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|
/* -*- mode: c++; tab-width: 4; indent-tabs-mode: nil; c-basic-offset: 4 -*- */
/*
Copyright (C) 2014 Fabien Le Floc'h
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 analytichestonengine.hpp
\brief analytic Heston expansion engine
*/
#include <ql/pricingengines/blackformula.hpp>
#include <ql/instruments/payoffs.hpp>
#include <ql/pricingengines/vanilla/hestonexpansionengine.hpp>
#if defined(QL_PATCH_MSVC)
#pragma warning(disable: 4180)
#endif
using std::exp;
using std::pow;
using std::log;
using std::sqrt;
namespace QuantLib {
HestonExpansionEngine::HestonExpansionEngine(
const ext::shared_ptr<HestonModel>& model,
HestonExpansionFormula formula)
: GenericModelEngine<HestonModel,
VanillaOption::arguments,
VanillaOption::results>(model),
formula_(formula) {
}
void HestonExpansionEngine::calculate() const
{
// this is a european option pricer
QL_REQUIRE(arguments_.exercise->type() == Exercise::European,
"not an European option");
// plain vanilla
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 Real riskFreeDiscount = process->riskFreeRate()->discount(
arguments_.exercise->lastDate());
const Real dividendDiscount = process->dividendYield()->discount(
arguments_.exercise->lastDate());
const Real spotPrice = process->s0()->value();
QL_REQUIRE(spotPrice > 0.0, "negative or null underlying given");
const Real strikePrice = payoff->strike();
const Real term = process->time(arguments_.exercise->lastDate());
//possible optimization:
// if term=lastTerm & model=lastModel & formula=lastApprox, reuse approx.
const Real forward = spotPrice*dividendDiscount/riskFreeDiscount;
Real vol;
switch(formula_) {
case LPP2: {
LPP2HestonExpansion expansion(model_->kappa(), model_->theta(),
model_->sigma(), model_->v0(),
model_->rho(), term);
vol = expansion.impliedVolatility(strikePrice, forward);
break;
}
case LPP3: {
LPP3HestonExpansion expansion(model_->kappa(), model_->theta(),
model_->sigma(), model_->v0(),
model_->rho(), term);
vol = expansion.impliedVolatility(strikePrice, forward);
break;
}
case Forde: {
FordeHestonExpansion expansion(model_->kappa(), model_->theta(),
model_->sigma(), model_->v0(),
model_->rho(), term);
vol = expansion.impliedVolatility(strikePrice, forward);
break;
}
default:
QL_FAIL("unknown expansion formula");
}
const Real price = blackFormula(payoff, forward, vol*sqrt(term),
riskFreeDiscount, 0);
results_.value = price;
}
LPP2HestonExpansion::LPP2HestonExpansion(Real kappa, Real theta, Real sigma,
Real v0, Real rho, Real term) {
ekt = exp(kappa*term);
e2kt = ekt*ekt;
e3kt = e2kt*ekt;
e4kt = e2kt*e2kt;
coeffs[0] = z0(term, kappa, theta, sigma, v0, rho);
coeffs[1] = z1(term, kappa, theta, sigma, v0, rho);
coeffs[2] = z2(term, kappa, theta, sigma, v0, rho);
}
static Real fastpow(Real x, int y) {
return pow(x,y);
}
Real LPP2HestonExpansion::impliedVolatility(const Real strike,
const Real forward) const {
Real x = log(strike/forward);
Real vol = coeffs[0]+x*(coeffs[1]+(x*coeffs[2]));
return std::max(1e-8,vol);
}
Real LPP2HestonExpansion::z0(Real t, Real kappa, Real theta,
Real delta, Real y, Real rho) const {
return (4*pow(delta,2)*kappa*(-theta - 4*ekt*(theta + kappa*t*(theta - y)) +
e2kt*((5 - 2*kappa*t)*theta - 2*y) + 2*y)*
((1 + ekt*(-1 + kappa*t))*theta + (-1 + ekt)*y) +
128*ekt*pow(kappa,3)*
pow((1 + ekt*(-1 + kappa*t))*theta + (-1 + ekt)*y,2) +
32*delta*ekt*pow(kappa,2)*rho*
((1 + ekt*(-1 + kappa*t))*theta + (-1 + ekt)*y)*
((2 + kappa*t + ekt*(-2 + kappa*t))*theta +
(-1 + ekt - kappa*t)*y) +
pow(delta,2)*ekt*pow(rho,2)*
(-theta + kappa*t*theta + (theta - y)/ekt + y)*
pow((2 + kappa*t + ekt*(-2 + kappa*t))*theta +
(-1 + ekt - kappa*t)*y,2) +
(48*pow(delta,2)*e2kt*pow(kappa,2)*pow(rho,2)*
pow((2 + kappa*t + ekt*(-2 + kappa*t))*theta +
(-1 + ekt - kappa*t)*y,2))/
((1 + ekt*(-1 + kappa*t))*theta + (-1 + ekt)*y) -
