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/* -*- mode: c++; tab-width: 4; indent-tabs-mode: nil; c-basic-offset: 4 -*- */
/*
Copyright (C) 2017 Klaus Spanderen
Copyright (C) 2022 Ignacio Anguita
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.
*/
#include <ql/math/functional.hpp>
#include <ql/pricingengines/vanilla/coshestonengine.hpp>
namespace QuantLib {
COSHestonEngine::COSHestonEngine(
const ext::shared_ptr<HestonModel>& model, Real L, Size N)
: GenericModelEngine<HestonModel,
VanillaOption::arguments,
VanillaOption::results>(model),
L_(L), N_(N),
kappa_(model_->kappa()),
theta_(model_->theta()),
sigma_(model_->sigma()),
rho_ (model_->rho()) ,
v0_ (model_->v0()) { }
void COSHestonEngine::update() {
kappa_ = model_->kappa();
theta_ = model_->theta();
sigma_ = model_->sigma();
rho_ = model_->rho();
v0_ = model_->v0();
GenericModelEngine<HestonModel,
VanillaOption::arguments,
VanillaOption::results>::update();
}
void COSHestonEngine::calculate() const {
// this is a european option pricer
QL_REQUIRE(arguments_.exercise->type() == Exercise::European,
"not an European option");
// plain vanilla
const ext::shared_ptr<PlainVanillaPayoff> payoff =
ext::dynamic_pointer_cast<PlainVanillaPayoff>(arguments_.payoff);
QL_REQUIRE(payoff, "non plain vanilla payoff given");
const ext::shared_ptr<HestonProcess> process = model_->process();
const Date maturityDate = arguments_.exercise->lastDate();
const Time maturity = process->time(maturityDate);
const Real cum1 = c1(maturity);
const Real w = std::sqrt(std::fabs(c2(maturity))
// the 4th order doesn't necessarily improve the precision
// + std::sqrt(std::fabs(c4(maturity)))
);
const Real k = payoff->strike();
const Real spot = process->s0()->value();
QL_REQUIRE(spot > 0.0, "negative or null underlying given");
const DiscountFactor df
= process->riskFreeRate()->discount(maturityDate);
const DiscountFactor qf
= process->dividendYield()->discount(maturityDate);
const Real fwd = spot*qf/df;
const Real x = std::log(fwd/k);
const Real a = x + cum1 - L_*w;
const Real b = x + cum1 + L_*w;
// Check if it exceeds the truncation bound
if (x >= b/2 || x <= a/2) {
//returns lower/upper bounds
if (payoff->optionType() == Option::Put)
results_.value = std::max(-spot*qf+k*df,0.0);
else if (payoff->optionType() == Option::Call)
results_.value = std::max(spot*qf-k*df,0.0);
else
QL_FAIL("unknown payoff type");
return;
}
const Real d = 1.0/(b-a);
const Real expA = std::exp(a);
Real s = chF(0, maturity).real()*(expA-1-a)*d;
for (Size n=1; n < N_; ++n) {
const Real r = n*M_PI*d;
const Real U_n = 2.0*d*( 1.0/(1.0 + r*r)
*(expA + r*std::sin(r*a) - std::cos(r*a)) - 1.0/r*std::sin(r*a));
s += U_n*(chF(r, maturity)
*std::exp(std::complex<Real>(0, r*(x-a)))).real();
}
if (payoff->optionType() == Option::Put)
results_.value = k*df*s;
else if (payoff->optionType() == Option::Call) {
const DiscountFactor qf
= process->dividendYield()->discount(maturityDate);
results_.value = spot*qf - k*df*(1-s);
}
else
QL_FAIL("unknown payoff type");
}
Real COSHestonEngine::muT(Time t) const {
