Real ExponentialJump1dMesher::jumpSizeDistribution(Real x, Time t) const {
const Real xmin = std::min(x, 1.0e-100);
return GaussLobattoIntegral(1000000, 1e-12)(
boost::bind(&ExponentialJump1dMesher::jumpSizeDensity, this, _1, t),
xmin, std::max(x, xmin));
}
Real ExtendedOrnsteinUhlenbeckProcess::expectation(
Time t0, Real x0, Time dt) const {
switch (discretization_) {
case MidPoint:
return ouProcess_->expectation(t0, x0, dt)
+ b_(t0+0.5*dt)*(1.0 - std::exp(-speed_*dt));
break;
case Trapezodial:
{
const Time t = t0+dt;
const Time u = t0;
const Real bt = b_(t);
const Real bu = b_(u);
const Real ex = std::exp(-speed_*dt);
return ouProcess_->expectation(t0, x0, dt)
+ bt-ex*bu - (bt-bu)/(speed_*dt)*(1-ex);
}
break;
case GaussLobatto:
return ouProcess_->expectation(t0, x0, dt)
+ speed_*std::exp(-speed_*(t0+dt))
* GaussLobattoIntegral(100000, intEps_)(
boost::lambda::bind(b_, boost::lambda::_1)
*boost::lambda::bind(std::ptr_fun<Real, Real>(std::exp),
speed_*boost::lambda::_1), t0, t0+dt);
break;
default:
QL_FAIL("unknown discretization scheme");
}
}
Real HestonRNDCalculator::cdf(Real x, Time t) const {
return GaussLobattoIntegral(
maxIntegrationIterations_, 0.1*integrationEps_)(
boost::bind(&CpxPv_Helper::p0,
CpxPv_Helper(getHestonParams(hestonProcess_), x_t(x,t),t),_1),
0.0, 1.0)/M_TWOPI + 0.5;
}
Real ExponentialJump1dMesher::jumpSizeDistribution(Real x) const {
const Real a = jumpIntensity_/beta_;
const Real xmin = std::min(x, QL_EPSILON);
const Real gammaValue
= std::exp(GammaFunction().logValue(jumpIntensity_/beta_));
const Real lowerEps =
(std::pow(xmin, a)/a - std::pow(xmin, a+1)/(a+1))/gammaValue;
return lowerEps + GaussLobattoIntegral(10000, 1e-12)(
boost::bind(&ExponentialJump1dMesher::jumpSizeDensity, this, _1),
xmin/eta_, std::max(x, xmin/eta_));
}
Real AnalyticPDFHestonEngine::Pv(Real x_t, Time t) const {
const Hestonp p = { model_->v0(),
model_->kappa(),
model_->theta(),
model_->sigma(),
model_->rho() };
Real xMax = (xMax_ != Null<Real>()) ? xMax_
: Brent().solve(boost::bind(&zero_pv, p, _1, x_t, t),
0.01, 1.0, 1.0);
return GaussLobattoIntegral(nIterations_, 0.1*eps_)
(boost::bind(&pv, p, _1, x_t, t), -xMax, xMax)/M_TWOPI;
}
void AnalyticPDFHestonEngine::calculate() const {
// this is an European option pricer
QL_REQUIRE(arguments_.exercise->type() == Exercise::European,
"not an European option");
const boost::shared_ptr<HestonProcess>& process = model_->process();
const Time t = process->time(arguments_.exercise->lastDate());
const Real xMax = 10 * std::sqrt(process->theta()*t);
results_.value = GaussLobattoIntegral(nIterations_, eps_)
(boost::bind(&AnalyticPDFHestonEngine::weightedPayoff, this,_1, t),
-xMax, xMax);
}
FdmHestonVarianceMesher::FdmHestonVarianceMesher(
Size size,
const boost::shared_ptr<HestonProcess> & process,
Time maturity, Size tAvgSteps, Real epsilon)
: Fdm1dMesher(size) {
std::vector<Real> vGrid(size, 0.0), pGrid(size, 0.0);
const Real df = 4*process->theta()*process->kappa()/
square<Real>()(process->sigma());
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)
/(square<Real>()(process->sigma())
*(1-std::exp(-process->kappa()*t)))*process->v0();
const Real k = square<Real>()(process->sigma())
*(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*InverseNonCentralChiSquareDistribution(
df, ncp, 1000, 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*InverseNonCentralChiSquareDistribution(
df, ncp, 1000, 1e-8)(p);
const Real vx = std::max(vTmp+minVStep, tmp);
p = NonCentralChiSquareDistribution(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");
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 = process->sigma()*
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)
? std::max(1.0, process->sigma()/process->kappa()) : 1.0);
std::sort(pGrid.begin(), pGrid.end());
volaEstimate_ = GaussLobattoIntegral(100000, 1e-4)(
boost::function1<Real, Real>(
compose(std::ptr_fun<Real, Real>(std::sqrt),
LinearInterpolation(pGrid.begin(), pGrid.end(),
vGrid.begin()))),
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>();
}