pow(delta,2)*pow(rho,2)*((1 + ekt*(-1 + kappa*t))*theta +
(-1 + ekt)*y)*pow((2 + kappa*t + ekt*(-2 + kappa*t))*
theta + (-1 + ekt - kappa*t)*y,2) +
2*pow(delta,2)*kappa*((1 + ekt*(-1 + kappa*t))*theta +
(-1 + ekt)*y)*(theta - 2*y +
e2kt*(-5*theta + 2*kappa*t*theta + 2*y +
8*pow(rho,2)*((-3 + kappa*t)*theta + y)) +
4*ekt*(theta + kappa*t*theta - kappa*t*y +
pow(rho,2)*((6 + kappa*t*(4 + kappa*t))*theta - (2 + kappa*t*(2 + kappa*t))*y))) -
(8*pow(delta,2)*pow(kappa,2)*((1 + ekt*(-1 + kappa*t))*theta +
(-1 + ekt)*y)*(theta - 2*y +
e2kt*(-5*theta + 2*kappa*t*theta + 2*y +
8*pow(rho,2)*((-3 + kappa*t)*theta + y)) +
4*ekt*(theta + kappa*t*theta - kappa*t*y +
pow(rho,2)*((6 + kappa*t*(4 + kappa*t))*theta - (2 + kappa*t*(2 + kappa*t))*y))))
/(-theta + kappa*t*theta + (theta - y)/ekt + y))/
(128.*e3kt*pow(kappa,5)*pow(t,2)*
pow((-theta + kappa*t*theta + (theta - y)/ekt + y)/(kappa*t),1.5));
}
Real LPP2HestonExpansion::z1(Real t, Real kappa, Real theta,
Real delta, Real y, Real rho) const {
return (delta*rho*(-(delta*fastpow(-1 + ekt,2)*rho*(4*theta - y)*y) +
2*ekt*fastpow(kappa,3)*fastpow(t,2)*theta*
((2 + 2*ekt + delta*rho*t)*theta - (2 + delta*rho*t)*y) -
2*(-1 + ekt)*kappa*(2*theta - y)*
((-1 + ekt)*(-2 + delta*rho*t)*theta +
(-2 + 2*ekt + delta*rho*t)*y) +
fastpow(kappa,2)*t*((-1 + ekt)*
(-4 + delta*rho*t + ekt*(-12 + delta*rho*t))*fastpow(theta,2) +
2*(-4 + 4*e2kt + delta*rho*t + 3*delta*ekt*rho*t)*theta*
y - (-4 + delta*rho*t + 2*ekt*(2 + delta*rho*t))*fastpow(y,2))))/
(8.*fastpow(kappa,2)*t*sqrt((-theta + kappa*t*theta + (theta - y)/ekt + y)/
(kappa*t))*fastpow((1 + ekt*(-1 + kappa*t))*theta + (-1 + ekt)*y,
2));
}
Real LPP2HestonExpansion::z2(Real t, Real kappa, Real theta,
Real delta, Real y, Real rho) const {
return (fastpow(delta,2)*sqrt((-theta + kappa*t*theta + (theta - y)/ekt + y)/(kappa*t))*
(-12*fastpow(rho,2)*fastpow((2 + kappa*t + ekt*(-2 + kappa*t))*theta +
(-1 + ekt - kappa*t)*y,2) +
(-theta + kappa*t*theta + (theta - y)/ekt + y)*
(theta - 2*y + e2kt*
(-5*theta + 2*kappa*t*theta + 2*y + 8*fastpow(rho,2)*((-3 + kappa*t)*theta + y)) +
4*ekt*(theta + kappa*t*theta - kappa*t*y +
fastpow(rho,2)*((6 + kappa*t*(4 + kappa*t))*theta - (2 + kappa*t*(2 + kappa*t))*y))))
)/(16.*e2kt*fastpow(-theta + kappa*t*theta + (theta - y)/ekt + y,
4));
}
FordeHestonExpansion::FordeHestonExpansion(Real kappa, Real theta,
Real sigma, Real v0,
Real rho, Real term) {
Real v0Sqrt = sqrt(v0);
Real rhoBarSquare = 1 - rho * rho;
Real sigma00 = v0Sqrt;
Real sigma01 = v0Sqrt * (rho * sigma / (4 * v0)); //term in x
Real sigma02 = v0Sqrt * ((1 - 5 * rho * rho / 2) / 24 * sigma * sigma/ (v0 * v0)); //term in x*x
Real a00 = -sigma * sigma / 12 * (1 - rho * rho / 4) + v0 * rho * sigma / 4 + kappa / 2 * (theta - v0);
Real a01 = rho * sigma / (24 * v0) * (sigma * sigma * rhoBarSquare - 2 * kappa * (theta + v0) + v0 * rho * sigma); //term in x
Real a02 = (176 * sigma * sigma - 480 * kappa * theta - 712 * rho * rho * sigma * sigma + 521 * rho * rho * rho * rho * sigma * sigma + 40 * sigma * rho * rho * rho * v0 + 1040 * kappa * theta * rho * rho - 80 * v0 * kappa * rho * rho) * sigma * sigma / (v0 * v0 * 7680);
coeffs[0] = sigma00*sigma00+a00*term;
coeffs[1] = sigma00*sigma01*2+a01*term;
coeffs[2] = sigma00*sigma02*2+sigma01*sigma01+a02*term;
coeffs[3] = sigma01*sigma02*2;
coeffs[4] = sigma02*sigma02;
}
Real FordeHestonExpansion::impliedVolatility(const Real strike,
const Real forward) const {
Real x = log(strike/forward);
Real var = coeffs[0]+x*(coeffs[1]+(x*(coeffs[2]+x*(coeffs[3]+(x*coeffs[4])))));
var = std::max(1e-8,var);
return sqrt(var);
}
Real LPP3HestonExpansion::z0(Real t, Real kappa, Real theta,
Real delta, Real y, Real rho) const {
return (96*pow(delta,2)*ekt*pow(kappa,3)*
(-theta - 4*ekt*(theta + kappa*t*(theta - y)) +
e2kt*((5 - 2*kappa*t)*theta - 2*y) + 2*y)*
((1 + ekt*(-1 + kappa*t))*theta + (-1 + ekt)*y) +
3072*e2kt*pow(kappa,5)*
