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/* -*- mode: c++; tab-width: 4; indent-tabs-mode: nil; c-basic-offset: 4 -*- */
/*
Copyright (C) 2013 Peter Caspers
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/models/gsrprocess.hpp>
namespace QuantLib {
GsrProcess::GsrProcess(const Array ×, const Array &vols,
const Array &reversions, const Real T)
: ForwardMeasureProcess1D(T), times_(times), vols_(vols),
reversions_(reversions), revZero_(reversions.size(), false) {
QL_REQUIRE(times.size() == vols.size() - 1,
"number of volatilities ("
<< vols.size() << ") compared to number of times ("
<< times_.size() << " must be bigger by one");
QL_REQUIRE(times.size() == reversions.size() - 1 ||
reversions.size() == 1,
"number of reversions ("
<< vols.size() << ") compared to number of times ("
<< times_.size() << " must be bigger by one, or exactly "
"1 reversion must be given");
for (int i = 0; i < ((int)times.size()) - 1; i++)
QL_REQUIRE(times[i] < times[i + 1], "times must be increasing ("
<< times[i] << "@" << i
<< " , " << times[i + 1]
<< "@" << i + 1 << ")");
for (int i = 0; i < (int)reversions.size(); i++)
// if (close(reversions[i], 0.0))
if (std::fabs(reversions[i]) < 1E-4)
revZero_[i] = true;
flushCache();
}
Real GsrProcess::x0() const { return 0.0; }
Real GsrProcess::drift(Time t, Real x) const {
QL_REQUIRE(t <= getForwardMeasureTime(),
"t (" << t
<< ") must not be greater than forward measure time ("
<< getForwardMeasureTime() << ")");
return y(t) - G(t, this->getForwardMeasureTime(), x) *
vol(lowerIndex(t)) * vol(lowerIndex(t)) -
rev(lowerIndex(t)) * x;
}
Real GsrProcess::diffusion(Time t, Real) const {
QL_REQUIRE(t <= getForwardMeasureTime(),
"t (" << t
<< ") must not be greater than forward measure time ("
<< getForwardMeasureTime() << ")");
return vol(lowerIndex(t));
}
Real GsrProcess::expectation(Time w, Real xw, Time dt) const {
Real t = w + dt;
QL_REQUIRE(t <= getForwardMeasureTime(),
"t (" << t
<< ") must not be greater than forward measure time ("
<< getForwardMeasureTime() << ")");
return expectationp1(w, xw, dt) + expectationp2(w, dt);
}
void GsrProcess::flushCache() const {
cache1_.clear();
cache2_.clear();
cache3_.clear();
cache4_.clear();
cache5_.clear();
}
Real GsrProcess::expectationp1(Time w, Real xw, Time dt) const {
Real t = w + dt;
std::pair<Real, Real> key;
key = std::make_pair(w, t);
std::map<std::pair<Real, Real>, Real>::const_iterator k =
cache1_.find(key);
if (k != cache1_.end())
return xw * (k->second);
// A(w,t)x(w)
Real res2 = 1.0;
for (int i = lowerIndex(w); i <= upperIndex(t) - 1; i++) {
res2 *= exp(-rev(i) * (cappedTime(i + 1, t) - flooredTime(i, w)));
}
cache1_.insert(std::make_pair(key, res2));
return res2 * xw;
}
Real GsrProcess::expectationp2(Time w, Time dt) const {
Real t = w + dt;
std::pair<Real, Real> key;
key = std::make_pair(w, t);
std::map<std::pair<Real, Real>, Real>::const_iterator k =
cache2_.find(key);
if (k != cache2_.end())
return k->second;
Real T = getForwardMeasureTime();
Real res = 0.0;
// \int A(s,t)y(s)
