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
185
186
187
188
189
190
191
192
193
194
195
196
197
198
199
200
201
202
203
204
205
206
207
208
209
210
211
212
213
|
/* -*- 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, 2019 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/math/functional.hpp>
#include <ql/math/distributions/normaldistribution.hpp>
#include <ql/math/distributions/chisquaredistribution.hpp>
#include <ql/math/interpolations/linearinterpolation.hpp>
#include <ql/math/integrals/gausslobattointegral.hpp>
#include <ql/termstructures/volatility/equityfx/localvoltermstructure.hpp>
#include <ql/methods/finitedifferences/meshers/fdmhestonvariancemesher.hpp>
#include <boost/accumulators/accumulators.hpp>
#include <boost/accumulators/statistics/mean.hpp>
#include <boost/accumulators/statistics/stats.hpp>
#include <set>
#include <algorithm>
namespace QuantLib {
namespace {
struct interpolated_volatility {
interpolated_volatility(const std::vector<Real>& pGrid,
const std::vector<Real>& vGrid)
: variance(pGrid.begin(), pGrid.end(), vGrid.begin()) {}
Real operator()(Real x) const {
return std::sqrt(variance(x, true));
}
LinearInterpolation variance;
};
}
FdmHestonVarianceMesher::FdmHestonVarianceMesher(
Size size,
const ext::shared_ptr<HestonProcess> & process,
Time maturity, Size tAvgSteps, Real epsilon,
Real mixingFactor)
: Fdm1dMesher(size) {
std::vector<Real> vGrid(size, 0.0), pGrid(size, 0.0);
const Real mixedSigma = process->sigma()*mixingFactor;
const Real df = 4*process->theta()*process->kappa()/squared(mixedSigma);
try {
std::multiset<std::pair<Real, Real> > grid;
for (Size l=1; l<=tAvgSteps; ++l) {
const Real t = (maturity*l)/tAvgSteps;
const Real ncp = 4*process->kappa()*std::exp(-process->kappa()*t)/(squared(mixedSigma)
*(1-std::exp(-process->kappa()*t)))*process->v0();
const Real k = squared(mixedSigma)
*(1-std::exp(-process->kappa()*t))/(4*process->kappa());
const Real qMin = 0.0; // v_min = 0.0;
const Real qMax = std::max(process->v0(),
k*InverseNonCentralCumulativeChiSquareDistribution(
df, ncp, 100, 1e-8)(1-epsilon));
const Real minVStep=(qMax-qMin)/(50*size);
Real ps,p = 0.0;
Real vTmp = qMin;
grid.insert(std::pair<Real, Real>(qMin, epsilon));
for (Size i=1; i < size; ++i) {
ps = (1 - epsilon - p)/(size-i);
p += ps;
const Real tmp = k*InverseNonCentralCumulativeChiSquareDistribution(
df, ncp, 100, 1e-8)(p);
const Real vx = std::max(vTmp+minVStep, tmp);
p = NonCentralCumulativeChiSquareDistribution(df, ncp)(vx/k);
vTmp=vx;
grid.insert(std::pair<Real, Real>(vx, p));
}
}
QL_REQUIRE(grid.size() == size*tAvgSteps,
"something wrong with the grid size");
const std::vector<std::pair<Real, Real> > tp(grid.begin(), grid.end());
for (Size i=0; i < size; ++i) {
const Size b = (i*tp.size())/size;
const Size e = ((i+1)*tp.size())/size;
for (Size j=b; j < e; ++j) {
vGrid[i]+=tp[j].first/(e-b);
pGrid[i]+=tp[j].second/(e-b);
}
}
}
catch (const Error&) {
// use default mesh
const Real vol = mixedSigma*
std::sqrt(process->theta()/(2*process->kappa()));
const Real mean = process->theta();
const Real upperBound = std::max(process->v0()+4*vol, mean+4*vol);
const Real lowerBound
= std::max(0.0, std::min(process->v0()-4*vol, mean-4*vol));
for (Size i=0; i < size; ++i) {
pGrid[i] = i/(size-1.0);
vGrid[i] = lowerBound + i*(upperBound-lowerBound)/(size-1.0);
}
}
Real skewHint = ((process->kappa() != 0.0)
? Real(std::max(1.0, mixedSigma/process->kappa())) : 1.0);
std::sort(pGrid.begin(), pGrid.end());
volaEstimate_ = GaussLobattoIntegral(100000, 1e-4)(
interpolated_volatility(pGrid, vGrid),
pGrid.front(), pGrid.back())*std::pow(skewHint, 1.5);
const Real v0 = process->v0();
for (Size i=1; i<vGrid.size(); ++i) {
if (vGrid[i-1] <= v0 && vGrid[i] >= v0) {
if (std::fabs(vGrid[i-1] - v0) < std::fabs(vGrid[i] - v0))
vGrid[i-1] = v0;
else
vGrid[i] = v0;
}
}
std::copy(vGrid.begin(), vGrid.end(), locations_.begin());
for (Size i=0; i < size-1; ++i) {
dminus_[i+1] = dplus_[i] = vGrid[i+1] - vGrid[i];
}
dplus_.back() = dminus_.front() = Null<Real>();
}
FdmHestonLocalVolatilityVarianceMesher::FdmHestonLocalVolatilityVarianceMesher(
Size size,
const ext::shared_ptr<HestonProcess>& process,
const ext::shared_ptr<LocalVolTermStructure>& leverageFct,
Time maturity, Size tAvgSteps, Real epsilon,
Real mixingFactor)
: Fdm1dMesher(size) {
const FdmHestonVarianceMesher mesher(
size, process, maturity, tAvgSteps, epsilon, mixingFactor);
for (Size i=0; i < size; ++i) {
dplus_[i] = mesher.dplus(i);
dminus_[i] = mesher.dminus(i);
locations_[i] = mesher.location(i);
}
volaEstimate_ = mesher.volaEstimate();
if (leverageFct != nullptr) {
typedef boost::accumulators::accumulator_set<
Real, boost::accumulators::stats<
boost::accumulators::tag::mean> >
accumulator_set;
accumulator_set acc;
const Real s0 = process->s0()->value();
acc(leverageFct->localVol(0, s0, true));
const Handle<YieldTermStructure> rTS = process->riskFreeRate();
const Handle<YieldTermStructure> qTS = process->dividendYield();
for (Size l=1; l <= tAvgSteps; ++l) {
const Real t = (maturity*l)/tAvgSteps;
const Real vol = volaEstimate_ * boost::accumulators::mean(acc);
const Real fwd = s0*qTS->discount(t)/rTS->discount(t);
const Size sAvgSteps = 50;
std::vector<Real> u(sAvgSteps), sig(sAvgSteps);
for (Size i=0; i < sAvgSteps; ++i) {
u[i] = epsilon + ((1-2*epsilon)/(sAvgSteps-1))*i;
const Real x = InverseCumulativeNormal()(u[i]);
const Real gf = x*vol*std::sqrt(t);
const Real f = fwd*std::exp(gf);
sig[i] = squared(leverageFct->localVol(t, f, true));
}
const Real leverageAvg =
GaussLobattoIntegral(10000, 1e-4)(
interpolated_volatility(u, sig), u.front(), u.back())
/ (1-2*epsilon);
acc(leverageAvg);
}
volaEstimate_ *= boost::accumulators::mean(acc);
}
}
}
|
