation.hpp>
#include <ql/math/interpolations/cubicinterpolation.hpp>
#include <ql/methods/finitedifferences/meshers/concentrating1dmesher.hpp>
#include <ql/methods/finitedifferences/meshers/fdmmeshercomposite.hpp>
#include <ql/methods/finitedifferences/operators/fdmlinearoplayout.hpp>
#include <ql/methods/finitedifferences/operators/firstderivativeop.hpp>
#include <ql/methods/finitedifferences/operators/secondderivativeop.hpp>
#include <ql/methods/finitedifferences/tridiagonaloperator.hpp>
#include <ql/pricingengines/blackcalculator.hpp>
#include <ql/termstructures/volatility/equityfx/andreasenhugevolatilityinterpl.hpp>
#include <ql/termstructures/yieldtermstructure.hpp>
#include <ql/timegrid.hpp>
#include <ql/utilities/null.hpp>
#include <cmath>
#include <limits>
#include <utility>
namespace QuantLib {
namespace {
struct close_enough_to {
Real y;
Size n;
explicit close_enough_to(Real y, Size n=42) : y(y), n(n) {}
bool operator()(Real x) const { return close_enough(x, y, n); }
};
}
class AndreasenHugeCostFunction : public CostFunction {
public:
AndreasenHugeCostFunction(
Array marketNPVs,
Array marketVegas,
Array lnMarketStrikes,
Array previousNPVs,
const ext::shared_ptr<FdmMesherComposite>& mesher,
Time dT,
AndreasenHugeVolatilityInterpl::InterpolationType interpolationType)
: marketNPVs_(std::move(marketNPVs)), marketVegas_(std::move(marketVegas)),
lnMarketStrikes_(std::move(lnMarketStrikes)), previousNPVs_(std::move(previousNPVs)),
mesher_(mesher), nGridPoints_(mesher->layout()->size()), dT_(dT),
interpolationType_((lnMarketStrikes_.size() > 1) ?
interpolationType :
AndreasenHugeVolatilityInterpl::PiecewiseConstant),
dxMap_(FirstDerivativeOp(0, mesher_)), dxxMap_(SecondDerivativeOp(0, mesher_)),
d2CdK2_(dxMap_.mult(Array(mesher->layout()->size(), -1.0)).add(dxxMap_)),
mapT_(0, mesher_) {}
Array d2CdK2(const Array& c) const {
return d2CdK2_.apply(c);
}
Array solveFor(Time dT, const Array& sig, const Array& b) const {
Array x(lnMarketStrikes_.size());
Interpolation sigInterpl;
switch (interpolationType_) {
case AndreasenHugeVolatilityInterpl::CubicSpline:
sigInterpl = CubicNaturalSpline(
lnMarketStrikes_.begin(), lnMarketStrikes_.end(),
sig.begin());
break;
case AndreasenHugeVolatilityInterpl::Linear:
sigInterpl = LinearInterpolation(
lnMarketStrikes_.begin(), lnMarketStrikes_.end(),
sig.begin());
break;
case AndreasenHugeVolatilityInterpl::PiecewiseConstant:
for (Size i=0; i < x.size()-1; ++i)
x[i] = 0.5*(lnMarketStrikes_[i] + lnMarketStrikes_[i+1]);
x.back() = lnMarketStrikes_.back();
sigInterpl = BackwardFlatInterpolation(
x.begin(), x.end(), sig.begin());
break;
default:
QL_FAIL("unknown interpolation type");
}
Array z(mesher_->layout()->size());
for (const auto& iter : *mesher_->layout()) {
const Size i = iter.index();
const Real lnStrike = mesher_->location(iter, 0);
const Real vol = sigInterpl(
std::min(std::max(lnStrike, lnMarketStrikes_.front()),
lnMarketStrikes_.back()), true);
z[i] = 0.5*vol*vol;
}
mapT_.axpyb(z, dxMap_, dxxMap_.mult(-z), Array());
return mapT_.mult(Array(z.size(), dT)).solve_splitting(b, 1.0);
}
Array apply(const Array& c) const {
return -mapT_.apply(c);
}
Array values(const Array& sig) const override {
Array newNPVs = solveFor(dT_, sig, previousNPVs_);
const std::vector<Real>& gridPoints =
mesher_->getFdm1dMeshers().front()->locations();
const MonotonicCubicNaturalSpline interpl(
