bivariatenormaldistribution.cpp

/* -*- 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
 Copyright (C) 2003 StatPro Italia srl
 Copyright (C) 2005 Gary Kennedy
 
 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/bivariatenormaldistribution.hpp>
#include <ql/math/integrals/gaussianquadratures.hpp>
 
namespace QuantLib {
 
    // Drezner 1978
 
    const Real BivariateCumulativeNormalDistributionDr78::x_[] = {
        0.24840615,
        0.39233107,
        0.21141819,
        0.03324666,
        0.00082485334
    };
 
    const Real BivariateCumulativeNormalDistributionDr78::y_[] = {
        0.10024215,
        0.48281397,
        1.06094980,
        1.77972940,
        2.66976040000
    };
 
    BivariateCumulativeNormalDistributionDr78::
    BivariateCumulativeNormalDistributionDr78(Real rho)
    : rho_(rho), rho2_(rho*rho) {
 
        QL_REQUIRE(rho>=-1.0,
                   "rho must be >= -1.0 (" << rho << " not allowed)");
        QL_REQUIRE(rho<=1.0,
                   "rho must be <= 1.0 (" << rho << " not allowed)");
    }
 
    Real BivariateCumulativeNormalDistributionDr78::operator()(Real a,
                                                               Real b) const {
 
        CumulativeNormalDistribution cumNormalDist;
        Real CumNormDistA = cumNormalDist(a);
        Real CumNormDistB = cumNormalDist(b);
        Real MaxCumNormDistAB = std::max(CumNormDistA, CumNormDistB);
        Real MinCumNormDistAB = std::min(CumNormDistA, CumNormDistB);
 
        if (1.0-MaxCumNormDistAB<1e-15)
            return MinCumNormDistAB;
 
        if (MinCumNormDistAB<1e-15)
            return MinCumNormDistAB;
 
        Real a1 = a / std::sqrt(2.0 * (1.0 - rho2_));
        Real b1 = b / std::sqrt(2.0 * (1.0 - rho2_));
 
        Real result=-1.0;
 
        if (a<=0.0 && b<=0 && rho_<=0) {
            Real sum=0.0;
            for (Size i=0; i<5; i++) {
                for (Size j=0;j<5; j++) {
                    sum += x_[i]*x_[j]*
                        std::exp(a1*(2.0*y_[i]-a1)+b1*(2.0*y_[j]-b1)
                                 +2.0*rho_*(y_[i]-a1)*(y_[j]-b1));
                }
            }
            result = std::sqrt(1.0 - rho2_)/M_PI*sum;
        } else if (a<=0 && b>=0 && rho_>=0) {
            BivariateCumulativeNormalDistributionDr78 bivCumNormalDist(-rho_);
            result= CumNormDistA - bivCumNormalDist(a, -b);
        } else if (a>=0.0 && b<=0.0 && rho_>=0.0) {
            BivariateCumulativeNormalDistributionDr78 bivCumNormalDist(-rho_);
            result= CumNormDistB - bivCumNormalDist(-a, b);
        } else if (a>=0.0 && b>=0.0 && rho_<=0.0) {
            result= CumNormDistA + CumNormDistB -1.0 + (*this)(-a, -b);
        } else if (a*b*rho_>0.0) {
            Real rho1 = (rho_*a-b)*(a>0.0 ? 1.0: -1.0)/
                std::sqrt(a*a-2.0*rho_*a*b+b*b);
            BivariateCumulativeNormalDistributionDr78 bivCumNormalDist(rho1);
 
            Real rho2 = (rho_*b-a)*(b>0.0 ? 1.0: -1.0)/
                std::sqrt(a*a-2.0*rho_*a*b+b*b);
            BivariateCumulativeNormalDistributionDr78 CBND2(rho2);
 
            Real delta = (1.0-(a>0.0 ? 1.0: -1.0)*(b>0.0 ? 1.0: -1.0))/4.0;
 
            result= bivCumNormalDist(a, 0.0) + CBND2(b, 0.0) - delta;
        } else {
            QL_FAIL("case not handled");
        }
 
        return result;
    }
 
 
    // West 2004
 
    namespace {
 
        class eqn3 { /* Relates to eqn3 Genz 2004 */
          public:
            eqn3(Real h, Real k, Real asr)
            : hk_(h * k), asr_(asr), hs_((h * h + k * k) / 2) {}
 