pow((1 + ekt*(-1 + kappa*t))*theta + (-1 + ekt)*y,2) +
96*pow(delta,3)*ekt*pow(kappa,2)*rho*
((1 + ekt*(-1 + kappa*t))*theta + (-1 + ekt)*y)*
(-2*theta - kappa*t*theta - 2*ekt*(2 + kappa*t)*
(2*theta + kappa*t*(theta - y)) + e2kt*((10 - 3*kappa*t)*theta - 3*y) +
3*y + 2*kappa*t*y) + 768*delta*e2kt*pow(kappa,4)*rho*
((1 + ekt*(-1 + kappa*t))*theta + (-1 + ekt)*y)*
((2 + kappa*t + ekt*(-2 + kappa*t))*theta +
(-1 + ekt - kappa*t)*y) +
6*pow(delta,3)*kappa*rho*(-theta - 4*ekt*(theta + kappa*t*(theta - y)) +
e2kt*((5 - 2*kappa*t)*theta - 2*y) + 2*y)*
((1 + ekt*(-1 + kappa*t))*theta + (-1 + ekt)*y)*
((2 + kappa*t + ekt*(-2 + kappa*t))*theta +
(-1 + ekt - kappa*t)*y) +
24*pow(delta,2)*e2kt*pow(kappa,2)*pow(rho,2)*
(-theta + kappa*t*theta + (theta - y)/ekt + y)*
pow((2 + kappa*t + ekt*(-2 + kappa*t))*theta +
(-1 + ekt - kappa*t)*y,2) +
(1152*pow(delta,2)*e3kt*pow(kappa,4)*pow(rho,2)*
pow((2 + kappa*t + ekt*(-2 + kappa*t))*theta +
(-1 + ekt - kappa*t)*y,2))/
((1 + ekt*(-1 + kappa*t))*theta + (-1 + ekt)*y) -
24*pow(delta,2)*ekt*pow(kappa,2)*pow(rho,2)*
((1 + ekt*(-1 + kappa*t))*theta + (-1 + ekt)*y)*
pow((2 + kappa*t + ekt*(-2 + kappa*t))*theta +
(-1 + ekt - kappa*t)*y,2) +
80*pow(delta,3)*ekt*kappa*pow(rho,3)*
pow((2 + kappa*t + ekt*(-2 + kappa*t))*theta +
(-1 + ekt - kappa*t)*y,3) +
pow(delta,3)*ekt*pow(rho,3)*
(-theta + kappa*t*theta + (theta - y)/ekt + y)*
pow((2 + kappa*t + ekt*(-2 + kappa*t))*theta +
(-1 + ekt - kappa*t)*y,3) -
(1440*pow(delta,3)*e3kt*pow(kappa,3)*pow(rho,3)*
pow((2 + kappa*t + ekt*(-2 + kappa*t))*theta +
(-1 + ekt - kappa*t)*y,3))/
pow((1 + ekt*(-1 + kappa*t))*theta + (-1 + ekt)*y,2) -
(528*pow(delta,3)*e2kt*pow(kappa,2)*pow(rho,3)*
pow((2 + kappa*t + ekt*(-2 + kappa*t))*theta +
(-1 + ekt - kappa*t)*y,3))/
((1 + ekt*(-1 + kappa*t))*theta + (-1 + ekt)*y) -
3*pow(delta,3)*pow(rho,3)*((1 + ekt*(-1 + kappa*t))*theta +
(-1 + ekt)*y)*pow((2 + kappa*t + ekt*(-2 + kappa*t))*
theta + (-1 + ekt - kappa*t)*y,3) +
384*pow(delta,3)*e2kt*pow(kappa,3)*rho*
((2 + kappa*t + 2*ekt*pow(2 + kappa*t,2) +
e2kt*(-10 + 3*kappa*t))*theta +
(-3 + 3*e2kt - 2*kappa*t - 2*ekt*kappa*t*(2 + kappa*t))*y) -
(576*pow(delta,3)*e2kt*pow(kappa,3)*rho*
((2 + kappa*t + ekt*(-2 + kappa*t))*theta +
(-1 + ekt - kappa*t)*y)*
((1 + e2kt*(-5 + 2*kappa*t + 4*pow(rho,2)*(-3 + kappa*t)) +
2*ekt*(2 + 2*kappa*t +
pow(rho,2)*(6 + 4*kappa*t + pow(kappa,2)*pow(t,2))))*theta +
2*(-1 + e2kt*(1 + 2*pow(rho,2)) -
ekt*(2*kappa*t +
pow(rho,2)*(2 + 2*kappa*t + pow(kappa,2)*pow(t,2))))*y))/
((1 + ekt*(-1 + kappa*t))*theta + (-1 + ekt)*y) +
pow(delta,3)*rho*((1 + ekt*(-1 + kappa*t))*theta +
(-1 + ekt)*y)*((2 + kappa*t + ekt*(-2 + kappa*t))*theta +
(-1 + ekt - kappa*t)*y)*
(theta*(12*ekt*pow(kappa,3)*pow(rho,2)*pow(t,2) +
8*pow(-1 + ekt,2)*pow(rho,2)*theta -
(-1 + ekt)*kappa*
(3 + 8*pow(rho,2)*t*theta + ekt*(15 + 8*pow(rho,2)*(9 + t*theta)))
+ 2*pow(kappa,2)*t*(pow(rho,2)*t*theta +
2*ekt*(3 + pow(rho,2)*(12 + t*theta)) +
e2kt*(3 + pow(rho,2)*(12 + t*theta)))) -
2*(6*ekt*pow(kappa,3)*pow(rho,2)*pow(t,2) +
4*pow(-1 + ekt,2)*pow(rho,2)*theta +
2*pow(kappa,2)*t*(pow(rho,2)*t*theta +
ekt*(3 + pow(rho,2)*(6 + t*theta))) -
(-1 + ekt)*kappa*
(3 + 6*pow(rho,2)*t*theta + ekt*(3 + 2*pow(rho,2)*(6 + t*theta))))*
y + 2*pow(rho,2)*pow(1 - ekt + kappa*t,2)*pow(y,2)) -
(40*pow(delta,3)*kappa*rho*((1 + ekt*(-1 + kappa*t))*theta +
(-1 + ekt)*y)*((2 + kappa*t + ekt*(-2 + kappa*t))*theta +
(-1 + ekt - kappa*t)*y)*
(theta*(12*ekt*pow(kappa,3)*pow(rho,2)*pow(t,2) +
8*pow(-1 + ekt,2)*pow(rho,2)*theta -
(-1 + ekt)*kappa*
(3 + 8*pow(rho,2)*t*theta +
ekt*(15 + 8*pow(rho,2)*(9 + t*theta))) +
2*pow(kappa,2)*t*(pow(rho,2)*t*theta +
2*ekt*(3 + pow(rho,2)*(12 + t*theta)) +
e2kt*(3 + pow(rho,2)*(12 + t*theta)))) -
2*(6*ekt*pow(kappa,3)*pow(rho,2)*pow(t,2) +
4*pow(-1 + ekt,2)*pow(rho,2)*theta +
2*pow(kappa,2)*t*(pow(rho,2)*t*theta +
ekt*(3 + pow(rho,2)*(6 + t*theta))) -
(-1 + ekt)*kappa*