gridPoints.begin(), gridPoints.end(), newNPVs.begin());
Array retVal(lnMarketStrikes_.size());
for (Size i=0; i < retVal.size(); ++i) {
const Real strike = lnMarketStrikes_[i];
retVal[i] = interpl(strike) - marketNPVs_[i];
}
return retVal;
}
Array vegaCalibrationError(const Array& sig) const {
return values(sig)/marketVegas_;
}
Array initialValues() const {
return Array(lnMarketStrikes_.size(), 0.25);
}
private:
const Array marketNPVs_, marketVegas_;
const Array lnMarketStrikes_, previousNPVs_;
const ext::shared_ptr<FdmMesherComposite> mesher_;
const Size nGridPoints_;
const Time dT_;
const AndreasenHugeVolatilityInterpl::InterpolationType
interpolationType_;
const FirstDerivativeOp dxMap_;
const TripleBandLinearOp dxxMap_;
const TripleBandLinearOp d2CdK2_;
mutable TripleBandLinearOp mapT_;
};
class CombinedCostFunction : public CostFunction {
public:
CombinedCostFunction(ext::shared_ptr<AndreasenHugeCostFunction> putCostFct,
ext::shared_ptr<AndreasenHugeCostFunction> callCostFct)
: putCostFct_(std::move(putCostFct)), callCostFct_(std::move(callCostFct)) {}
Array values(const Array& sig) const override {
if ((putCostFct_ != nullptr) && (callCostFct_ != nullptr)) {
const Array pv = putCostFct_->values(sig);
const Array cv = callCostFct_->values(sig);
Array retVal(pv.size() + cv.size());
std::copy(pv.begin(), pv.end(), retVal.begin());
std::copy(cv.begin(), cv.end(), retVal.begin() + cv.size());
return retVal;
} else if (putCostFct_ != nullptr)
return putCostFct_->values(sig);
else if (callCostFct_ != nullptr)
return callCostFct_->values(sig);
else
QL_FAIL("internal error: cost function not set");
}
Array initialValues() const {
if ((putCostFct_ != nullptr) && (callCostFct_ != nullptr))
return 0.5*( putCostFct_->initialValues()
+ callCostFct_->initialValues());
else if (putCostFct_ != nullptr)
return putCostFct_->initialValues();
else if (callCostFct_ != nullptr)
return callCostFct_->initialValues();
else
QL_FAIL("internal error: cost function not set");
}
private:
const ext::shared_ptr<AndreasenHugeCostFunction> putCostFct_;
const ext::shared_ptr<AndreasenHugeCostFunction> callCostFct_;
};
AndreasenHugeVolatilityInterpl::AndreasenHugeVolatilityInterpl(
const CalibrationSet& calibrationSet,
Handle<Quote> spot,
Handle<YieldTermStructure> rTS,
Handle<YieldTermStructure> qTS,
InterpolationType interplationType,
CalibrationType calibrationType,
Size nGridPoints,
Real _minStrike,
Real _maxStrike,
ext::shared_ptr<OptimizationMethod> optimizationMethod,
const EndCriteria& endCriteria)
: spot_(std::move(spot)), rTS_(std::move(rTS)), qTS_(std::move(qTS)),
interpolationType_(interplationType), calibrationType_(calibrationType),
nGridPoints_(nGridPoints), minStrike_(_minStrike), maxStrike_(_maxStrike),
optimizationMethod_(std::move(optimizationMethod)), endCriteria_(endCriteria) {
QL_REQUIRE(nGridPoints > 2 && !calibrationSet.empty(), "undefined grid or calibration set");
std::set<Real> strikes;
std::set<Date> expiries;
calibrationSet_.reserve(calibrationSet.size());
for (const auto& i : calibrationSet) {
const ext::shared_ptr<Exercise> exercise = i.first->exercise();
QL_REQUIRE(exercise->type() == Exercise::European,
"European option required");
const Date expiry = exercise->lastDate();
expiries.insert(expiry);
const ext::shared_ptr<PlainVanillaPayoff> payoff =
ext::dynamic_pointer_cast<PlainVanillaPayoff>(i.first->payoff());
