Real blackFormulaImpliedStdDev(Option::Type optionType,
Real strike,
Real forward,
Real blackPrice,
Real discount,
Real displacement,
Real guess,
Real accuracy,
Natural maxIterations)
{
checkParameters(strike, forward, displacement);
QL_REQUIRE(discount>0.0,
"discount (" << discount << ") must be positive");
QL_REQUIRE(blackPrice>=0.0,
"option price (" << blackPrice << ") must be non-negative");
// check the price of the "other" option implied by put-call paity
Real otherOptionPrice = blackPrice - optionType*(forward-strike)*discount;
QL_REQUIRE(otherOptionPrice>=0.0,
"negative " << Option::Type(-1*optionType) <<
" price (" << otherOptionPrice <<
") implied by put-call parity. No solution exists for " <<
optionType << " strike " << strike <<
", forward " << forward <<
", price " << blackPrice <<
", deflator " << discount);
// solve for the out-of-the-money option which has
// greater vega/price ratio, i.e.
// it is numerically more robust for implied vol calculations
if (optionType==Option::Put && strike>forward) {
optionType = Option::Call;
blackPrice = otherOptionPrice;
}
if (optionType==Option::Call && strike<forward) {
optionType = Option::Put;
blackPrice = otherOptionPrice;
}
strike = strike + displacement;
forward = forward + displacement;
if (guess==Null<Real>())
guess = blackFormulaImpliedStdDevApproximation(
optionType, strike, forward, blackPrice, discount, displacement);
else
QL_REQUIRE(guess>=0.0,
"stdDev guess (" << guess << ") must be non-negative");
BlackImpliedStdDevHelper f(optionType, strike, forward,
blackPrice/discount);
NewtonSafe solver;
solver.setMaxEvaluations(maxIterations);
Real minSdtDev = 0.0, maxStdDev = 24.0; // 24 = 300% * sqrt(60)
Real stdDev = solver.solve(f, accuracy, guess, minSdtDev, maxStdDev);
QL_ENSURE(stdDev>=0.0,
"stdDev (" << stdDev << ") must be non-negative");
return stdDev;
}
boost::shared_ptr<SmileSection>
Gaussian1dSwaptionVolatility::smileSectionImpl(Time optionTime,
Time swapLength) const {
DateHelper hlp(*this, optionTime);
NewtonSafe newton;
Date d(static_cast<BigInteger>(newton.solve(
hlp, 0.1,
365.25 * optionTime + static_cast<Real>(referenceDate().serialNumber()),
1.0)));
Period tenor(
static_cast<Integer>(Rounding(0).operator()(swapLength * 12.0)),
Months);
d = indexBase_->fixingCalendar().adjust(d);
return smileSectionImpl(d, tenor);
}
Volatility CapFloor::impliedVolatility(Real targetValue,
const Handle<YieldTermStructure>& d,
Volatility guess,
Real accuracy,
Natural maxEvaluations,
Volatility minVol,
Volatility maxVol) const {
//calculate();
QL_REQUIRE(!isExpired(), "instrument expired");
ImpliedVolHelper f(*this, d, targetValue);
//Brent solver;
NewtonSafe solver;
solver.setMaxEvaluations(maxEvaluations);
return solver.solve(f, accuracy, guess, minVol, maxVol);
}
Real solveImpl(const F& f,
Real xAccuracy) const {
/* The implementation of the algorithm was inspired by
Press, Teukolsky, Vetterling, and Flannery,
"Numerical Recipes in C", 2nd edition,
Cambridge University Press
*/
Real froot, dfroot, dx;
froot = f(root_);
dfroot = f.derivative(root_);
QL_REQUIRE(dfroot != Null<Real>(),
"Newton requires function's derivative");
++evaluationNumber_;
while (evaluationNumber_<=maxEvaluations_) {
dx = froot/dfroot;
root_ -= dx;
// jumped out of brackets, switch to NewtonSafe
if ((xMin_-root_)*(root_-xMax_) < 0.0) {
NewtonSafe s;
s.setMaxEvaluations(maxEvaluations_-evaluationNumber_);
return s.solve(f, xAccuracy, root_+dx, xMin_, xMax_);
}
if (std::fabs(dx) < xAccuracy) {
f(root_);
++evaluationNumber_;
return root_;
}
froot = f(root_);
dfroot = f.derivative(root_);
++evaluationNumber_;
}
QL_FAIL("maximum number of function evaluations ("
<< maxEvaluations_ << ") exceeded");
}
boost::shared_ptr<SmileSection>
SingleSabrSwaptionVolatility::smileSectionImpl(Time optionTime,
Time swapLength) const {
std::vector<Real> params(4);
params[0] = alpha_;
params[1] = beta_;
params[2] = nu_;
params[3] = rho_;
DateHelper hlp(*this,optionTime);
NewtonSafe newton;
Date d(static_cast<BigInteger>(newton.solve(
hlp, 0.1, 365.25 * optionTime +
static_cast<Real>(referenceDate().serialNumber()),
1.0)));
Period tenor(static_cast<Integer>(Rounding(0).operator()(swapLength*12.0)), Months);
d = indexBase_->fixingCalendar().adjust(d);
Real forward = indexBase_->clone(tenor)->fixing(d);
boost::shared_ptr<SmileSection> raw(new SabrSmileSection(optionTime,forward,params,shift_));
// make it arbitrage free
//boost::shared_ptr<SmileSection> af(new KahaleSmileSection(raw));
//return af;
return raw;
}