return std::log( model_->process()->dividendYield()->discount(t)
/ model_->process()->riskFreeRate()->discount(t));
}
std::complex<Real> COSHestonEngine::chF(Real u, Real t)
const {
const Real sigma2 = sigma_*sigma_;
const std::complex<Real> D = std::sqrt(
squared(std::complex<Real>(kappa_, -rho_*sigma_*u))
+ std::complex<Real>(u*u, u)*sigma2);
const std::complex<Real> g(kappa_, -rho_*sigma_*u);
const std::complex<Real> G = (g-D)/(g+D);
return std::exp(
v0_/(sigma2)*(1.0-std::exp(-D*t))/(1.0-G*std::exp(-D*t))
*(g-D) + kappa_*theta_/sigma2*((g-D)*t
-2.0*std::log((1.0-G*std::exp(-D*t))/(1.0-G)))
);
}
/*
Mathematica program to calculate the n-th cumulant
d[z_] := Sqrt[(kappa -i*rho*sigma*z)^2 + (z*z+i*z)*sigma^2]
g[z_] := (kappa -i*rho*sigma*z - d[z])/(kappa -i*rho*sigma*z + d[z])
phi[z_] := Exp[ v0/(sigma^2)*(1-Exp[-d[z]*t])/(1-g[z]*Exp[-d[z]*t])
*(kappa -i*rho*sigma*z - d[z]) + kappa*theta/sigma^2
*((kappa -i*rho*sigma*z-d[z])*t
-2*Log[(1-g[z]*Exp[-d[z]*t])/(1-g[z]) ]) ]
e[z_] := Log[phi[-i*z]]
// for C++ formatting see
// http://mathematica.stackexchange.com/questions/114175/cform-not-getting-exp-pow-log-functions
cpp = RawBoxes[Replace[ToBoxes@#,
InterpretationBox[a_, b_, c___] :> With[{aa =
StringReplace[ a, {"Sqrt" -> "std::sqrt", "Power(E," -> "std::exp(",
"Power" -> "std::pow"}]}, aa], {0, Infinity}]] &;
c[n_] := cpp@CForm[FullSimplify[Derivative[n][e][0],
kappa > 0 && theta > 0 && v0 > 0 && sigma > 0 &&
rho [Element] {-1, 1} && i^2 == -1]]
*/
Real COSHestonEngine::c1(Time t) const {
return (-theta_ + std::exp(kappa_*t)
*( theta_ - kappa_*t*theta_ -
v0_) + v0_)/(2*std::exp(kappa_*t)*kappa_);
}
Real COSHestonEngine::c2(Time t) const {
const Real sigma2 = sigma_*sigma_;
const Real kappa2 = kappa_*kappa_;
const Real kappa3 = kappa2*kappa_;
return (sigma2*(theta_ - 2*v0_) +
std::exp(2*kappa_*t)*(8*kappa3*t*theta_ -
8*kappa2*(theta_ + rho_*sigma_*t*theta_ - v0_) +
sigma2*(-5*theta_ + 2*v0_) + 2*kappa_*sigma_*(8*rho_*theta_ +
sigma_*t*theta_ - 4*rho_*v0_)) +
4*std::exp(kappa_*t)*(sigma2*theta_ -
2*kappa2*(-1 + rho_*sigma_*t)*(theta_ - v0_) +
kappa_*sigma_*(sigma_*t*(theta_ - v0_) + 2*rho_*(-2*theta_ +
v0_))))/(8.*std::exp(2*kappa_*t)*kappa3);
}
Real COSHestonEngine::c3(Time t) const {
const Real sigma2 = sigma_*sigma_;
const Real sigma3 = sigma2*sigma_;
const Real kappa2 = kappa_*kappa_;
const Real kappa3 = kappa2*kappa_;
const Real kappa4 = kappa3*kappa_;
const Real rho2 = rho_*rho_;
return
-(sigma_*(sigma3*(theta_ - 3*v0_) +
std::exp(3*kappa_*t)*(2*(-11*sigma3 -
24*kappa4*rho_*t + 3*kappa_*sigma2*(20*rho_ +
sigma_*t) - 6*kappa2*sigma_*(5 + 3*rho_*(4*rho_ + sigma_*t)) +
12*kappa3*(sigma_*t + 2*rho_*(2 + rho_*sigma_*t)))*theta_ -
6*(2*kappa_*rho_ - sigma_)*(4*kappa2 - 4*kappa_*rho_*sigma_ +
sigma2)*v0_) + 6*std::exp(kappa_*t)*sigma_*(-2*kappa2*(-1 +
rho_*sigma_*t)*(theta_ - 2*v0_) + sigma2*(theta_ - v0_) +
kappa_*sigma_*(-4*rho_*theta_ + sigma_*t*theta_ + 6*rho_*v0_ - 2*sigma_*t*v0_)) +
3*std::exp(2*kappa_*t)*(2*kappa_*sigma2*(-16*rho_*theta_ +
sigma_*t*(3*theta_ - v0_)) + 8*kappa4*rho_*t*(-2 +
rho_*sigma_*t)*(theta_ - v0_) + sigma3*(5*theta_ + v0_) +