QL_REQUIRE(payoff, "plain vanilla payoff required");
const Real strike = payoff->strike();
strikes.insert(strike);
calibrationSet_.emplace_back(ext::make_shared<VanillaOption>(payoff, exercise), i.second);
registerWith(i.second);
}
strikes_.assign(strikes.begin(), strikes.end());
expiries_.assign(expiries.begin(), expiries.end());
dT_.resize(expiries_.size());
expiryTimes_.resize(expiries_.size());
calibrationMatrix_ = std::vector< std::vector<Size> >(
expiries.size(), std::vector<Size>(strikes.size(), Null<Size>()));
for (Size i=0; i < calibrationSet.size(); ++i) {
const Date expiry =
calibrationSet[i].first->exercise()->lastDate();
const Size l = std::distance(expiries.begin(), expiries.lower_bound(expiry));
const Real strike =
ext::dynamic_pointer_cast<PlainVanillaPayoff>(
calibrationSet[i].first->payoff())->strike();
const Size k = std::distance(strikes_.begin(),
std::find_if(strikes_.begin(), strikes_.end(),
close_enough_to(strike)));
calibrationMatrix_[l][k] = i;
}
registerWith(spot_);
registerWith(rTS_);
registerWith(qTS_);
}
ext::shared_ptr<AndreasenHugeCostFunction>
AndreasenHugeVolatilityInterpl::buildCostFunction(
Size iExpiry, Option::Type optionType,
const Array& previousNPVs) const {
if (calibrationType_ != CallPut
&& ( (calibrationType_ == Call && optionType ==Option::Put)
|| (calibrationType_ == Put && optionType ==Option::Call)))
return ext::shared_ptr<AndreasenHugeCostFunction>();
const Time expiryTime = expiryTimes_[iExpiry];
const DiscountFactor discount = rTS_->discount(expiryTime);
const Real fwd = spot_->value()*qTS_->discount(expiryTime)/discount;
Size null = Null<Size>();
const Size nOptions = std::count_if(
calibrationMatrix_[iExpiry].begin(),
calibrationMatrix_[iExpiry].end(),
[=](Size n){ return n != null; });
Array lnMarketStrikes(nOptions),
marketNPVs(nOptions), marketVegas(nOptions);
// calculate undiscounted market prices
for (Size j=0, k=0; j < strikes_.size(); ++j) {
const Size idx = calibrationMatrix_[iExpiry][j];
if (idx != null) {
const Volatility vol = calibrationSet_[idx].second->value();
const Real stdDev = vol*std::sqrt(expiryTime);
const BlackCalculator calculator(
optionType, strikes_[j], fwd, stdDev, discount);
const Real npv = calculator.value();
const Real vega = calculator.vega(expiryTime);
marketNPVs[k] = npv/(discount*fwd);
marketVegas[k] = vega/(discount*fwd);
lnMarketStrikes[k++] = std::log(strikes_[j]/fwd);
}
}
return ext::make_shared<AndreasenHugeCostFunction>(
marketNPVs,
marketVegas,
lnMarketStrikes,
previousNPVs,
mesher_,
dT_[iExpiry],
interpolationType_);
}
void AndreasenHugeVolatilityInterpl::performCalculations() const {
QL_REQUIRE(maxStrike() > minStrike(),
"max strike must be greater than min strike");
const DayCounter dc = rTS_->dayCounter();
for (Size i=0; i < expiryTimes_.size(); ++i) {
expiryTimes_[i] =
dc.yearFraction(rTS_->referenceDate(), expiries_[i]);
dT_[i] = expiryTimes_[i] - ( (i==0)? 0.0 : expiryTimes_[i-1]);
}
mesher_ =
ext::make_shared<FdmMesherComposite>(
ext::make_shared<Concentrating1dMesher>(
std::log(minStrike()/spot_->value()),
std::log(maxStrike()/spot_->value()),
nGridPoints_,
std::pair<Real, Real>(0.0, 0.025)));
gridPoints_ = mesher_->locations(0);
gridInFwd_ = Exp(gridPoints_)*spot_->value();
localVolCache_.clear();
calibrationResults_.clear();
avgError_ = 0.0;
minError_ = std::numeric_limits<Real>::max();
maxError_ = 0.0;
