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/* -*- mode: c++; tab-width: 4; indent-tabs-mode: nil; c-basic-offset: 4 -*- */
/*
Copyright (C) 2002, 2003 Ferdinando Ametrano
Copyright (C) 2000, 2001, 2002, 2003 RiskMap srl
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/distributions/normaldistribution.hpp>
namespace QuantLib {
Real CumulativeNormalDistribution::operator()(Real z) const {
//QL_REQUIRE(!(z >= average_ && 2.0*average_-z > average_),
// "not a real number. ");
z = (z - average_) / sigma_;
Real result = 0.5 * ( 1.0 + errorFunction_( z*M_SQRT_2 ) );
if (result<=1e-8) { //todo: investigate the threshold level
// Asymptotic expansion for very negative z following (26.2.12)
// on page 408 in M. Abramowitz and A. Stegun,
// Pocketbook of Mathematical Functions, ISBN 3-87144818-4.
Real sum=1.0, zsqr=z*z, i=1.0, g=1.0, x, y,
a=QL_MAX_REAL, lasta;
do {
lasta=a;
x = (4.0*i-3.0)/zsqr;
y = x*((4.0*i-1)/zsqr);
a = g*(x-y);
sum -= a;
g *= y;
++i;
a = std::fabs(a);
} while (lasta>a && a>=std::fabs(sum*QL_EPSILON));
result = -gaussian_(z)/z*sum;
}
return result;
}
#if !defined(QL_PATCH_SOLARIS)
const CumulativeNormalDistribution InverseCumulativeNormal::f_;
#endif
// Coefficients for the rational approximation.
const Real InverseCumulativeNormal::a1_ = -3.969683028665376e+01;
const Real InverseCumulativeNormal::a2_ = 2.209460984245205e+02;
const Real InverseCumulativeNormal::a3_ = -2.759285104469687e+02;
const Real InverseCumulativeNormal::a4_ = 1.383577518672690e+02;
const Real InverseCumulativeNormal::a5_ = -3.066479806614716e+01;
const Real InverseCumulativeNormal::a6_ = 2.506628277459239e+00;
const Real InverseCumulativeNormal::b1_ = -5.447609879822406e+01;
const Real InverseCumulativeNormal::b2_ = 1.615858368580409e+02;
const Real InverseCumulativeNormal::b3_ = -1.556989798598866e+02;
const Real InverseCumulativeNormal::b4_ = 6.680131188771972e+01;
const Real InverseCumulativeNormal::b5_ = -1.328068155288572e+01;
const Real InverseCumulativeNormal::c1_ = -7.784894002430293e-03;
const Real InverseCumulativeNormal::c2_ = -3.223964580411365e-01;
const Real InverseCumulativeNormal::c3_ = -2.400758277161838e+00;
const Real InverseCumulativeNormal::c4_ = -2.549732539343734e+00;
const Real InverseCumulativeNormal::c5_ = 4.374664141464968e+00;
const Real InverseCumulativeNormal::c6_ = 2.938163982698783e+00;
const Real InverseCumulativeNormal::d1_ = 7.784695709041462e-03;
const Real InverseCumulativeNormal::d2_ = 3.224671290700398e-01;
const Real InverseCumulativeNormal::d3_ = 2.445134137142996e+00;
const Real InverseCumulativeNormal::d4_ = 3.754408661907416e+00;
// Limits of the approximation regions
const Real InverseCumulativeNormal::x_low_ = 0.02425;
const Real InverseCumulativeNormal::x_high_= 1.0 - x_low_;
Real InverseCumulativeNormal::operator()(Real x) const {
QL_REQUIRE(x > 0.0 && x < 1.0,
"InverseCumulativeNormal(" << x
<< ") undefined: must be 0 < x < 1");
Real z, r;
if (x < x_low_) {
// Rational approximation for the lower region 0<x<u_low
z = std::sqrt(-2.0*std::log(x));
z = (((((c1_*z+c2_)*z+c3_)*z+c4_)*z+c5_)*z+c6_) /
((((d1_*z+d2_)*z+d3_)*z+d4_)*z+1.0);
} else if (x <= x_high_) {
// Rational approximation for the central region u_low<=x<=u_high
z = x - 0.5;
r = z*z;
z = (((((a1_*r+a2_)*r+a3_)*r+a4_)*r+a5_)*r+a6_)*z /
(((((b1_*r+b2_)*r+b3_)*r+b4_)*r+b5_)*r+1.0);
} else {
// Rational approximation for the upper region u_high<x<1
z = std::sqrt(-2.0*std::log(1.0-x));
z = -(((((c1_*z+c2_)*z+c3_)*z+c4_)*z+c5_)*z+c6_) /
((((d1_*z+d2_)*z+d3_)*z+d4_)*z+1.0);
}
// The relative error of the approximation has absolute value less
// than 1.15e-9. One iteration of Halley's rational method (third
// order) gives full machine precision.
