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
|
/* -*- mode: c++; tab-width: 4; indent-tabs-mode: nil; c-basic-offset: 4 -*- */
/*
Copyright (C) 2010 Adrian O' Neill
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/experimental/variancegamma/fftengine.hpp>
#include <ql/exercise.hpp>
#include <ql/math/interpolations/linearinterpolation.hpp>
#include <ql/math/fastfouriertransform.hpp>
#include <complex>
namespace QuantLib {
FFTEngine::FFTEngine(
const boost::shared_ptr<StochasticProcess1D>& process, Real logStrikeSpacing)
: process_(process), lambda_(logStrikeSpacing) {
registerWith(process_);
}
void FFTEngine::calculate() const
{
QL_REQUIRE(arguments_.exercise->type() == Exercise::European,
"not an European Option");
boost::shared_ptr<StrikedTypePayoff> payoff =
boost::dynamic_pointer_cast<StrikedTypePayoff>(arguments_.payoff);
QL_REQUIRE(payoff, "non-striked payoff given");
ResultMap::const_iterator r1 = resultMap_.find(arguments_.exercise->lastDate());
if (r1 != resultMap_.end())
{
PayoffResultMap::const_iterator r2 = r1->second.find(payoff);
if (r2 != r1->second.end())
{
results_.value = r2->second;
return;
}
}
// Option not precalculated - do entire FFT for one option. Not very efficient - call precalculate!
calculateUncached(payoff, arguments_.exercise);
}
void FFTEngine::update()
{
// Process has changed so cached values may no longer be correct
resultMap_.clear();
// Call base class implementation
VanillaOption::engine::update();
}
void FFTEngine::calculateUncached(boost::shared_ptr<StrikedTypePayoff> payoff,
boost::shared_ptr<Exercise> exercise) const
{
boost::shared_ptr<VanillaOption> option(new VanillaOption(payoff, exercise));
std::vector<boost::shared_ptr<Instrument> > optionList;
optionList.push_back(option);
boost::shared_ptr<FFTEngine> tempEngine(clone().release());
tempEngine->precalculate(optionList);
option->setPricingEngine(tempEngine);
results_.value = option->NPV();
}
void FFTEngine::precalculate(const std::vector<boost::shared_ptr<Instrument> >& optionList) {
// Group payoffs by expiry date
// as with FFT we can compute a bunch of these at once
resultMap_.clear();
typedef std::vector<boost::shared_ptr<StrikedTypePayoff> > PayoffList;
typedef std::map<Date, PayoffList> PayoffMap;
PayoffMap payoffMap;
for (std::vector<boost::shared_ptr<Instrument> >::const_iterator optIt = optionList.begin();
optIt != optionList.end(); optIt++)
{
boost::shared_ptr<VanillaOption> option = boost::dynamic_pointer_cast<VanillaOption>(*optIt);
QL_REQUIRE(option, "instrument must be option");
QL_REQUIRE(option->exercise()->type() == Exercise::European,
"not an European Option");
boost::shared_ptr<StrikedTypePayoff> payoff =
boost::dynamic_pointer_cast<StrikedTypePayoff>(option->payoff());
QL_REQUIRE(payoff, "non-striked payoff given");
payoffMap[option->exercise()->lastDate()].push_back(payoff);
}
std::complex<Real> i1(0, 1);
Real alpha = 1.25;
for (PayoffMap::const_iterator payIt = payoffMap.begin(); payIt != payoffMap.end(); payIt++)
{
Date expiryDate = payIt->first;
// Calculate n large enough for maximum strike, and round up to a power of 2
Real maxStrike = 0.0;
for (PayoffList::const_iterator it = payIt->second.begin();
it != payIt->second.end(); it++)
{
boost::shared_ptr<StrikedTypePayoff> payoff = *it;
if (payoff->strike() > maxStrike)
maxStrike = payoff->strike();
}
Real nR = 2.0 * (std::log(maxStrike) + lambda_) / lambda_;
Size log2_n = (static_cast<Size>((std::log(nR) / std::log(2.0))) + 1);
Size n = 1 << log2_n;
// Strike range (equation 19,20)
Real b = n * lambda_ / 2.0;
// Grid spacing (equation 23)
Real eta = 2.0 * M_PI / (lambda_ * n);
// Discount factor
Real df = discountFactor(expiryDate);
Real div = dividendYield(expiryDate);
// Input to fourier transform
std::vector<std::complex<Real> > fti;
fti.resize(n);
// Precalculate any discount factors etc.
precalculateExpiry(expiryDate);
for (Size i=0; i<n; i++)
{
Real v_j = eta * i;
Real sw = eta * (3.0 + ((i % 2) == 0 ? -1.0 : 1.0) - ((i == 0) ? 1.0 : 0.0)) / 3.0;
std::complex<Real> psi = df * complexFourierTransform(v_j - (alpha + 1)* i1);
psi = psi / (alpha*alpha + alpha - v_j*v_j + i1 * (2 * alpha + 1.0) * v_j);
fti[i] = std::exp(i1 * b * v_j) * sw * psi;
}
// Perform fft
std::vector<std::complex<Real> > results(n);
FastFourierTransform fft(log2_n);
fft.transform(fti.begin(), fti.end(), results.begin());
// Call prices
std::vector<Real> prices, strikes;
prices.resize(n);
strikes.resize(n);
for (Size i=0; i<n; i++)
{
Real k_u = -b + lambda_ * i;
prices[i] = (std::exp(-alpha * k_u) / M_PI) * results[i].real();
strikes[i] = std::exp(k_u);
}
for (PayoffList::const_iterator it = payIt->second.begin();
it != payIt->second.end(); it++)
{
boost::shared_ptr<StrikedTypePayoff> payoff = *it;
Real callPrice = LinearInterpolation(strikes.begin(), strikes.end(), prices.begin())(payoff->strike());
switch (payoff->optionType())
{
case Option::Call:
resultMap_[expiryDate][payoff] = callPrice;
break;
case Option::Put:
resultMap_[expiryDate][payoff] = callPrice - process_->x0() * div + payoff->strike() * df;
break;
default:
QL_FAIL("Invalid option type");
}
}
}
}
}
|