calibrationResults_.reserve(expiries_.size());
Array npvPuts(nGridPoints_);
Array npvCalls(nGridPoints_);
for (Size i=0; i < nGridPoints_; ++i) {
const Real strike = std::exp(gridPoints_[i]);
npvPuts[i] = PlainVanillaPayoff(Option::Put, strike)(1.0);
npvCalls[i]= PlainVanillaPayoff(Option::Call, strike)(1.0);
}
for (Size i=0; i < expiries_.size(); ++i) {
const ext::shared_ptr<AndreasenHugeCostFunction> putCostFct =
buildCostFunction(i, Option::Put, npvPuts);
const ext::shared_ptr<AndreasenHugeCostFunction> callCostFct =
buildCostFunction(i, Option::Call, npvCalls);
CombinedCostFunction costFunction(putCostFct, callCostFct);
PositiveConstraint positiveConstraint;
Problem problem(costFunction,
positiveConstraint, costFunction.initialValues());
optimizationMethod_->minimize(problem, endCriteria_);
const Array& sig = problem.currentValue();
const SingleStepCalibrationResult calibrationResult = {
npvPuts, npvCalls, sig,
(calibrationType_ == Call)? callCostFct : putCostFct
};
calibrationResults_.push_back(calibrationResult);
Array vegaDiffs(sig.size());
switch (calibrationType_) {
case CallPut: {
const Array vegaPutDiffs =
putCostFct->vegaCalibrationError(sig);
const Array vegaCallDiffs =
callCostFct->vegaCalibrationError(sig);
const Real fwd = spot_->value()*
qTS_->discount(expiryTimes_[i])/rTS_->discount(expiryTimes_[i]);
for (Size j=0; j < vegaDiffs.size(); ++j)
vegaDiffs[j] = std::fabs(
(fwd > gridInFwd_[j])? vegaPutDiffs[j] : vegaCallDiffs[j]);
}
break;
case Put:
vegaDiffs = Abs(putCostFct->vegaCalibrationError(sig));
break;
case Call:
vegaDiffs = Abs(callCostFct->vegaCalibrationError(sig));
break;
default:
QL_FAIL("unknown calibration type");
}
avgError_ +=
std::accumulate(vegaDiffs.begin(), vegaDiffs.end(), Real(0.0));
minError_ = std::min(minError_,
*std::min_element(vegaDiffs.begin(), vegaDiffs.end()));
maxError_ = std::max(maxError_,
*std::max_element(vegaDiffs.begin(), vegaDiffs.end()));
if (putCostFct != nullptr)
npvPuts = putCostFct->solveFor(dT_[i], sig, npvPuts);
if (callCostFct != nullptr)
npvCalls= callCostFct->solveFor(dT_[i], sig, npvCalls);
}
avgError_ /= calibrationSet_.size();
}
Date AndreasenHugeVolatilityInterpl::maxDate() const {
return expiries_.back();
}
Real AndreasenHugeVolatilityInterpl::minStrike() const {
return (minStrike_ == Null<Real>())
? 1/8.0*strikes_.front() : minStrike_;
}
Real AndreasenHugeVolatilityInterpl::maxStrike() const {
return (maxStrike_ == Null<Real>())
? 8.0*strikes_.back() : maxStrike_;
}
Real AndreasenHugeVolatilityInterpl::fwd(Time t) const {
return spot_->value()*qTS_->discount(t)/rTS_->discount(t);
}
const Handle<YieldTermStructure>&
AndreasenHugeVolatilityInterpl::riskFreeRate() const {
return rTS_;
}
ext::tuple<Real, Real, Real>
AndreasenHugeVolatilityInterpl::calibrationError() const {
calculate();
return ext::make_tuple(minError_, maxError_, avgError_);
}
Size AndreasenHugeVolatilityInterpl::getExerciseTimeIdx(Time t) const {
return std::min<Size>(expiryTimes_.size()-1,
std::distance(expiryTimes_.begin(),
std::upper_bound(
expiryTimes_.begin(), expiryTimes_.end(), t)));
}
Real AndreasenHugeVolatilityInterpl::getCacheValue(
Real strike, const TimeValueCacheType::const_iterator& f) const {
const Real fwd = ext::get<0>(f->second);
const Real k = std::log(strike / fwd);
const Real s = std::max(gridPoints_[1],
std::min(*(gridPoints_.end()-2), k));
return (*(ext::get<2>(f->second)))(s);
}
