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
|
/* -*- mode: c++; tab-width: 4; indent-tabs-mode: nil; c-basic-offset: 4 -*- */
/*
Copyright (C) 2006, 2007 Banca Profilo S.p.A.
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/processes/hullwhiteprocess.hpp>
namespace QuantLib {
HullWhiteProcess::HullWhiteProcess(const Handle<YieldTermStructure>& h,
Real a,
Real sigma)
: process_(new OrnsteinUhlenbeckProcess(
a, sigma, h->forwardRate(0.0,0.0,Continuous,NoFrequency))),
h_(h), a_(a), sigma_(sigma) {
QL_REQUIRE(a_ >= 0.0, "negative a given");
QL_REQUIRE(sigma_ >= 0.0, "negative sigma given");
}
Real HullWhiteProcess::x0() const {
return process_->x0();
}
Real HullWhiteProcess::drift(Time t, Real x) const {
Real alpha_drift = sigma_*sigma_/(2*a_)*(1-std::exp(-2*a_*t));
Real shift = 0.0001;
Real f = h_->forwardRate(t, t, Continuous, NoFrequency);
Real fup = h_->forwardRate(t+shift, t+shift, Continuous, NoFrequency);
Real f_prime = (fup-f)/shift;
alpha_drift += a_*f+f_prime;
return process_->drift(t, x) + alpha_drift;
}
Real HullWhiteProcess::diffusion(Time t, Real x) const{
return process_->diffusion(t, x);
}
Real HullWhiteProcess::expectation(Time t0, Real x0, Time dt) const {
return process_->expectation(t0, x0, dt)
+ alpha(t0 + dt) - alpha(t0)*std::exp(-a_*dt);
}
Real HullWhiteProcess::stdDeviation(Time t0, Real x0, Time dt) const{
return process_->stdDeviation(t0, x0, dt);
}
Real HullWhiteProcess::variance(Time t0, Real x0, Time dt) const{
return process_->variance(t0, x0, dt);
}
Real HullWhiteProcess::alpha(Time t) const {
Real alfa = a_ > QL_EPSILON ?
(sigma_/a_)*(1 - std::exp(-a_*t)) :
sigma_*t;
alfa *= 0.5*alfa;
alfa += h_->forwardRate(t, t, Continuous, NoFrequency);
return alfa;
}
Real HullWhiteProcess::a() const {
return a_;
}
Real HullWhiteProcess::sigma() const {
return sigma_;
}
HullWhiteForwardProcess::HullWhiteForwardProcess(
const Handle<YieldTermStructure>& h,
Real a,
Real sigma)
: process_(new OrnsteinUhlenbeckProcess(
a, sigma, h->forwardRate(0.0,0.0,Continuous,NoFrequency))),
h_(h), a_(a), sigma_(sigma) {}
Real HullWhiteForwardProcess::x0() const {
return process_->x0();
}
Real HullWhiteForwardProcess::drift(Time t, Real x) const {
Real alpha_drift = sigma_*sigma_/(2*a_)*(1-std::exp(-2*a_*t));
Real shift = 0.0001;
Real f = h_->forwardRate(t, t, Continuous, NoFrequency);
Real fup = h_->forwardRate(t+shift, t+shift, Continuous, NoFrequency);
Real f_prime = (fup-f)/shift;
alpha_drift += a_*f+f_prime;
return process_->drift(t, x) + alpha_drift - B(t, T_)*sigma_*sigma_;
}
Real HullWhiteForwardProcess::diffusion(Time t, Real x) const{
return process_->diffusion(t, x);
}
Real HullWhiteForwardProcess::expectation(Time t0, Real x0,
Time dt) const {
return process_->expectation(t0, x0, dt)
+ alpha(t0 + dt) - alpha(t0)*std::exp(-a_*dt)
- M_T(t0, t0+dt, T_);
}
Real HullWhiteForwardProcess::stdDeviation(Time t0, Real x0,
Time dt) const {
return process_->stdDeviation(t0, x0, dt);
}
Real HullWhiteForwardProcess::variance(Time t0, Real x0, Time dt) const{
return process_->variance(t0, x0, dt);
}
Real HullWhiteForwardProcess::alpha(Time t) const {
Real alfa = a_ > QL_EPSILON ?
(sigma_/a_)*(1 - std::exp(-a_*t)) :
sigma_*t;
alfa *= 0.5*alfa;
alfa += h_->forwardRate(t, t, Continuous, NoFrequency);
return alfa;
}
Real HullWhiteForwardProcess::M_T(Real s, Real t, Real T) const {
if (a_ > QL_EPSILON) {
Real coeff = (sigma_*sigma_)/(a_*a_);
Real exp1 = std::exp(-a_*(t-s));
Real exp2 = std::exp(-a_*(T-t));
Real exp3 = std::exp(-a_*(T+t-2.0*s));
return coeff*(1-exp1)-0.5*coeff*(exp2-exp3);
} else {
// low-a algebraic limit
Real coeff = (sigma_*sigma_)/2.0;
return coeff*(t-s)*(2.0*T-t-s);
}
}
Real HullWhiteForwardProcess::B(Time t, Time T) const {
return a_ > QL_EPSILON ?
1/a_ * (1-std::exp(-a_*(T-t))) :
T-t;
}
Real HullWhiteForwardProcess::a() const {
return a_;
}
Real HullWhiteForwardProcess::sigma() const {
return sigma_;
}
}
|
