1 /* -*- mode: c++; tab-width: 4; indent-tabs-mode: nil; c-basic-offset: 4 -*- */ 2 3 /* 4 Copyright (C) 2015 Johannes Göttker-Schnetmann 5 Copyright (C) 2015 Klaus Spanderen 6 7 This file is part of QuantLib, a free-software/open-source library 8 for financial quantitative analysts and developers - http://quantlib.org/ 9 10 QuantLib is free software: you can redistribute it and/or modify it 11 under the terms of the QuantLib license. You should have received a 12 copy of the license along with this program; if not, please email 13 <quantlib-dev@lists.sf.net>. The license is also available online at 14 <http://quantlib.org/license.shtml>. 15 16 This program is distributed in the hope that it will be useful, but WITHOUT 17 ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or FITNESS 18 FOR A PARTICULAR PURPOSE. See the license for more details. 19 */ 20 21 /*! \file hestonslvprocess.cpp 22 \brief Heston stochastic local volatility process 23 */ 24 25 #include <ql/math/functional.hpp> 26 #include <ql/math/distributions/normaldistribution.hpp> 27 #include <ql/experimental/processes/hestonslvprocess.hpp> 28 #include <ql/methods/finitedifferences/utilities/squarerootprocessrndcalculator.hpp> 29 30 namespace QuantLib { 31 HestonSLVProcess(const ext::shared_ptr<HestonProcess> & hestonProcess,const ext::shared_ptr<LocalVolTermStructure> & leverageFct,const Real mixingFactor)32 HestonSLVProcess::HestonSLVProcess( 33 const ext::shared_ptr<HestonProcess>& hestonProcess, 34 const ext::shared_ptr<LocalVolTermStructure>& leverageFct, 35 const Real mixingFactor) 36 : mixingFactor_(mixingFactor), 37 hestonProcess_(hestonProcess), 38 leverageFct_(leverageFct) { 39 registerWith(hestonProcess); 40 update(); 41 }; 42 update()43 void HestonSLVProcess::update() { 44 v0_ = hestonProcess_->v0(); 45 kappa_ = hestonProcess_->kappa(); 46 theta_ = hestonProcess_->theta(); 47 sigma_ = hestonProcess_->sigma(); 48 rho_ = hestonProcess_->rho(); 49 mixedSigma_ = mixingFactor_ * sigma_; 50 } 51 drift(Time t,const Array & x) const52 Disposable<Array> HestonSLVProcess::drift(Time t, const Array& x) const { 53 Array tmp(2); 54 55 const Volatility vol = 56 std::max(1e-8, std::sqrt(x[1])*leverageFct_->localVol(t, x[0], true)); 57 58 tmp[0] = riskFreeRate()->forwardRate(t, t, Continuous) 59 - dividendYield()->forwardRate(t, t, Continuous) 60 - 0.5*vol*vol; 61 62 tmp[1] = kappa_*(theta_ - x[1]); 63 64 return tmp; 65 } 66 diffusion(Time t,const Array & x) const67 Disposable<Matrix> HestonSLVProcess::diffusion(Time t, const Array& x) 68 const { 69 70 const Real vol = 71 std::max(1e-8, std::sqrt(x[1])*leverageFct_->localVol(t, x[0], true)); 72 73 const Real sigma2 = mixedSigma_ * std::sqrt(x[1]); 74 const Real sqrhov = std::sqrt(1.0 - rho_*rho_); 75 76 Matrix tmp(2,2); 77 tmp[0][0] = vol; tmp[0][1] = 0.0; 78 tmp[1][0] = rho_*sigma2; tmp[1][1] = sqrhov*sigma2; 79 80 return tmp; 81 } 82 evolve(Time t0,const Array & x0,Time dt,const Array & dw) const83 Disposable<Array> HestonSLVProcess::evolve( 84 Time t0, const Array& x0, Time dt, const Array& dw) const { 85 Array retVal(2); 86 87 const Real ex = std::exp(-kappa_*dt); 88 89 const Real m = theta_+(x0[1]-theta_)*ex; 90 const Real s2 = x0[1]*mixedSigma_*mixedSigma_*ex/kappa_*(1-ex) 91 + theta_*mixedSigma_*mixedSigma_/(2*kappa_)*(1-ex)*(1-ex); 92 const Real psi = s2/(m*m); 93 94 if (psi < 1.5) { 95 const Real b2 = 2/psi-1+std::sqrt(2/psi*(2/psi-1)); 96 const Real b = std::sqrt(b2); 97 const Real a = m/(1+b2); 98 99 retVal[1] = a*(b+dw[1])*(b+dw[1]); 100 } 101 else { 102 const Real p = (psi-1)/(psi+1); 103 const Real beta = (1-p)/m; 104 const Real u = CumulativeNormalDistribution()(dw[1]); 105 106 retVal[1] = ((u <= p) ? 0.0 : std::log((1-p)/(1-u))/beta); 107 } 108 109 const Real mu = riskFreeRate()->forwardRate(t0, t0+dt, Continuous) 110 - dividendYield()->forwardRate(t0, t0+dt, Continuous); 111 112 const Real rho1 = std::sqrt(1-rho_*rho_); 113 114 const Volatility l_0 = leverageFct_->localVol(t0, x0[0], true); 115 const Real v_0 = 0.5*(x0[1]+retVal[1])*l_0*l_0; 116 117 retVal[0] = x0[0]*std::exp(mu*dt - 0.5*v_0*dt 118 + rho_/mixedSigma_*l_0 * ( 119 retVal[1] - kappa_*theta_*dt 120 + 0.5*(x0[1]+retVal[1])*kappa_*dt - x0[1]) 121 + rho1*std::sqrt(v_0*dt)*dw[0]); 122 123 return retVal; 124 } 125 } 126