hestonslvprocess.cpp

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/* -*- mode: c++; tab-width: 4; indent-tabs-mode: nil; c-basic-offset: 4 -*- */
 
/*
 Copyright (C) 2015 Johannes Göttker-Schnetmann
 Copyright (C) 2015 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 hestonslvprocess.cpp
    \brief Heston stochastic local volatility process
*/
 
#include <ql/processes/hestonslvprocess.hpp>
#include <ql/math/distributions/normaldistribution.hpp>
#include <ql/methods/finitedifferences/utilities/squarerootprocessrndcalculator.hpp>
#include <utility>
 
namespace QuantLib {
 
    HestonSLVProcess::HestonSLVProcess(const ext::shared_ptr<HestonProcess>& hestonProcess,
                                       ext::shared_ptr<LocalVolTermStructure> leverageFct,
                                       const Real mixingFactor)
    : mixingFactor_(mixingFactor), hestonProcess_(hestonProcess),
      leverageFct_(std::move(leverageFct)) {
        registerWith(hestonProcess);
        setParameters();
    };
 
    void HestonSLVProcess::update() {
        setParameters();
        StochasticProcess::update();
    }
 
    Array HestonSLVProcess::drift(Time t, const Array& x) const {
        Array tmp(2);
 
        const Volatility vol =
           std::max(1e-8, std::sqrt(x[1])*leverageFct_->localVol(t, x[0], true));
 
        tmp[0] = riskFreeRate()->forwardRate(t, t, Continuous).rate()
               - dividendYield()->forwardRate(t, t, Continuous).rate()
               - 0.5*vol*vol;
 
        tmp[1] = kappa_*(theta_ - x[1]);
 
        return tmp;
    }
 
    Matrix HestonSLVProcess::diffusion(Time t, const Array& x) const {
 
        const Real vol =
            std::max(1e-8, std::sqrt(x[1])*leverageFct_->localVol(t, x[0], true));
 
        const Real sigma2 = mixedSigma_ * std::sqrt(x[1]);
        const Real sqrhov = std::sqrt(1.0 - rho_*rho_);
 
        Matrix tmp(2,2);
        tmp[0][0] = vol;          tmp[0][1] = 0.0;
        tmp[1][0] = rho_*sigma2;  tmp[1][1] = sqrhov*sigma2;
 
        return tmp;
    }
 
    Array HestonSLVProcess::evolve(
        Time t0, const Array& x0, Time dt, const Array& dw) const {
        Array retVal(2);
 
        const Real ex = std::exp(-kappa_*dt);
 
        const Real m  =  theta_+(x0[1]-theta_)*ex;
        const Real s2 =  x0[1]*mixedSigma_*mixedSigma_*ex/kappa_*(1-ex)
                       + theta_*mixedSigma_*mixedSigma_/(2*kappa_)*(1-ex)*(1-ex);
        const Real psi = s2/(m*m);
 
        if (psi < 1.5) {
            const Real b2 = 2/psi-1+std::sqrt(2/psi*(2/psi-1));
            const Real b  = std::sqrt(b2);
            const Real a  = m/(1+b2);
 
            retVal[1] = a*(b+dw[1])*(b+dw[1]);
        }
        else {
            const Real p = (psi-1)/(psi+1);
            const Real beta = (1-p)/m;
            const Real u = CumulativeNormalDistribution()(dw[1]);
 
            retVal[1] = ((u <= p) ? Real(0.0) : std::log((1-p)/(1-u))/beta);
        }
 
        const Real mu = riskFreeRate()->forwardRate(t0, t0+dt, Continuous).rate()
             - dividendYield()->forwardRate(t0, t0+dt, Continuous).rate();
 
        const Real rho1 = std::sqrt(1-rho_*rho_);
 
        const Volatility l_0 = leverageFct_->localVol(t0, x0[0], true);
        const Real v_0 = 0.5*(x0[1]+retVal[1])*l_0*l_0;
 
        retVal[0] = x0[0]*std::exp(mu*dt - 0.5*v_0*dt
            + rho_/mixedSigma_*l_0 * (
                  retVal[1] - kappa_*theta_*dt
                  + 0.5*(x0[1]+retVal[1])*kappa_*dt - x0[1])
            + rho1*std::sqrt(v_0*dt)*dw[0]);
 
        return retVal;
    }
 
    void HestonSLVProcess::setParameters() {
        v0_    = hestonProcess_->v0();
        kappa_ = hestonProcess_->kappa();
        theta_ = hestonProcess_->theta();
        sigma_ = hestonProcess_->sigma();
        rho_   = hestonProcess_->rho();
        mixedSigma_ = mixingFactor_ * sigma_;
    }
 
}