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
|
/* -*- mode: c++; tab-width: 4; indent-tabs-mode: nil; c-basic-offset: 4 -*- */
/*
Copyright (C) 2014 Michal Kaut
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/distributions/bivariatestudenttdistribution.hpp>
namespace QuantLib {
namespace {
Real epsilon = 1.0e-8;
Real sign(Real val) {
return val == 0.0 ? 0.0
: (val < 0.0 ? -1.0 : 1.0);
}
/* unlike the atan2 function in C++ that gives results in
[-pi,pi], this returns a value in [0, 2*pi]
*/
Real arctan(Real x, Real y) {
Real res = std::atan2(x, y);
return res >= 0.0 ? res : res + 2 * M_PI;
}
// function x(m,h,k) defined on top of page 155
Real f_x(Real m, Real h, Real k, Real rho) {
Real unCor = 1 - rho*rho;
Real sub = std::pow(h - rho * k, 2);
Real denom = sub + unCor * (m + k*k);
if (denom < epsilon)
return 0.0; // limit case for rho = +/-1.0
return sub / (sub + unCor * (m + k*k));
}
// this calculates the cdf
Real P_n(Real h, Real k, Natural n, Real rho) {
Real unCor = 1.0 - rho*rho;
Real div = 4 * std::sqrt(n * M_PI);
Real xHK = f_x(n, h, k, rho);
Real xKH = f_x(n, k, h, rho);
Real divH = 1 + h*h / n;
Real divK = 1 + k*k / n;
Real sgnHK = sign(h - rho * k);
Real sgnKH = sign(k - rho * h);
if (n % 2 == 0) { // n is even, equation (10)
// first line of (10)
Real res = arctan(std::sqrt(unCor), -rho) / M_TWOPI;
// second line of (10)
Real dgM = 2 * (1 - xHK); // multiplier for dgj
Real gjM = sgnHK * 2 / M_PI; // multiplier for g_j
// initializations for j = 1:
Real f_j = std::sqrt(M_PI / divK);
Real g_j = 1 + gjM * arctan(std::sqrt(xHK), std::sqrt(1 - xHK));
Real sum = f_j * g_j;
if (n >= 4) {
// different formulas for j = 2:
f_j *= 0.5 / divK; // (2 - 1.5) / (Real) (2 - 1) / divK;
Real dgj = gjM * std::sqrt(xHK * (1 - xHK));
g_j += dgj;
sum += f_j * g_j;
// and then the loop for the rest of the j's:
for (Natural j = 3; j <= n / 2; ++j) {
f_j *= (j - 1.5) / (Real) (j - 1) / divK;
dgj *= (Real) (j - 2) / (2 * j - 3) * dgM;
g_j += dgj;
sum += f_j * g_j;
}
}
res += k / div * sum;
// third line of (10)
dgM = 2 * (1 - xKH);
gjM = sgnKH * 2 / M_PI;
// initializations for j = 1:
f_j = std::sqrt(M_PI / divH);
g_j = 1 + gjM * arctan(std::sqrt(xKH), std::sqrt(1 - xKH));
sum = f_j * g_j;
if (n >= 4) {
// different formulas for j = 2:
f_j *= 0.5 / divH; // (2 - 1.5) / (Real) (2 - 1) / divK;
Real dgj = gjM * std::sqrt(xKH * (1 - xKH));
g_j += dgj;
sum += f_j * g_j;
// and then the loop for the rest of the j's:
for (Natural j = 3; j <= n / 2; ++j) {
f_j *= (j - 1.5) / (Real) (j - 1) / divH;
dgj *= (Real) (j - 2) / (2 * j - 3) * dgM;
g_j += dgj;
sum += f_j * g_j;
}
}
res += h / div * sum;
return res;
} else { // n is odd, equation (11)
// first line of (11)
Real hk = h * k;
Real hkcn = hk + rho * n;
Real sqrtExpr = std::sqrt(h*h - 2 * rho * hk + k*k + n * unCor);
Real res = arctan(std::sqrt(Real(n)) * (-(h + k) * hkcn - (hk - n) * sqrtExpr),
(hk - n) * hkcn - n * (h + k) * sqrtExpr ) / M_TWOPI;
if (n > 1) {
// second line of (11)
Real mult = (1 - xHK) / 2;
// initializations for j = 1:
Real f_j = 2 / std::sqrt(M_PI) / divK;
Real dgj = sgnHK * std::sqrt(xHK);
Real g_j = 1 + dgj;
Real sum = f_j * g_j;
// and then the loop for the rest of the j's:
for (Natural j = 2; j <= (n - 1) / 2; ++j) {
f_j *= (Real) (j - 1) / (j - 0.5) / divK;
dgj *= (Real) (2 * j - 3) / (j - 1) * mult;
g_j += dgj;
sum += f_j * g_j;
}
res += k / div * sum;
// third line of (11)
mult = (1 - xKH) / 2;
// initializations for j = 1:
f_j = 2 / std::sqrt(M_PI) / divH;
dgj = sgnKH * std::sqrt(xKH);
g_j = 1 + dgj;
sum = f_j * g_j;
// and then the loop for the rest of the j's:
for (Natural j = 2; j <= (n - 1) / 2; ++j) {
f_j *= (Real) (j - 1) / (j - 0.5) / divH;
dgj *= (Real) (2 * j - 3) / (j - 1) * mult;
g_j += dgj;
sum += f_j * g_j;
}
res += h / div * sum;
}
return res;
}
}
}
BivariateCumulativeStudentDistribution::
BivariateCumulativeStudentDistribution(Natural n,
Real rho)
: n_(n), rho_(rho) {}
Real BivariateCumulativeStudentDistribution::operator()(Real x,
Real y) const {
return P_n(x, y, n_, rho_);
}
}
|
