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/* -*- mode: c++; tab-width: 4; indent-tabs-mode: nil; c-basic-offset: 4 -*- */
/*
Copyright (C) 2017 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.
*/
/*! \file gbsmrndcalculator.hpp
\brief risk neutral terminal density calculator for the
Black-Scholes-Merton model with skew dependent volatility
*/
#include <ql/math/distributions/normaldistribution.hpp>
#include <ql/math/solvers1d/brent.hpp>
#include <ql/methods/finitedifferences/utilities/gbsmrndcalculator.hpp>
#include <ql/pricingengines/blackcalculator.hpp>
#include <ql/processes/blackscholesprocess.hpp>
#include <utility>
namespace QuantLib {
GBSMRNDCalculator::GBSMRNDCalculator(ext::shared_ptr<GeneralizedBlackScholesProcess> process)
: process_(std::move(process)) {}
Real GBSMRNDCalculator::pdf(Real k, Time t) const {
const Real dk = 1e-3*k;
return (cdf(k+dk, t) - cdf(k-dk, t)) / (2*dk);
}
Real GBSMRNDCalculator::cdf(Real k, Time t) const {
const Handle<BlackVolTermStructure> volTS
= process_->blackVolatility();
const Real dk = 1e-3*k;
const Real dvol_dk
= (volTS->blackVol(t, k+dk) - volTS->blackVol(t, k-dk)) / (2*dk);
const DiscountFactor dR
= process_->riskFreeRate()->discount(t, true);
const DiscountFactor dD
= process_->dividendYield()->discount(t, true);
const Real forward = process_->x0() * dD / dR;
const Real stdDev = std::sqrt(
process_->blackVolatility()->blackVariance(t, k, true));
if (forward <= k) {
const BlackCalculator calc(Option::Call, k, forward, stdDev, dR);
return 1.0 + ( calc.strikeSensitivity()
+ calc.vega(t) * dvol_dk) /dR;
}
else {
const BlackCalculator calc(Option::Put, k, forward, stdDev, dR);
return ( calc.strikeSensitivity()
+ calc.vega(t) * dvol_dk) /dR;
}
}
Real GBSMRNDCalculator::invcdf(Real q, Time t) const {
const Real fwd = process_->x0()
/ process_->riskFreeRate()->discount(t, true)
* process_->dividendYield()->discount(t, true);
const Volatility atmVariance = std::sqrt(
process_->blackVolatility()->blackVariance(t, fwd, true));
const Real atmX = InverseCumulativeNormal()(q);
const Real guess = fwd*std::exp(atmVariance*atmX);
Real lower = guess;
while (guess/lower < 65535.0 && cdf(lower, t) > q)
lower*=0.5;
Real upper = guess;
while (upper/guess < 65535.0 && cdf(upper, t) < q) upper*=2;
QL_REQUIRE(guess/lower < 65535.0 && upper/guess < 65535.0,
"Could not find an start interval with ("
<< lower << ", " << upper << ") -> ("
<< cdf(lower, t) << ", " << cdf(upper, t) << ")");
return Brent().solve(
[&](Real _k) -> Real { return cdf(_k, t) - q; },
1e-10, 0.5*(lower+upper), lower, upper);
}
}
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