(3 + 6*pow(rho,2)*t*theta + ekt*(3 + 2*pow(rho,2)*(6 + t*theta)))
)*y + 2*pow(rho,2)*pow(1 - ekt + kappa*t,2)*pow(y,2)))/
(-theta + kappa*t*theta + (theta - y)/ekt + y) -
12*pow(delta,3)*kappa*rho*((1 + ekt*(-1 + kappa*t))*theta +
(-1 + ekt)*y)*(2*theta + kappa*t*theta - y - kappa*t*y +
ekt*((-2 + kappa*t)*theta + y))*
(theta - 2*y + e2kt*
(-5*theta + 2*kappa*t*theta + 2*y + 4*pow(rho,2)*((-3 + kappa*t)*theta + y)) +
2*ekt*(2*(theta + kappa*t*(theta - y)) +
pow(rho,2)*((6 + kappa*t*(4 + kappa*t))*theta - (2 + kappa*t*(2 + kappa*t))*y))) +
(288*pow(delta,3)*pow(kappa,2)*rho*
((1 + ekt*(-1 + kappa*t))*theta + (-1 + ekt)*y)*
(2*theta + kappa*t*theta - y - kappa*t*y + ekt*((-2 + kappa*t)*theta + y))*
(theta - 2*y + e2kt*
(-5*theta + 2*kappa*t*theta + 2*y + 4*pow(rho,2)*((-3 + kappa*t)*theta + y)) +
2*ekt*(2*(theta + kappa*t*(theta - y)) +
pow(rho,2)*((6 + kappa*t*(4 + kappa*t))*theta - (2 + kappa*t*(2 + kappa*t))*y))))
/(-theta + kappa*t*theta + (theta - y)/ekt + y) +
48*pow(delta,2)*ekt*pow(kappa,3)*
((1 + ekt*(-1 + kappa*t))*theta + (-1 + ekt)*y)*
(theta - 2*y + e2kt*
(-5*theta + 2*kappa*t*theta + 2*y + 8*pow(rho,2)*((-3 + kappa*t)*theta + y)) +
4*ekt*(theta + kappa*t*theta - kappa*t*y +
pow(rho,2)*((6 + kappa*t*(4 + kappa*t))*theta - (2 + kappa*t*(2 + kappa*t))*y))) -
(192*pow(delta,2)*ekt*pow(kappa,4)*
((1 + ekt*(-1 + kappa*t))*theta + (-1 + ekt)*y)*
(theta - 2*y + e2kt*
(-5*theta + 2*kappa*t*theta + 2*y + 8*pow(rho,2)*((-3 + kappa*t)*theta + y)) +
4*ekt*(theta + kappa*t*theta - kappa*t*y +
pow(rho,2)*((6 + kappa*t*(4 + kappa*t))*theta - (2 + kappa*t*(2 + kappa*t))*y))))
/(-theta + kappa*t*theta + (theta - y)/ekt + y) +
3*pow(delta,3)*kappa*rho*((1 + ekt*(-1 + kappa*t))*theta +
(-1 + ekt)*y)*((2 + kappa*t + ekt*(-2 + kappa*t))*theta +
(-1 + ekt - kappa*t)*y)*
(theta - 2*y + e2kt*
(-5*theta + 2*kappa*t*theta + 2*y + 8*pow(rho,2)*((-3 + kappa*t)*theta + y)) +
4*ekt*(theta + kappa*t*theta - kappa*t*y +
pow(rho,2)*((6 + kappa*t*(4 + kappa*t))*theta - (2 + kappa*t*(2 + kappa*t))*y))) -
(12*pow(delta,3)*pow(kappa,2)*rho*
((1 + ekt*(-1 + kappa*t))*theta + (-1 + ekt)*y)*
((2 + kappa*t + ekt*(-2 + kappa*t))*theta +
(-1 + ekt - kappa*t)*y)*
(theta - 2*y + e2kt*
(-5*theta + 2*kappa*t*theta + 2*y + 8*pow(rho,2)*((-3 + kappa*t)*theta + y)) +
4*ekt*(theta + kappa*t*theta - kappa*t*y +
pow(rho,2)*((6 + kappa*t*(4 + kappa*t))*theta - (2 + kappa*t*(2 + kappa*t))*y))))
/(-theta + kappa*t*theta + (theta - y)/ekt + y) +
4*pow(delta,3)*kappa*rho*((1 + ekt*(-1 + kappa*t))*theta +
(-1 + ekt)*y)*(3*(theta - 2*y)*((2 + kappa*t)*theta - (1 + kappa*t)*y) +
3*ekt*(6*pow(theta,2) + theta*y - 2*pow(y,2) +
kappa*(13*t*pow(theta,2) + theta*(8 - 18*t*y) + 4*y*(-3 + t*y)) +
4*pow(kappa,2)*t*(theta + t*pow(theta,2) - 2*t*theta*y + y*(-2 + t*y))) +
3*e3kt*(10*pow(theta,2) +
2*pow(kappa,2)*t*theta*(6 + 8*pow(rho,2) + t*theta) - 9*theta*y + 2*pow(y,2) +
kappa*(-9*t*pow(theta,2) + 4*(3 + 4*pow(rho,2))*y +
theta*(-40 - 64*pow(rho,2) + 4*t*y))) +
e2kt*(-54*pow(theta,2) +
8*pow(kappa,4)*pow(rho,2)*pow(t,3)*(theta - y) + 39*theta*y - 6*pow(y,2) +
24*pow(kappa,3)*pow(t,2)*(theta + 2*pow(rho,2)*theta - (1 + pow(rho,2))*y) +
6*pow(kappa,2)*t*(3*t*pow(theta,2) - 8*(1 + pow(rho,2))*y +
theta*(16 + 24*pow(rho,2) - 3*t*y)) -
3*kappa*(5*t*pow(theta,2) + 2*y*(8*pow(rho,2) + 3*t*y) -
theta*(32 + 64*pow(rho,2) + 17*t*y)))) -
(48*pow(delta,3)*pow(kappa,2)*rho*
((1 + ekt*(-1 + kappa*t))*theta + (-1 + ekt)*y)*
(3*(theta - 2*y)*((2 + kappa*t)*theta - (1 + kappa*t)*y) +
3*ekt*(6*pow(theta,2) + theta*y - 2*pow(y,2) +
kappa*(13*t*pow(theta,2) + theta*(8 - 18*t*y) + 4*y*(-3 + t*y)) +
4*pow(kappa,2)*t*(theta + t*pow(theta,2) - 2*t*theta*y + y*(-2 + t*y))) +
3*e3kt*(10*pow(theta,2) +
2*pow(kappa,2)*t*theta*(6 + 8*pow(rho,2) + t*theta) - 9*theta*y +
2*pow(y,2) + kappa*(-9*t*pow(theta,2) + 4*(3 + 4*pow(rho,2))*y +
theta*(-40 - 64*pow(rho,2) + 4*t*y))) +
e2kt*(-54*pow(theta,2) +