8*kappa3*(-(rho_*(4 + sigma2*t*t)*theta_) + 2*sigma_*t*(theta_ - v0_) +
2*rho2*sigma_*t*(2*theta_ - v0_) + rho_*(2 +
sigma2*t*t)*v0_) + 2*kappa2*sigma_*((8
+ 24*rho2 - 16*rho_*sigma_*t + sigma2*t*t)*theta_ - (8*rho2 -
8*rho_*sigma_*t + sigma2*t*t)*v0_))))/(16.*std::exp(3*kappa_*t)*
kappa_*kappa4);
}
Real COSHestonEngine::c4(Time t) const {
const Real sigma2 = sigma_*sigma_;
const Real sigma3 = sigma2*sigma_;
const Real sigma4 = sigma2*sigma2;
const Real kappa2 = kappa_*kappa_;
const Real kappa3 = kappa2*kappa_;
const Real kappa4 = kappa2*kappa2;
const Real kappa5 = kappa2*kappa3;
const Real kappa6 = kappa3*kappa3;
const Real kappa7 = kappa4*kappa3;
const Real rho2 = rho_*rho_;
const Real rho3 = rho2*rho_;
const Time t2 = t*t;
const Time t3 = t2*t;
return
(sigma2*(3*sigma4*(theta_ - 4*v0_) +
3*std::exp(4*kappa_*t)*((-93*sigma4 +
64*kappa5*(t + 4*rho2*t) +
4*kappa_*sigma3*(176*rho_ + 5*sigma_*t) -
32*kappa2*sigma2*(11 + 50*rho2 +
5*rho_*sigma_*t) + 32*kappa3*sigma_*(3*sigma_*t + 4*rho_*(10 +
8*rho2 + 3*rho_*sigma_*t)) - 32*kappa4*(5 +
4*rho_*(6*rho_ + (3 + 2*rho2)*sigma_*t)))*theta_ +
4*(4*kappa2 - 4*kappa_*rho_*sigma_ +
sigma2)*(4*kappa2*(1 + 4*rho2) -
20*kappa_*rho_*sigma_ + 5*sigma2)*v0_) +
24*std::exp(kappa_*t)*sigma2*(-2*kappa2*(-1 +
rho_*sigma_*t)*(theta_ - 3*v0_) + sigma2*(theta_ - 2*v0_) +
kappa_*sigma_*(-4*rho_*theta_ + sigma_*t*theta_ + 10*rho_*v0_ -
3*sigma_*t*v0_)) + 12*std::exp(2*kappa_*t)*(sigma4*(7*theta_ -
4*v0_) + 8*kappa4*(1 + 2*rho_*sigma_*t*(-2 +
rho_*sigma_*t))*(theta_ - 2*v0_) +
2*kappa_*sigma3*(-24*rho_*theta_ + 5*sigma_*t*theta_ +
20*rho_*v0_ - 6*sigma_*t*v0_) + 4*kappa2*sigma2*((6
+ 20*rho2 - 14*rho_*sigma_*t +
sigma2*t2)*theta_ - 2*(3 + 12*rho2 -
10*rho_*sigma_*t + sigma2*t2)*v0_) +
8*kappa3*sigma_*((3*sigma_*t + 2*rho_*(-4 + sigma_*t*(4*rho_ -
sigma_*t)))*theta_ + 2*(-3*sigma_*t + 2*rho_*(3 + sigma_*t*(-3*rho_ +
sigma_*t)))*v0_)) -
8*std::exp(3*kappa_*t)*(16*kappa6*rho2*t2*(-3 + rho_*sigma_*t)*(theta_ - v0_) - 3*sigma4*(7*theta_ +
2*v0_) + 2*kappa3*sigma_*((192*(rho_ + rho3) -
6*(9 + 40*rho2)*sigma_*t + 42*rho_*sigma2*t2 -
sigma3*t3)*theta_ + (-48*rho3 + 18*(1
+ 4*rho2)*sigma_*t - 24*rho_*sigma2*t2
+ sigma3*t3)*v0_) + 12*kappa4*((-4 -
24*rho2 + 8*rho_*(4 + 3*rho2)*sigma_*t - (3 +
14*rho2)*sigma2*t2 + rho_*sigma3*t3)*theta_ + (8*rho2 -
8*rho_*(2 + rho2)*sigma_*t + (3 +
8*rho2)*sigma2*t2 - rho_*sigma3*t3)*v0_) -
6*kappa2*sigma2*((15 + 80*rho2 -
35*rho_*sigma_*t + 2*sigma2*t2)*theta_ + (3 +
sigma_*t*(7*rho_ - sigma_*t))*v0_) + 24*kappa5*t*((-2 +
rho_*(4*sigma_*t + rho_*(-8 + sigma_*t*(4*rho_ - sigma_*t))))*theta_ + (2 +
rho_*(-4*sigma_*t + rho_*(4 + sigma_*t*(-2*rho_ + sigma_*t))))*v0_) +
3*kappa_*sigma3*(sigma_*t*(-9*theta_ + v0_) + 10*rho_*(6*theta_
+ v0_)))))/(64.*std::exp(4*kappa_*t)*kappa7);
}
Real COSHestonEngine::mu(Time t) const {
return c1(t);
}
Real COSHestonEngine::var(Time t) const {
return c2(t);
}
Real COSHestonEngine::skew(Time t) const {
return c3(t)/std::pow(c2(t), 1.5);
}
Real COSHestonEngine::kurtosis(Time t) const {
return c4(t)/squared(c2(t));
}
}
|