            Real operator()(Real x) const {
                Real sn = std::sin(asr_ * (-x + 1) * 0.5);
                return std::exp((sn * hk_ - hs_) / (1.0 - sn * sn));
            }
          private:
            Real hk_, asr_, hs_;
        };
 
        class eqn6 { /* Relates to eqn6 Genz 2004 */
          public:
            eqn6(Real a, Real c, Real d, Real bs, Real hk)
            : a_(a), c_(c), d_(d), bs_(bs), hk_(hk) {}
            Real operator()(Real x) const {
                Real xs = a_ * (-x + 1);
                xs = std::fabs(xs*xs);
                Real rs = std::sqrt(1 - xs);
                Real asr = -(bs_ / xs + hk_) / 2;
                if (asr > -100.0) {
                    return (a_ * std::exp(asr) *
                            (std::exp(-hk_ * (1 - rs) / (2 * (1 + rs))) / rs -
                             (1 + c_ * xs * (1 + d_ * xs))));
                } else {
                    return 0.0;
                }
            }
          private:
            Real a_, c_, d_, bs_, hk_;
        };
 
    }
 
    BivariateCumulativeNormalDistributionWe04DP::
    BivariateCumulativeNormalDistributionWe04DP(Real rho)
    : correlation_(rho) {
 
        QL_REQUIRE(rho>=-1.0,
                   "rho must be >= -1.0 (" << rho << " not allowed)");
        QL_REQUIRE(rho<=1.0,
                   "rho must be <= 1.0 (" << rho << " not allowed)");
    }
 
 
    Real BivariateCumulativeNormalDistributionWe04DP::operator()(
                                                       Real x, Real y) const {
 
        /* The implementation is described at section 2.4 "Hybrid
           Numerical Integration Algorithms" of "Numerical Computation
           of Rectangular Bivariate an Trivariate Normal and t
           Probabilities", Genz (2004), Statistics and Computing 14,
           151-160. (available at
           www.sci.wsu.edu/math/faculty/henz/homepage)
 
           The Gauss-Legendre quadrature have been extracted to
           TabulatedGaussLegendre (x,w zero-based)
 
           Tthe functions ot be integrated numerically have been moved
           to classes eqn3 and eqn6
 
           Change some magic numbers to M_PI */
 
        TabulatedGaussLegendre gaussLegendreQuad(20);
        if (std::fabs(correlation_) < 0.3) {
            gaussLegendreQuad.order(6);
        } else if (std::fabs(correlation_) < 0.75) {
            gaussLegendreQuad.order(12);
        }
 
        Real h = -x;
        Real k = -y;
        Real hk = h * k;
        Real BVN = 0.0;
 
        if (std::fabs(correlation_) < 0.925)
        {
            if (std::fabs(correlation_) > 0)
            {
                Real asr = std::asin(correlation_);
                eqn3 f(h,k,asr);
                BVN = gaussLegendreQuad(f);
                BVN *= asr * (0.25 / M_PI);
            }
            BVN += cumnorm_(-h) * cumnorm_(-k);
        }
        else
        {
            if (correlation_ < 0)
            {
                k *= -1;
                hk *= -1;
            }
            if (std::fabs(correlation_) < 1)
            {
                Real Ass = (1 - correlation_) * (1 + correlation_);
                Real a = std::sqrt(Ass);
                Real bs = (h-k)*(h-k);
                Real c = (4 - hk) / 8;
                Real d = (12 - hk) / 16;
                Real asr = -(bs / Ass + hk) / 2;
                if (asr > -100)
                {
                    BVN = a * std::exp(asr) *
                        (1 - c * (bs - Ass) * (1 - d * bs / 5) / 3 +
                         c * d * Ass * Ass / 5);
                }
                if (-hk < 100)
                {
                    Real B = std::sqrt(bs);
                    BVN -= std::exp(-hk / 2) * 2.506628274631 *
                        cumnorm_(-B / a) * B *
                        (1 - c * bs * (1 - d * bs / 5) / 3);
                }
                a /= 2;
                eqn6 f(a,c,d,bs,hk);
                BVN += gaussLegendreQuad(f);
                BVN /= (-2.0 * M_PI);
            }
 
            if (correlation_ > 0) {
                BVN += cumnorm_(-std::max(h, k));
            } else {
                BVN *= -1;
                if (k > h) {
                    // evaluate cumnorm where it is most precise, that
                    // is in the lower tail because of double accuracy
                    // around 0.0 vs around 1.0
                    if (h >= 0) {
                        BVN += cumnorm_(-h) - cumnorm_(-k);
                    } else {
                        BVN += cumnorm_(k) - cumnorm_(h);
                    }
                }
            }
        }
        return BVN;
    }
 
}