8*pow(kappa,4)*pow(rho,2)*pow(t,3)*(theta - y) + 39*theta*y - 6*pow(y,2) +
24*pow(kappa,3)*pow(t,2)*
(theta + 2*pow(rho,2)*theta - (1 + pow(rho,2))*y) +
6*pow(kappa,2)*t*(3*t*pow(theta,2) - 8*(1 + pow(rho,2))*y +
theta*(16 + 24*pow(rho,2) - 3*t*y)) -
3*kappa*(5*t*pow(theta,2) + 2*y*(8*pow(rho,2) + 3*t*y) -
theta*(32 + 64*pow(rho,2) + 17*t*y)))))/
(-theta + kappa*t*theta + (theta - y)/ekt + y) +
(240*pow(delta,3)*e2kt*pow(kappa,2)*rho*
((2 + kappa*t + ekt*(-2 + kappa*t))*theta +
(-1 + ekt - kappa*t)*y)*
(12*ekt*pow(kappa,3)*pow(rho,2)*pow(t,2)*(theta - y) +
2*pow(-1 + ekt,2)*pow(rho,2)*pow(-2*theta + y,2) -
(-1 + ekt)*kappa*
(8*(1 + ekt)*pow(rho,2)*t*pow(theta,2) +
2*y*(-3 - 3*ekt*(1 + 4*pow(rho,2)) + 2*pow(rho,2)*t*y) +
theta*(3 - 12*pow(rho,2)*t*y + ekt*(15 + pow(rho,2)*(72 - 4*t*y)))
) + 2*pow(kappa,2)*t*(e2kt*theta*
(3 + pow(rho,2)*(12 + t*theta)) + pow(rho,2)*t*pow(theta - y,2) +
2*ekt*(pow(rho,2)*t*pow(theta,2) - 3*(y + 2*pow(rho,2)*y) +
theta*(3 + pow(rho,2)*(12 - t*y))))))/
((1 + ekt*(-1 + kappa*t))*theta + (-1 + ekt)*y))/
(3072.*e4kt*pow(kappa,7)*pow(t,2)*
pow((-theta + kappa*t*theta + (theta - y)/ekt + y)/(kappa*t),1.5));
}
Real LPP3HestonExpansion::z1(Real t, Real kappa, Real theta,
Real delta, Real y, Real rho) const {
return (delta*(768*e2kt*pow(kappa,4)*rho*
((2 + kappa*t + ekt*(-2 + kappa*t))*theta +
(-1 + ekt - kappa*t)*y) -
(576*delta*e2kt*pow(kappa,3)*pow(rho,2)*
pow((2 + kappa*t + ekt*(-2 + kappa*t))*theta +
(-1 + ekt - kappa*t)*y,2))/
((1 + ekt*(-1 + kappa*t))*theta + (-1 + ekt)*y) -
10*pow(delta,2)*pow(rho,3)*pow((2 + kappa*t + ekt*(-2 + kappa*t))*
theta + (-1 + ekt - kappa*t)*y,3) +
(6*pow(delta,2)*kappa*pow(rho,3)*
pow((2 + kappa*t + ekt*(-2 + kappa*t))*theta +
(-1 + ekt - kappa*t)*y,3))/
(-theta + kappa*t*theta + (theta - y)/ekt + y) -
(3360*pow(delta,2)*e3kt*pow(kappa,3)*pow(rho,3)*
pow((2 + kappa*t + ekt*(-2 + kappa*t))*theta +
(-1 + ekt - kappa*t)*y,3))/
pow((1 + ekt*(-1 + kappa*t))*theta + (-1 + ekt)*y,3) -
(288*pow(delta,2)*e2kt*pow(kappa,2)*pow(rho,3)*
pow((2 + kappa*t + ekt*(-2 + kappa*t))*theta +
(-1 + ekt - kappa*t)*y,3))/
pow((1 + ekt*(-1 + kappa*t))*theta + (-1 + ekt)*y,2) +
(234*pow(delta,2)*ekt*kappa*pow(rho,3)*
pow((2 + kappa*t + ekt*(-2 + kappa*t))*theta +
(-1 + ekt - kappa*t)*y,3))/
((1 + ekt*(-1 + kappa*t))*theta + (-1 + ekt)*y) -
96*delta*ekt*pow(kappa,3)*
((1 + 4*ekt*(1 + kappa*t) + e2kt*(-5 + 2*kappa*t))*theta +
2*(-1 + e2kt - 2*ekt*kappa*t)*y) -
12*pow(delta,2)*kappa*rho*((2 + kappa*t + ekt*(-2 + kappa*t))*theta +
(-1 + ekt - kappa*t)*y)*
((1 + 4*ekt*(1 + kappa*t) + e2kt*(-5 + 2*kappa*t))*theta +
2*(-1 + e2kt - 2*ekt*kappa*t)*y) -
192*pow(delta,2)*ekt*pow(kappa,2)*rho*
((2 + kappa*t + 2*ekt*pow(2 + kappa*t,2) +
e2kt*(-10 + 3*kappa*t))*theta +
(-3 + 3*e2kt - 2*kappa*t - 2*ekt*kappa*t*(2 + kappa*t))*y)
- (12*pow(delta,2)*ekt*pow(kappa,2)*rho*
((2 + kappa*t + ekt*(-2 + kappa*t))*theta +
(-1 + ekt - kappa*t)*y)*
((1 + e2kt*(-5 + 2*kappa*t + 8*pow(rho,2)*(-3 + kappa*t)) +
4*ekt*(1 + kappa*t +
pow(rho,2)*(6 + 4*kappa*t + pow(kappa,2)*pow(t,2))))*theta +
2*(-1 + e2kt*(1 + 4*pow(rho,2)) -
2*ekt*(kappa*t +
pow(rho,2)*(2 + 2*kappa*t + pow(kappa,2)*pow(t,2))))*y))/
((1 + ekt*(-1 + kappa*t))*theta + (-1 + ekt)*y) +
(576*pow(delta,2)*ekt*pow(kappa,2)*rho*
((2 + kappa*t + ekt*(-2 + kappa*t))*theta +
(-1 + ekt - kappa*t)*y)*
((1 + e2kt*(-5 + 2*kappa*t + 4*pow(rho,2)*(-3 + kappa*t)) +
2*ekt*(2 + 2*kappa*t +
pow(rho,2)*(6 + 4*kappa*t + pow(kappa,2)*pow(t,2))))*theta +
2*(-1 + e2kt*(1 + 2*pow(rho,2)) -
ekt*(2*kappa*t +
pow(rho,2)*(2 + 2*kappa*t + pow(kappa,2)*pow(t,2))))*y))/
((1 + ekt*(-1 + kappa*t))*theta + (-1 + ekt)*y) +
(5*pow(delta,2)*rho*((1 + ekt*(-1 + kappa*t))*theta +
(-1 + ekt)*y)*
((2 + kappa*t + ekt*(-2 + kappa*t))*theta +
(-1 + ekt - kappa*t)*y)*
(theta*(12*ekt*pow(kappa,3)*pow(rho,2)*pow(t,2) +
8*pow(-1 + ekt,2)*pow(rho,2)*theta -
(-1 + ekt)*kappa*