for (int k = lowerIndex(w); k <= upperIndex(t) - 1; k++) {
// l<k
for (int l = 0; l <= k - 1; l++) {
Real res2 = 1.0;
// alpha_l
res2 *= revZero(l) ? vol(l) * vol(l) * (time2(l + 1) - time2(l))
: vol(l) * vol(l) / (2.0 * rev(l)) *
(1.0 - exp(-2.0 * rev(l) *
(time2(l + 1) - time2(l))));
// zeta_i (i>k)
for (int i = k + 1; i <= upperIndex(t) - 1; i++)
res2 *= exp(-rev(i) * (cappedTime(i + 1, t) - time2(i)));
// beta_j (j<k)
for (int j = l + 1; j <= k - 1; j++)
res2 *= exp(-2.0 * rev(j) * (time2(j + 1) - time2(j)));
// zeta_k beta_k
res2 *=
revZero(k)
? 2.0 * time2(k) - flooredTime(k, w) -
cappedTime(k + 1, t) -
2.0 * (time2(k) - cappedTime(k + 1, t))
: (exp(rev(k) * (2.0 * time2(k) - flooredTime(k, w) -
cappedTime(k + 1, t))) -
exp(2.0 * rev(k) *
(time2(k) - cappedTime(k + 1, t)))) /
rev(k);
// add to sum
res += res2;
}
// l=k
Real res2 = 1.0;
// alpha_k zeta_k
res2 *=
revZero(k)
? vol(k) * vol(k) / 4.0 *
(4.0 * pow(cappedTime(k + 1, t) - time2(k), 2.0) -
(pow(flooredTime(k, w) - 2.0 * time2(k) +
cappedTime(k + 1, t),
2.0) +
pow(cappedTime(k + 1, t) - flooredTime(k, w), 2.0)))
: vol(k) * vol(k) / (2.0 * rev(k) * rev(k)) *
(exp(-2.0 * rev(k) *
(cappedTime(k + 1, t) - time2(k))) +
1.0 -
(exp(-rev(k) * (flooredTime(k, w) - 2.0 * time2(k) +
cappedTime(k + 1, t))) +
exp(-rev(k) *
(cappedTime(k + 1, t) - flooredTime(k, w)))));
// zeta_i (i>k)
for (int i = k + 1; i <= upperIndex(t) - 1; i++)
res2 *= exp(-rev(i) * (cappedTime(i + 1, t) - time2(i)));
// no beta_j in this case ...
res += res2;
}
// int -A(s,t) \sigma^2 G(s,T)
for (int k = lowerIndex(w); k <= upperIndex(t) - 1; k++) {
Real res2 = 0.0;
// l>k
for (int l = k + 1; l <= upperIndex(T) - 1; l++) {
Real res3 = 1.0;
// eta_l
res3 *= revZero(l)
? cappedTime(l + 1, T) - time2(l)
: (1.0 - exp(-rev(l) *
(cappedTime(l + 1, T) - time2(l)))) /
rev(l);
// zeta_i (i>k)
for (int i = k + 1; i <= upperIndex(t) - 1; i++)
res3 *= exp(-rev(i) * (cappedTime(i + 1, t) - time2(i)));
// gamma_j (j>k)
for (int j = k + 1; j <= l - 1; j++)
res3 *= exp(-rev(j) * (time2(j + 1) - time2(j)));
// zeta_k gamma_k
res3 *=
revZero(k)
? (cappedTime(k + 1, t) - time2(k + 1) -
(2.0 * flooredTime(k, w) - cappedTime(k + 1, t) -
time2(k + 1))) /
2.0
: (exp(rev(k) * (cappedTime(k + 1, t) - time2(k + 1))) -
exp(rev(k) *
(2.0 * flooredTime(k, w) - cappedTime(k + 1, t) -
time2(k + 1)))) /
(2.0 * rev(k));
// add to sum
res2 += res3;
}
// l=k
Real res3 = 1.0;
// eta_k zeta_k
res3 *=
revZero(k)
? (-pow(cappedTime(k + 1, t) - cappedTime(k + 1, T), 2.0) -
2.0 *
pow(cappedTime(k + 1, t) - flooredTime(k, w), 2.0) +
pow(2.0 * flooredTime(k, w) - cappedTime(k + 1, T) -
cappedTime(k + 1, t),
2.0)) /
4.0
: (2.0 - exp(rev(k) * (cappedTime(k + 1, t) -
cappedTime(k + 1, T))) -
(2.0 * exp(-rev(k) *
(cappedTime(k + 1, t) - flooredTime(k, w))) -
exp(rev(k) *
(2.0 * flooredTime(k, w) - cappedTime(k + 1, T) -
cappedTime(k + 1, t))))) /
(2.0 * rev(k) * rev(k));
// zeta_i (i>k)
for (int i = k + 1; i <= upperIndex(t) - 1; i++)
res3 *= exp(-rev(i) * (cappedTime(i + 1, t) - time2(i)));
// no gamma_j in this case ...
