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/* -*- mode: c++; tab-width: 4; indent-tabs-mode: nil; c-basic-offset: 4 -*- */
/*
Copyright (C) 2020 Lew Wei Hao
Copyright (C) 2021 Magnus Mencke
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 coxingersollrossprocess.hpp
\brief CoxIngersollRoss process
*/
#ifndef quantlib_coxingersollross_process_hpp
#define quantlib_coxingersollross_process_hpp
#include <ql/stochasticprocess.hpp>
#include <ql/math/distributions/normaldistribution.hpp>
namespace QuantLib {
//! CoxIngersollRoss process class
/*! This class describes the CoxIngersollRoss process governed by
\f[
dx(t) = k (\theta - x(t)) dt + \sigma \sqrt{x(t)} dW(t).
\f]
The process is discretized using the Quadratic Exponential scheme.
For details see Leif Andersen,
Efficient Simulation of the Heston Stochastic Volatility Model.
\ingroup processes
*/
class CoxIngersollRossProcess : public StochasticProcess1D {
public:
CoxIngersollRossProcess(Real speed,
Volatility vol,
Real x0 = 0.0,
Real level = 0.0);
//@{
Real drift(Time t, Real x) const override;
Real diffusion(Time t, Real x) const override;
Real expectation(Time t0, Real x0, Time dt) const override;
Real stdDeviation(Time t0, Real x0, Time dt) const override;
//@}
Real x0() const override;
Real speed() const;
Real volatility() const;
Real level() const;
Real variance(Time t0, Real x0, Time dt) const override;
Real evolve (Time t0,
Real x0,
Time dt,
Real dw) const override;
private:
Real x0_, speed_, level_;
Volatility volatility_;
};
// inline
inline Real CoxIngersollRossProcess::x0() const {
return x0_;
}
inline Real CoxIngersollRossProcess::speed() const {
return speed_;
}
inline Real CoxIngersollRossProcess::volatility() const {
return volatility_;
}
inline Real CoxIngersollRossProcess::level() const {
return level_;
}
inline Real CoxIngersollRossProcess::drift(Time, Real x) const {
return speed_ * (level_ - x);
}
inline Real CoxIngersollRossProcess::diffusion(Time, Real) const {
return volatility_;
}
inline Real CoxIngersollRossProcess::expectation(Time, Real x0,
Time dt) const {
return level_ + (x0 - level_) * std::exp(-speed_*dt);
}
inline Real CoxIngersollRossProcess::stdDeviation(Time t, Real x0,
Time dt) const {
return std::sqrt(variance(t,x0,dt));
}
inline Real CoxIngersollRossProcess::evolve (Time t0,
Real x0,
Time dt,
Real dw) const {
Real result;
const Real ex = std::exp(-speed_*dt);
const Real m = level_+(x0-level_)*ex;
const Real s2 = x0*volatility_*volatility_*ex/speed_*(1-ex)
+ level_*volatility_*volatility_/(2*speed_)*(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);
result = a*(b+dw)*(b+dw);
}
else {
const Real p = (psi-1)/(psi+1);
const Real beta = (1-p)/m;
const Real u = CumulativeNormalDistribution()(dw);
result = ((u <= p) ? 0.0 : Real(std::log((1-p)/(1-u))/beta));
}
return result;
}
}
#endif
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