1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
181
182
183
184
|
/* -*- mode: c++; tab-width: 4; indent-tabs-mode: nil; c-basic-offset: 4 -*- */
/*
Copyright (C) 2008 Andreas Gaida
Copyright (C) 2008 Ralph Schreyer
Copyright (C) 2008 Klaus Spanderen
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
<http://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/methods/finitedifferences/meshers/fdmmesher.hpp>
#include <ql/methods/finitedifferences/operators/fdmlinearoplayout.hpp>
#include <ql/methods/finitedifferences/operators/secondderivativeop.hpp>
#include <ql/methods/finitedifferences/operators/secondderivativeop.hpp>
#include <ql/methods/finitedifferences/operators/fdmhestonhullwhiteop.hpp>
#include <ql/methods/finitedifferences/operators/secondordermixedderivativeop.hpp>
namespace QuantLib {
FdmHestonHullWhiteEquityPart::FdmHestonHullWhiteEquityPart(
const boost::shared_ptr<FdmMesher>& mesher,
const boost::shared_ptr<YieldTermStructure>& qTS)
: rates_(mesher->locations(2)),
varianceValues_(0.5*mesher->locations(1)),
dxMap_ (FirstDerivativeOp(0, mesher)),
dxxMap_(SecondDerivativeOp(0, mesher).mult(0.5*mesher->locations(1))),
mapT_ (0, mesher),
mesher_ (mesher),
qTS_(qTS) {
// on the boundary s_min and s_max the second derivative
// d²V/dS² is zero and due to Ito's Lemma the variance term
// in the drift should vanish.
boost::shared_ptr<FdmLinearOpLayout> layout = mesher_->layout();
FdmLinearOpIterator endIter = layout->end();
for (FdmLinearOpIterator iter = layout->begin(); iter != endIter;
++iter) {
if ( iter.coordinates()[0] == 0
|| iter.coordinates()[0] == layout->dim()[0]-1) {
varianceValues_[iter.index()] = 0.0;
}
}
volatilityValues_ = Sqrt(2*varianceValues_);
}
void FdmHestonHullWhiteEquityPart::setTime(Time t1, Time t2) {
const Rate q = qTS_->forwardRate(t1, t2, Continuous).rate();
mapT_.axpyb(rates_-varianceValues_-q, dxMap_, dxxMap_, Array());
}
const TripleBandLinearOp& FdmHestonHullWhiteEquityPart::getMap() const {
return mapT_;
}
FdmHestonHullWhiteVariancePart::FdmHestonHullWhiteVariancePart(
const boost::shared_ptr<FdmMesher>& mesher,
Real sigma, Real kappa, Real theta)
: dyMap_(SecondDerivativeOp(1, mesher)
.mult(0.5*sigma*sigma*mesher->locations(1))
.add(FirstDerivativeOp(1, mesher)
.mult(kappa*(theta - mesher->locations(1))))) {
}
const TripleBandLinearOp& FdmHestonHullWhiteVariancePart::getMap() const {
return dyMap_;
}
FdmHestonHullWhiteRatesPart::FdmHestonHullWhiteRatesPart(
const boost::shared_ptr<FdmMesher>& mesher,
const boost::shared_ptr<HullWhiteProcess>& hwProcess)
: rates_(mesher->locations(2)),
dzMap_(FirstDerivativeOp(2, mesher)),
dzzMap_(SecondDerivativeOp(2, mesher)
.mult(0.5*hwProcess->sigma()*hwProcess->sigma()
*Array(mesher->layout()->size(), 1.))
.add(-mesher->locations(2))),
mapT_(2, mesher),
hwProcess_(hwProcess) {
}
void FdmHestonHullWhiteRatesPart::setTime(Time t1, Time t2) {
const Time dt = t2-t1;
const Array drift = (rates_*(std::exp(-hwProcess_->a()*dt)-1.0)
+ hwProcess_->expectation(t1, 0.0, dt))/dt;
mapT_.axpyb(drift, dzMap_, dzzMap_, Array());
}
const TripleBandLinearOp& FdmHestonHullWhiteRatesPart::getMap() const {
return mapT_;
}
FdmHestonHullWhiteOp::FdmHestonHullWhiteOp(
const boost::shared_ptr<FdmMesher>& mesher,
const boost::shared_ptr<HestonProcess>& hestonProcess,
const boost::shared_ptr<HullWhiteProcess>& hwProcess,
Real equityShortRateCorrelation)
: rates_(mesher->locations(2)),
v0_(hestonProcess->v0()),
kappa_(hestonProcess->kappa()),
theta_(hestonProcess->theta()),
sigma_(hestonProcess->sigma()),
rho_(hestonProcess->rho()),
hwProcess_(hwProcess),
hestonCorrMap_(SecondOrderMixedDerivativeOp(0, 1, mesher)
.mult(rho_*sigma_*mesher->locations(1))),
equityIrCorrMap_(SecondOrderMixedDerivativeOp(0, 2, mesher)
.mult(Sqrt(mesher->locations(1))
* hwProcess->sigma()
* equityShortRateCorrelation)),
dyMap_(mesher, sigma_, kappa_, theta_),
dxMap_(mesher, hestonProcess->dividendYield().currentLink()),
dzMap_(mesher, hwProcess) {
QL_REQUIRE( equityShortRateCorrelation*equityShortRateCorrelation
+hestonProcess->rho()*hestonProcess->rho() <= 1.0,
"correlation matrix has negative eigenvalues");
}
void FdmHestonHullWhiteOp::setTime(Time t1, Time t2) {
dxMap_.setTime(t1, t2);
dzMap_.setTime(t1, t2);
}
Size FdmHestonHullWhiteOp::size() const {
return 3;
}
Disposable<Array> FdmHestonHullWhiteOp::apply(const Array& u) const {
return dyMap_.getMap().apply(u) + dxMap_.getMap().apply(u)
+ dzMap_.getMap().apply(u)
+ hestonCorrMap_.apply(u) + equityIrCorrMap_.apply(u);
}
Disposable<Array>
FdmHestonHullWhiteOp::apply_direction(Size direction,
const Array& r) const {
if (direction == 0)
return dxMap_.getMap().apply(r);
else if (direction == 1)
return dyMap_.getMap().apply(r);
else if (direction == 2)
return dzMap_.getMap().apply(r);
else
QL_FAIL("direction too large");
}
Disposable<Array> FdmHestonHullWhiteOp::apply_mixed(const Array& r) const {
return hestonCorrMap_.apply(r) + equityIrCorrMap_.apply(r);
}
Disposable<Array>
FdmHestonHullWhiteOp::solve_splitting(Size direction, const Array& r,
Real a) const {
if (direction == 0) {
return dxMap_.getMap().solve_splitting(r, a, 1.0);
}
else if (direction == 1) {
return dyMap_.getMap().solve_splitting(r, a, 1.0);
}
else if (direction == 2) {
return dzMap_.getMap().solve_splitting(r, a, 1.0);
}
else
QL_FAIL("direction too large");
}
Disposable<Array> FdmHestonHullWhiteOp::preconditioner(const Array& r,
Real dt) const {
return solve_splitting(0, r, dt);
}
}
|