(3 + 8*pow(rho,2)*t*theta +
ekt*(15 + 8*pow(rho,2)*(9 + t*theta))) +
2*pow(kappa,2)*t*(pow(rho,2)*t*theta +
2*ekt*(3 + pow(rho,2)*(12 + t*theta)) +
e2kt*(3 + pow(rho,2)*(12 + t*theta)))) -
2*(6*ekt*pow(kappa,3)*pow(rho,2)*pow(t,2) +
4*pow(-1 + ekt,2)*pow(rho,2)*theta +
2*pow(kappa,2)*t*(pow(rho,2)*t*theta +
ekt*(3 + pow(rho,2)*(6 + t*theta))) -
(-1 + ekt)*kappa*
(3 + 6*pow(rho,2)*t*theta +
ekt*(3 + 2*pow(rho,2)*(6 + t*theta))))*y +
2*pow(rho,2)*pow(1 - ekt + kappa*t,2)*pow(y,2)))/
(ekt*(-theta + kappa*t*theta + (theta - y)/ekt + y)) -
(48*pow(delta,2)*kappa*rho*((1 + ekt*(-1 + kappa*t))*theta +
(-1 + ekt)*y)*
(2*theta + kappa*t*theta - y - kappa*t*y +
ekt*((-2 + kappa*t)*theta + y))*
(theta - 2*y + e2kt*
(-5*theta + 2*kappa*t*theta + 2*y + 4*pow(rho,2)*((-3 + kappa*t)*theta + y)) +
2*ekt*(2*(theta + kappa*t*(theta - y)) +
pow(rho,2)*((6 + kappa*t*(4 + kappa*t))*theta - (2 + kappa*t*(2 + kappa*t))*y))
))/(ekt*(-theta + kappa*t*theta + (theta - y)/ekt + y)) +
(96*delta*pow(kappa,3)*((1 + ekt*(-1 + kappa*t))*theta +
(-1 + ekt)*y)*
(theta - 2*y + e2kt*
(-5*theta + 2*kappa*t*theta + 2*y + 8*pow(rho,2)*((-3 + kappa*t)*theta + y)) +
4*ekt*(theta + kappa*t*theta - kappa*t*y +
pow(rho,2)*((6 + kappa*t*(4 + kappa*t))*theta - (2 + kappa*t*(2 + kappa*t))*y))
))/(-theta + kappa*t*theta + (theta - y)/ekt + y) +
(9*pow(delta,2)*kappa*rho*((1 + ekt*(-1 + kappa*t))*theta +
(-1 + ekt)*y)*
((2 + kappa*t + ekt*(-2 + kappa*t))*theta +
(-1 + ekt - kappa*t)*y)*
(theta - 2*y + e2kt*
(-5*theta + 2*kappa*t*theta + 2*y + 8*pow(rho,2)*((-3 + kappa*t)*theta + y)) +
4*ekt*(theta + kappa*t*theta - kappa*t*y +
pow(rho,2)*((6 + kappa*t*(4 + kappa*t))*theta - (2 + kappa*t*(2 + kappa*t))*y))
))/(ekt*(-theta + kappa*t*theta + (theta - y)/ekt + y)) -
(48*pow(delta,2)*ekt*pow(kappa,2)*rho*
(3*(theta - 2*y)*((2 + kappa*t)*theta - (1 + kappa*t)*y) +
3*ekt*(6*pow(theta,2) + theta*y - 2*pow(y,2) +
kappa*(13*t*pow(theta,2) + theta*(8 - 18*t*y) + 4*y*(-3 + t*y)) +
4*pow(kappa,2)*t*(theta + t*pow(theta,2) - 2*t*theta*y + y*(-2 + t*y))) +
3*e3kt*(10*pow(theta,2) +
2*pow(kappa,2)*t*theta*(6 + 8*pow(rho,2) + t*theta) - 9*theta*y +
2*pow(y,2) + kappa*(-9*t*pow(theta,2) + 4*(3 + 4*pow(rho,2))*y +
theta*(-40 - 64*pow(rho,2) + 4*t*y))) +
e2kt*(-54*pow(theta,2) +
8*pow(kappa,4)*pow(rho,2)*pow(t,3)*(theta - y) + 39*theta*y -
6*pow(y,2) + 24*pow(kappa,3)*pow(t,2)*
(theta + 2*pow(rho,2)*theta - (1 + pow(rho,2))*y) +
6*pow(kappa,2)*t*(3*t*pow(theta,2) - 8*(1 + pow(rho,2))*y +
theta*(16 + 24*pow(rho,2) - 3*t*y)) -
3*kappa*(5*t*pow(theta,2) + 2*y*(8*pow(rho,2) + 3*t*y) -
theta*(32 + 64*pow(rho,2) + 17*t*y)))))/
((1 + ekt*(-1 + kappa*t))*theta + (-1 + ekt)*y) +
(12*pow(delta,2)*kappa*rho*((1 + ekt*(-1 + kappa*t))*theta +
(-1 + ekt)*y)*
(3*(theta - 2*y)*((2 + kappa*t)*theta - (1 + kappa*t)*y) +
3*ekt*(6*pow(theta,2) + theta*y - 2*pow(y,2) +
kappa*(13*t*pow(theta,2) + theta*(8 - 18*t*y) + 4*y*(-3 + t*y)) +
4*pow(kappa,2)*t*(theta + t*pow(theta,2) - 2*t*theta*y + y*(-2 + t*y))) +
3*e3kt*(10*pow(theta,2) +
2*pow(kappa,2)*t*theta*(6 + 8*pow(rho,2) + t*theta) - 9*theta*y +
2*pow(y,2) + kappa*(-9*t*pow(theta,2) + 4*(3 + 4*pow(rho,2))*y +
theta*(-40 - 64*pow(rho,2) + 4*t*y))) +
e2kt*(-54*pow(theta,2) +
8*pow(kappa,4)*pow(rho,2)*pow(t,3)*(theta - y) + 39*theta*y -
6*pow(y,2) + 24*pow(kappa,3)*pow(t,2)*
(theta + 2*pow(rho,2)*theta - (1 + pow(rho,2))*y) +
6*pow(kappa,2)*t*(3*t*pow(theta,2) - 8*(1 + pow(rho,2))*y +
theta*(16 + 24*pow(rho,2) - 3*t*y)) -
3*kappa*(5*t*pow(theta,2) + 2*y*(8*pow(rho,2) + 3*t*y) -
theta*(32 + 64*pow(rho,2) + 17*t*y)))))/
(ekt*(-theta + kappa*t*theta + (theta - y)/ekt + y)) +
(240*pow(delta,2)*e2kt*pow(kappa,2)*rho*
((2 + kappa*t + ekt*(-2 + kappa*t))*theta +
(-1 + ekt - kappa*t)*y)*
(12*ekt*pow(kappa,3)*pow(rho,2)*pow(t,2)*(theta - y) +
2*pow(-1 + ekt,2)*pow(rho,2)*pow(-2*theta + y,2) -
(-1 + ekt)*kappa*
(8*(1 + ekt)*pow(rho,2)*t*pow(theta,2) +
