gbsmrndcalculator.cpp

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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
 <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.
*/
 
/*! \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);
    }
}