res2 += res3;
// add to main accumulator
res += -vol(k) * vol(k) * res2;
}
cache2_.insert(std::make_pair(key, res));
return res;
}
Real GsrProcess::stdDeviation(Time t0, Real x0, Time dt) const {
return sqrt(variance(t0, x0, dt));
}
Real GsrProcess::variance(Time w, Real, Time dt) const {
Real t = w + dt;
QL_REQUIRE(t <= getForwardMeasureTime(),
"t (" << t
<< ") must not be greater than forward measure time ("
<< getForwardMeasureTime() << ")");
std::pair<Real, Real> key;
key = std::make_pair(w, t);
std::map<std::pair<Real, Real>, Real>::const_iterator k =
cache3_.find(key);
if (k != cache3_.end())
return k->second;
Real res = 0.0;
for (int k = lowerIndex(w); k <= upperIndex(t) - 1; k++) {
Real res2 = vol(k) * vol(k);
// zeta_k^2
res2 *= revZero(k)
? -(flooredTime(k, w) - cappedTime(k + 1, t))
: (1.0 - exp(2.0 * rev(k) * (flooredTime(k, w) -
cappedTime(k + 1, t)))) /
(2.0 * rev(k));
// zeta_i (i>k)
for (int i = k + 1; i <= upperIndex(t) - 1; i++) {
res2 *= exp(-2.0 * rev(i) * (cappedTime(i + 1, t) - time2(i)));
}
res += res2;
}
cache3_.insert(std::make_pair(key, res));
return res;
}
Real GsrProcess::sigma(Time t) const { return vol(lowerIndex(t)); }
Real GsrProcess::reversion(Time t) const { return rev(lowerIndex(t)); }
Real GsrProcess::y(Time t) const {
QL_REQUIRE(t >= 0.0 && t <= getForwardMeasureTime(),
"y(t) should be called with t (" << t << ") in Range [0,"
<< getForwardMeasureTime()
<< "].");
Real key;
key = t;
std::map<Real, Real>::const_iterator k = cache4_.find(key);
if (k != cache4_.end())
return k->second;
Real res = 0.0;
for (int i = 0; i <= upperIndex(t) - 1; i++) {
Real res2 = 1.0;
for (int j = i + 1; j <= upperIndex(t) - 1; j++) {
res2 *= exp(-2.0 * rev(j) * (cappedTime(j + 1, t) - time2(j)));
}
res2 *= revZero(i)
? vol(i) * vol(i) * (cappedTime(i + 1, t) - time2(i))
: (vol(i) * vol(i) / (2.0 * rev(i)) *
(1.0 - exp(-2.0 * rev(i) *
(cappedTime(i + 1, t) - time2(i)))));
res += res2;
}
cache4_.insert(std::make_pair(key, res));
return res;
}
Real GsrProcess::G(Time t, Time w, Real) const {
QL_REQUIRE(w >= t, "G(t,w) should be called with w ("
<< w << ") not lesser than t (" << t << ")");
QL_REQUIRE(t >= 0.0 && w <= getForwardMeasureTime(),
"G(t,w) should be called with (t,w)=("
<< t << "," << w << ") in Range [0,"
<< getForwardMeasureTime() << "].");
std::pair<Real, Real> key;
key = std::make_pair(w, t);
std::map<std::pair<Real, Real>, Real>::const_iterator k =
cache5_.find(key);
if (k != cache5_.end())
return k->second;
Real res = 0.0;
for (int i = lowerIndex(t); i <= upperIndex(w) - 1; i++) {
Real res2 = 1.0;
for (int j = lowerIndex(t); j <= i - 1; j++) {
res2 *= exp(-rev(j) * (time2(j + 1) - flooredTime(j, t)));
}
res2 *= revZero(i) ? cappedTime(i + 1, w) - flooredTime(i, t)
: (1.0 - exp(-rev(i) * (cappedTime(i + 1, w) -
flooredTime(i, t)))) /
rev(i);
res += res2;
}
cache5_.insert(std::make_pair(key, res));
return res;
}
const int GsrProcess::lowerIndex(Time t) const {
return std::upper_bound(times_.begin(), times_.end(), t) -
times_.begin();
}
const int GsrProcess::upperIndex(Time t) const {
if (t < QL_EPSILON)
return 0;
return std::upper_bound(times_.begin(), times_.end(), t - QL_EPSILON) -
times_.begin() + 1;
}
const Real GsrProcess::cappedTime(Size index, Real cap) const {
return cap != Null<Real>() ? std::min(cap, time2(index)) : time2(index);
}
const Real GsrProcess::flooredTime(Size index, Real floor) const {
return floor != Null<Real>() ? std::max(floor, time2(index))
: time2(index);
}
const Real GsrProcess::time2(Size index) const {
if (index == 0)
return 0.0;
if (index > times_.size())
return getForwardMeasureTime(); // FIXME how to ensure that forward
// measure time is geq all times
// given
return times_[index - 1];
}
const Real GsrProcess::vol(Size index) const {
if (index >= vols_.size())
return vols_.back();
return vols_[index];
}
const Real GsrProcess::rev(Size index) const {
if (index >= reversions_.size())
return reversions_.back();
return reversions_[index];
}
const bool GsrProcess::revZero(Size index) const {
if (index >= revZero_.size())
return revZero_.back();
return revZero_[index];
}
}
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