Array AndreasenHugeVolatilityInterpl::getPriceSlice(
Time t, Option::Type optionType) const {
const Size iu = getExerciseTimeIdx(t);
return calibrationResults_[iu].costFunction->solveFor(
(iu == 0) ? t : t-expiryTimes_[iu-1],
calibrationResults_[iu].sigmas,
(optionType == Option::Call)? calibrationResults_[iu].callNPVs
: calibrationResults_[iu].putNPVs);
}
Real AndreasenHugeVolatilityInterpl::optionPrice(
Time t, Real strike, Option::Type optionType) const {
TimeValueCacheType::const_iterator f = priceCache_.find(t);
const DiscountFactor df = rTS_->discount(t);
if (f != priceCache_.end()) {
const Real fwd = ext::get<0>(f->second);
Real price = getCacheValue(strike, f);
if (optionType == Option::Put
&& (calibrationType_ == Call || calibrationType_ == CallPut))
price = price + strike/fwd - 1.0;
else if (optionType == Option::Call && calibrationType_ == Put)
price = 1.0 - strike/fwd + price;
return price*df*fwd;
}
calculate();
ext::shared_ptr<Array> prices(
ext::make_shared<Array>(gridPoints_));
switch (calibrationType_) {
case Put:
(*prices) = getPriceSlice(t, Option::Put);
break;
case Call:
case CallPut:
(*prices) = getPriceSlice(t, Option::Call);
break;
default:
QL_FAIL("unknown calibration type");
}
Real fwd = spot_->value()*qTS_->discount(t)/df;
priceCache_[t] = ext::make_tuple(
fwd, prices,
ext::make_shared<CubicNaturalSpline>(
gridPoints_.begin()+1, gridPoints_.end()-1,
prices->begin()+1));
return this->optionPrice(t, strike, optionType);
}
Array AndreasenHugeVolatilityInterpl::getLocalVolSlice(
Time t, Option::Type optionType) const {
const Size iu = getExerciseTimeIdx(t);
const Array& previousNPVs =
(optionType == Option::Call)? calibrationResults_[iu].callNPVs
: calibrationResults_[iu].putNPVs;
const ext::shared_ptr<AndreasenHugeCostFunction> costFunction
= calibrationResults_[iu].costFunction;
const Time dt = (iu == 0) ? t : t-expiryTimes_[iu-1];
const Array& sig = calibrationResults_[iu].sigmas;
const Array cAtJ = costFunction->solveFor(dt, sig, previousNPVs);
const Array dCdT =
costFunction->solveFor(dt, sig,
costFunction->apply(
costFunction->solveFor(dt, sig, previousNPVs)));
const Array d2CdK2 = costFunction->d2CdK2(cAtJ);
Array localVol = Sqrt(2*dCdT/d2CdK2);
for (Size i=1; i < localVol.size()-1; ++i)
if (!std::isfinite(localVol[i]) || localVol[i] < 0.0)
localVol[i] = 0.25;
return localVol;
}
Volatility AndreasenHugeVolatilityInterpl::localVol(Time t, Real strike)
const {
TimeValueCacheType::const_iterator f = localVolCache_.find(t);
if (f != localVolCache_.end())
return getCacheValue(strike, f);
calculate();
ext::shared_ptr<Array> localVol(
ext::make_shared<Array>(gridPoints_.size()));
switch (calibrationType_) {
case CallPut: {
const Array putLocalVol = getLocalVolSlice(t, Option::Put);
const Array callLocalVol = getLocalVolSlice(t, Option::Call);
for (Size i=0, n=localVol->size(); i < n; ++i)
(*localVol)[i] =
(gridPoints_[i] > 0.0)? callLocalVol[i] : putLocalVol[i];
}
break;
case Put:
(*localVol) = getLocalVolSlice(t, Option::Put);
break;
case Call:
(*localVol) = getLocalVolSlice(t, Option::Call);
break;
default:
QL_FAIL("unknown calibration type");
}
Real fwd = spot_->value()*qTS_->discount(t)/rTS_->discount(t);
localVolCache_[t] = ext::make_tuple(
fwd, localVol,
ext::make_shared<LinearInterpolation>(
gridPoints_.begin()+1, gridPoints_.end()-1,
localVol->begin()+1));
return this->localVol(t, strike);
}
}