2*y*(-3 - 3*ekt*(1 + 4*pow(rho,2)) + 2*pow(rho,2)*t*y) +
theta*(3 - 12*pow(rho,2)*t*y +
ekt*(15 + pow(rho,2)*(72 - 4*t*y)))) +
2*pow(kappa,2)*t*(e2kt*theta*(3 + pow(rho,2)*(12 + t*theta)) +
pow(rho,2)*t*pow(theta - y,2) +
2*ekt*(pow(rho,2)*t*pow(theta,2) - 3*(y + 2*pow(rho,2)*y) +
theta*(3 + pow(rho,2)*(12 - t*y))))))/
pow((1 + ekt*(-1 + kappa*t))*theta + (-1 + ekt)*y,2) -
(120*pow(delta,2)*ekt*kappa*rho*
((2 + kappa*t + ekt*(-2 + kappa*t))*theta +
(-1 + ekt - kappa*t)*y)*
(12*ekt*pow(kappa,3)*pow(rho,2)*pow(t,2)*(theta - y) +
2*pow(-1 + ekt,2)*pow(rho,2)*pow(-2*theta + y,2) -
(-1 + ekt)*kappa*
(8*(1 + ekt)*pow(rho,2)*t*pow(theta,2) +
2*y*(-3 - 3*ekt*(1 + 4*pow(rho,2)) + 2*pow(rho,2)*t*y) +
theta*(3 - 12*pow(rho,2)*t*y +
ekt*(15 + pow(rho,2)*(72 - 4*t*y)))) +
2*pow(kappa,2)*t*(e2kt*theta*(3 + pow(rho,2)*(12 + t*theta)) +
pow(rho,2)*t*pow(theta - y,2) +
2*ekt*(pow(rho,2)*t*pow(theta,2) - 3*(y + 2*pow(rho,2)*y) +
theta*(3 + pow(rho,2)*(12 - t*y))))))/
((1 + ekt*(-1 + kappa*t))*theta + (-1 + ekt)*y)))/
(1536.*e3kt*pow(kappa,6)*pow(t,2)*
pow((-theta + kappa*t*theta + (theta - y)/ekt + y)/(kappa*t),1.5));
}
Real LPP3HestonExpansion::z2(Real t, Real kappa, Real theta,
Real delta, Real y, Real rho) const{
return (pow(delta,2)*(8*e3kt*pow(kappa,5)*pow(rho,2)*pow(t,4)*(2 + delta*rho*t)*
pow(theta,2)*(theta - y) - delta*pow(-1 + ekt,3)*rho*
(2*(-1 + ekt*(-5 + 24*pow(rho,2)))*pow(theta,3) +
(7 + ekt*(3 + 56*pow(rho,2)))*pow(theta,2)*y -
3*(1 + ekt*(-3 + 8*pow(rho,2)))*theta*pow(y,2) +
2*(-1 + ekt*(-1 + 2*pow(rho,2)))*pow(y,3)) -
pow(-1 + ekt,2)*kappa*
((-4 + delta*rho*t - 8*ekt*
(2 - 12*pow(rho,2) - 4*delta*rho*t + 25*delta*pow(rho,3)*t) +
e2kt*(20 - 96*pow(rho,2) + 3*delta*rho*t + 56*delta*pow(rho,3)*t)
)*pow(theta,3) - 2*(-8 + 2*delta*rho*t +
e2kt*(24 - 80*pow(rho,2) - 9*delta*rho*t +
24*delta*pow(rho,3)*t) -
4*ekt*(4 - 20*pow(rho,2) - 10*delta*rho*t + 39*delta*pow(rho,3)*t)
)*pow(theta,2)*y + (5*(-4 + delta*rho*t) +
ekt*(-16 + 80*pow(rho,2) + 57*delta*rho*t -
140*delta*pow(rho,3)*t) +
2*e2kt*(18 - 40*pow(rho,2) - 3*delta*rho*t +
6*delta*pow(rho,3)*t))*theta*pow(y,2) +
2*(4 + e2kt*(-4 + 8*pow(rho,2)) - delta*rho*t +
ekt*rho*(-8*rho - 7*delta*t + 14*delta*pow(rho,2)*t))*pow(y,3)) +
ekt*(-1 + ekt)*pow(kappa,2)*t*
((-24 + 128*pow(rho,2) + 9*delta*rho*t - 144*delta*pow(rho,3)*t -
4*ekt*(6 - 8*pow(rho,2) - 9*delta*rho*t + 6*delta*pow(rho,3)*t) +
e2kt*(48 - 160*pow(rho,2) - 9*delta*rho*t +
24*delta*pow(rho,3)*t))*pow(theta,3) -
(-72 + 320*pow(rho,2) + 27*delta*rho*t - 360*delta*pow(rho,3)*t -
ekt*rho*(160*rho - 81*delta*t + 348*delta*pow(rho,2)*t) +
2*e2kt*(36 - 80*pow(rho,2) - 3*delta*rho*t +
6*delta*pow(rho,3)*t))*pow(theta,2)*y -
2*(32 - 128*pow(rho,2) + 12*e2kt*(-1 + 2*pow(rho,2)) -
15*delta*rho*t + 144*delta*pow(rho,3)*t +
2*ekt*(-10 + 52*pow(rho,2) - 13*delta*rho*t +
58*delta*pow(rho,3)*t))*theta*pow(y,2) +
4*(4 - 16*pow(rho,2) - 3*delta*rho*t + 18*delta*pow(rho,3)*t +
ekt*(-4 + 16*pow(rho,2) - 2*delta*rho*t + 11*delta*pow(rho,3)*t))*
pow(y,3)) - 4*e2kt*pow(kappa,4)*pow(t,3)*theta*
(2*e2kt*(-1 + 2*pow(rho,2))*pow(theta,2) +
pow(rho,2)*(4 + 13*delta*rho*t)*pow(theta - y,2) +
ekt*((-4 + 16*pow(rho,2) - 2*delta*rho*t + 9*delta*pow(rho,3)*t)*
pow(theta,2) + (4 - 32*pow(rho,2) + 2*delta*rho*t - 19*delta*pow(rho,3)*t)*
theta*y + 4*pow(rho,2)*(2 + delta*rho*t)*pow(y,2))) -
2*ekt*pow(kappa,3)*pow(t,2)*
(-4*pow(rho,2)*(-4 + 3*delta*rho*t)*pow(theta - y,3) +
e3kt*pow(theta,2)*
((18 - 40*pow(rho,2) - delta*rho*t + 2*delta*pow(rho,3)*t)*theta +
12*(-1 + 2*pow(rho,2))*y) +
2*ekt*((-9 + 36*pow(rho,2) + 19*delta*pow(rho,3)*t)*pow(theta,3) +
2*(9 - 30*pow(rho,2) + 7*delta*pow(rho,3)*t)*pow(theta,2)*y +
(-8 + 20*pow(rho,2) + delta*rho*t - 46*delta*pow(rho,3)*t)*theta*pow(y,2) +
pow(rho,2)*(4 + 13*delta*rho*t)*pow(y,3)) +
e2kt*(8*theta*y*(-3*theta + 2*y) +
delta*rho*t*theta*(7*pow(theta,2) - 23*theta*y + 8*pow(y,2)) -
