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/* -*- mode: c++; tab-width: 4; indent-tabs-mode: nil; c-basic-offset: 4 -*- */
/*
Copyright (C) 2001, 2002, 2003 Sadruddin Rejeb
Copyright (C) 2006 Chiara Fornarola
Copyright (C) 2007 StatPro Italia srl
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/models/shortrate/onefactormodels/hullwhite.hpp>
#include <ql/methods/lattices/trinomialtree.hpp>
#include <ql/pricingengines/blackformula.hpp>
using std::exp;
using std::sqrt;
namespace QuantLib {
HullWhite::HullWhite(const Handle<YieldTermStructure>& termStructure,
Real a, Real sigma)
: Vasicek(termStructure->forwardRate(0.0, 0.0, Continuous, NoFrequency),
a, 0.0, sigma, 0.0),
TermStructureConsistentModel(termStructure) {
b_ = NullParameter();
lambda_ = NullParameter();
HullWhite::generateArguments();
registerWith(termStructure);
}
ext::shared_ptr<Lattice> HullWhite::tree(const TimeGrid& grid) const {
TermStructureFittingParameter phi(termStructure());
ext::shared_ptr<ShortRateDynamics> numericDynamics(
new Dynamics(phi, a(), sigma()));
ext::shared_ptr<TrinomialTree> trinomial(
new TrinomialTree(numericDynamics->process(), grid));
ext::shared_ptr<ShortRateTree> numericTree(
new ShortRateTree(trinomial, numericDynamics, grid));
typedef TermStructureFittingParameter::NumericalImpl NumericalImpl;
ext::shared_ptr<NumericalImpl> impl =
ext::dynamic_pointer_cast<NumericalImpl>(phi.implementation());
impl->reset();
for (Size i=0; i<(grid.size() - 1); i++) {
Real discountBond = termStructure()->discount(grid[i+1]);
const Array& statePrices = numericTree->statePrices(i);
Size size = numericTree->size(i);
Time dt = numericTree->timeGrid().dt(i);
Real dx = trinomial->dx(i);
Real x = trinomial->underlying(i,0);
Real value = 0.0;
for (Size j=0; j<size; j++) {
value += statePrices[j]*std::exp(-x*dt);
x += dx;
}
value = std::log(value/discountBond)/dt;
impl->set(grid[i], value);
}
return numericTree;
}
Real HullWhite::A(Time t, Time T) const {
DiscountFactor discount1 = termStructure()->discount(t);
DiscountFactor discount2 = termStructure()->discount(T);
Rate forward = termStructure()->forwardRate(t, t,
Continuous, NoFrequency);
Real temp = sigma()*B(t,T);
Real value = B(t,T)*forward - 0.25*temp*temp*B(0.0,2.0*t);
return std::exp(value)*discount2/discount1;
}
void HullWhite::generateArguments() {
phi_ = FittingParameter(termStructure(), a(), sigma());
}
Real HullWhite::discountBondOption(Option::Type type, Real strike,
Time maturity,
Time bondMaturity) const {
Real _a = a();
Real v;
if (_a < std::sqrt(QL_EPSILON)) {
v = sigma()*B(maturity, bondMaturity)* std::sqrt(maturity);
} else {
v = sigma()*B(maturity, bondMaturity)*
std::sqrt(0.5*(1.0 - std::exp(-2.0*_a*maturity))/_a);
}
Real f = termStructure()->discount(bondMaturity);
Real k = termStructure()->discount(maturity)*strike;
return blackFormula(type, k, f, v);
}
Real HullWhite::discountBondOption(Option::Type type, Real strike,
Time maturity, Time bondStart,
Time bondMaturity) const {
Real _a = a();
Real v;
if (_a < std::sqrt(QL_EPSILON)) {
v = sigma()*B(bondStart, bondMaturity)* std::sqrt(maturity);
} else {
Real c = exp(-2.0*_a*(bondStart-maturity))
- exp(-2.0*_a*bondStart)
-2.0*(exp(-_a*(bondStart+bondMaturity-2.0*maturity))
- exp(-_a*(bondStart+bondMaturity)))
+ exp(-2.0*_a*(bondMaturity-maturity))
- exp(-2.0*_a*bondMaturity);
// The above should always be positive, but due to
// numerical errors it can be a very small negative number.
// We floor it at 0 to avoid NaNs.
v = sigma()/(_a*sqrt(2.0*_a)) * sqrt(std::max(c, 0.0));
}
Real f = termStructure()->discount(bondMaturity);
Real k = termStructure()->discount(bondStart)*strike;
return blackFormula(type, k, f, v);
}
Rate HullWhite::convexityBias(Real futuresPrice,
Time t,
Time T,
Real sigma,
Real a) {
QL_REQUIRE(futuresPrice>=0.0,
"negative futures price (" << futuresPrice << ") not allowed");
QL_REQUIRE(t>=0.0,
"negative t (" << t << ") not allowed");
QL_REQUIRE(T>=t,
"T (" << T << ") must not be less than t (" << t << ")");
QL_REQUIRE(sigma>=0.0,
"negative sigma (" << sigma << ") not allowed");
QL_REQUIRE(a>=0.0,
"negative a (" << a << ") not allowed");
auto temp = [a](Real x) { return a < QL_EPSILON ? x : (1.0-std::exp(-a*x)) / a; };
Time deltaT = (T-t);
Real tempDeltaT = temp(deltaT);
Real halfSigmaSquare = sigma*sigma/2.0;
// lambda adjusts for the fact that the underlying is an interest rate
Real lambda = temp(2.0*t) * tempDeltaT;
Real tempT = temp(t);
// phi is the MtM adjustment
Real phi = tempT * tempT;
// the adjustment
Real z = halfSigmaSquare * (lambda + phi);
Rate futureRate = (100.0-futuresPrice)/100.0;
return deltaT < QL_EPSILON ? z : (1.0-std::exp(-z*tempDeltaT)) * (futureRate + 1.0/deltaT);
}
}
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