// #define REFINE_TO_FULL_MACHINE_PRECISION_USING_HALLEYS_METHOD
#ifdef REFINE_TO_FULL_MACHINE_PRECISION_USING_HALLEYS_METHOD
// error (f_(z) - x) divided by the cumulative's derivative
r = (f_(z) - x) * M_SQRT2 * M_SQRTPI * exp(0.5 * z*z);
// Halley's method
z -= r/(1+0.5*z*r);
#endif
return average_ + z*sigma_;
}
const Real MoroInverseCumulativeNormal::a0_ = 2.50662823884;
const Real MoroInverseCumulativeNormal::a1_ =-18.61500062529;
const Real MoroInverseCumulativeNormal::a2_ = 41.39119773534;
const Real MoroInverseCumulativeNormal::a3_ =-25.44106049637;
const Real MoroInverseCumulativeNormal::b0_ = -8.47351093090;
const Real MoroInverseCumulativeNormal::b1_ = 23.08336743743;
const Real MoroInverseCumulativeNormal::b2_ =-21.06224101826;
const Real MoroInverseCumulativeNormal::b3_ = 3.13082909833;
const Real MoroInverseCumulativeNormal::c0_ = 0.3374754822726147;
const Real MoroInverseCumulativeNormal::c1_ = 0.9761690190917186;
const Real MoroInverseCumulativeNormal::c2_ = 0.1607979714918209;
const Real MoroInverseCumulativeNormal::c3_ = 0.0276438810333863;
const Real MoroInverseCumulativeNormal::c4_ = 0.0038405729373609;
const Real MoroInverseCumulativeNormal::c5_ = 0.0003951896511919;
const Real MoroInverseCumulativeNormal::c6_ = 0.0000321767881768;
const Real MoroInverseCumulativeNormal::c7_ = 0.0000002888167364;
const Real MoroInverseCumulativeNormal::c8_ = 0.0000003960315187;
Real MoroInverseCumulativeNormal::operator()(Real x) const {
QL_REQUIRE(x > 0.0 && x < 1.0,
"MoroInverseCumulativeNormal(" << x
<< ") undefined: must be 0<x<1");
Real result;
Real temp=x-0.5;
if (std::fabs(temp) < 0.42) {
// Beasley and Springer, 1977
result=temp*temp;
result=temp*
(((a3_*result+a2_)*result+a1_)*result+a0_) /
((((b3_*result+b2_)*result+b1_)*result+b0_)*result+1.0);
} else {
// improved approximation for the tail (Moro 1995)
if (x<0.5)
result = x;
else
result=1.0-x;
result = std::log(-std::log(result));
result = c0_+result*(c1_+result*(c2_+result*(c3_+result*
(c4_+result*(c5_+result*(c6_+result*
(c7_+result*c8_)))))));
if (x<0.5)
result=-result;
}
return average_ + result*sigma_;
}
}
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