8*pow(rho,2)*(6*pow(theta,3) - 18*pow(theta,2)*y + 11*theta*pow(y,2) -
pow(y,3)) + 4*delta*pow(rho,3)*t*
(-13*pow(theta,3) + 31*pow(theta,2)*y - 14*theta*pow(y,2) + pow(y,3))))))/
(64.*pow(kappa,2)*t*sqrt((-theta + kappa*t*theta + (theta - y)/ekt + y)/
(kappa*t))*pow((1 + ekt*(-1 + kappa*t))*theta + (-1 + ekt)*y,
4));
}
Real LPP3HestonExpansion::z3(Real t, Real kappa, Real theta,
Real delta, Real y, Real rho) const {
return (pow(delta,3)*ekt*rho*((-15*(2 + kappa*t) +
3*e4kt*(50 - 79*kappa*t + 35*pow(kappa,2)*pow(t,2) -
6*pow(kappa,3)*pow(t,3) +
8*pow(rho,2)*(-18 + 15*kappa*t - 6*pow(kappa,2)*pow(t,2) +
pow(kappa,3)*pow(t,3))) +
ekt*(-3*(20 + 86*kappa*t + 29*pow(kappa,2)*pow(t,2)) +
pow(rho,2)*(432 + 936*kappa*t + 552*pow(kappa,2)*pow(t,2) +
92*pow(kappa,3)*pow(t,3))) +
e2kt*(360 + 324*kappa*t - 261*pow(kappa,2)*pow(t,2) -
48*pow(kappa,3)*pow(t,3) -
4*pow(rho,2)*(324 + 378*kappa*t - 12*pow(kappa,2)*pow(t,2) -
2*pow(kappa,3)*pow(t,3) + 23*pow(kappa,4)*pow(t,4))) +
e3kt*(3*(-140 + 62*kappa*t + 81*pow(kappa,2)*pow(t,2) -
38*pow(kappa,3)*pow(t,3) + 8*pow(kappa,4)*pow(t,4)) +
4*pow(rho,2)*(324 + 54*kappa*t - 114*pow(kappa,2)*pow(t,2) +
77*pow(kappa,3)*pow(t,3) - 19*pow(kappa,4)*pow(t,4) +
2*pow(kappa,5)*pow(t,5))))*pow(theta,3) +
(15*(7 + 4*kappa*t) + 3*e4kt*
(-79 + 70*kappa*t - 18*pow(kappa,2)*pow(t,2) +
24*pow(rho,2)*(5 - 4*kappa*t + pow(kappa,2)*pow(t,2))) -
3*ekt*(26 - 200*kappa*t - 87*pow(kappa,2)*pow(t,2) +
4*pow(rho,2)*(30 + 142*kappa*t + 115*pow(kappa,2)*pow(t,2) +
23*pow(kappa,3)*pow(t,3))) +
2*e2kt*(3*(-66 - 195*kappa*t + 63*pow(kappa,2)*pow(t,2) +
16*pow(kappa,3)*pow(t,3)) +
4*pow(rho,2)*(135 + 390*kappa*t - 9*pow(kappa,2)*pow(t,2) -
48*pow(kappa,3)*pow(t,3) + 23*pow(kappa,4)*pow(t,4))) +
e3kt*(606 + 300*kappa*t - 585*pow(kappa,2)*pow(t,2) +
210*pow(kappa,3)*pow(t,3) - 24*pow(kappa,4)*pow(t,4) -
4*pow(rho,2)*(270 + 282*kappa*t - 345*pow(kappa,2)*pow(t,2) +
153*pow(kappa,3)*pow(t,3) - 29*pow(kappa,4)*pow(t,4) +
2*pow(kappa,5)*pow(t,5))))*pow(theta,2)*y +
(-93 - 75*kappa*t + 3*e4kt*
(35 - 18*kappa*t + 24*pow(rho,2)*(-2 + kappa*t)) +
3*ekt*(58 - 123*kappa*t - 86*pow(kappa,2)*pow(t,2) +
4*pow(rho,2)*(12 + 80*kappa*t + 92*pow(kappa,2)*pow(t,2) +
23*pow(kappa,3)*pow(t,3))) +
e3kt*(-3*(74 + 137*kappa*t - 100*pow(kappa,2)*pow(t,2) +
16*pow(kappa,3)*pow(t,3)) -
16*pow(rho,2)*(-27 - 51*kappa*t + 45*pow(kappa,2)*pow(t,2) -
12*pow(kappa,3)*pow(t,3) + pow(kappa,4)*pow(t,4))) +
e2kt*(36 + 909*kappa*t - 42*pow(kappa,2)*pow(t,2) -
60*pow(kappa,3)*pow(t,3) -
4*pow(rho,2)*(108 + 462*kappa*t + 96*pow(kappa,2)*pow(t,2) -
117*pow(kappa,3)*pow(t,3) + 23*pow(kappa,4)*pow(t,4))))*theta*pow(y,2)
+ 2*(9 + 3*e4kt*(-3 + 4*pow(rho,2)) + 15*kappa*t +
e2kt*(-3*kappa*t*(33 + 10*kappa*t) +
pow(rho,2)*(36 + 192*kappa*t + 96*pow(kappa,2)*pow(t,2) -
46*pow(kappa,3)*pow(t,3))) +
e3kt*(18 + 57*kappa*t - 12*pow(kappa,2)*pow(t,2) -
2*pow(rho,2)*(18 + 48*kappa*t - 21*pow(kappa,2)*pow(t,2) +
2*pow(kappa,3)*pow(t,3))) +
ekt*(3*(-6 + 9*kappa*t + 14*pow(kappa,2)*pow(t,2)) -
2*pow(rho,2)*(6 + 48*kappa*t + 69*pow(kappa,2)*pow(t,2) +
23*pow(kappa,3)*pow(t,3))))*pow(y,3)))/
(96.*kappa*t*sqrt((-theta + kappa*t*theta + (theta - y)/ekt + y)/(kappa*t))*
pow((1 + ekt*(-1 + kappa*t))*theta + (-1 + ekt)*y,5));
}
LPP3HestonExpansion::LPP3HestonExpansion(Real kappa, Real theta, Real sigma,
Real v0, Real rho, Real term) {
ekt = exp(kappa*term);
e2kt = ekt*ekt;
e3kt = e2kt*ekt;
e4kt = e2kt*e2kt;
coeffs[0] = z0(term, kappa, theta, sigma, v0, rho);
coeffs[1] = z1(term, kappa, theta, sigma, v0, rho);
coeffs[2] = z2(term, kappa, theta, sigma, v0, rho);
coeffs[3] = z3(term, kappa, theta, sigma, v0, rho);
}
Real LPP3HestonExpansion::impliedVolatility(const Real strike,
const Real forward) const {
Real x = log(strike/forward);
Real vol = coeffs[0]+x*(coeffs[1]+x*(coeffs[2]+x*(coeffs[3])));
return std::max(1e-8,vol);
}
}
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