KienitzLawsonSwayneSabrPdeDensity Class Reference

#include <qle/models/kienitzlawsonswaynesabrpdedensity.hpp>

Collaboration diagram for KienitzLawsonSwayneSabrPdeDensity:

Public Member Functions
	KienitzLawsonSwayneSabrPdeDensity (const Real alpha, const Real beta, const Real nu, const Real rho, const Real forward, const Real expiryTime, const Real displacement, const Size zSteps, const Size tSteps, const Real nStdDev)
Real	alpha () const
Real	beta () const
Real	nu () const
Real	rho () const
Real	forward () const
Real	expiryTime () const
Real	displacement () const
Size	zSteps () const
Size	tSteps () const
Real	nStdDev () const
std::vector< Real >	callPrices (const std::vector< Real > &strikes) const
std::vector< Real >	putPrices (const std::vector< Real > &strikes) const
Real	inverseCumulative (const Real p) const
const Array &	sabrProb () const
Real	zMin () const
Real	zMax () const
Real	hh () const
Real	pL () const
Real	pR () const
Real	theta0 () const
Real	yf (const Real f) const
Real	fy (const Real y) const
Real	yz (const Real z) const
Real	zy (const Real y) const

Private Member Functions
void	calculate ()
void	PDE_method (const Array &fm, const Array &Ccm, const Array &Gamma_Vec, const Real dt, const Size Nj)
void	solveTimeStep_LS (const Array &fm, const Array &cm, const Array &Em, const Real dt, const Array &PP_in, const Real PL_in, const Real PR_in, const Size n, Array &PP_out, Real &PL_out, Real &PR_out)
void	tridag (const Array &a, const Array &b, const Array &c, const Array &R, Array &u, const Size n, const bool firstLastRZero)

Private Attributes
const Real	alpha_
const Real	beta_
const Real	nu_
const Real	rho_
const Real	forward_
const Real	expiryTime_
const Real	displacement_
const Size	zSteps_
const Size	tSteps_
const Real	nStdDev_
Real	zMin_
Real	zMax_
Real	hh_
Real	pL_
Real	pR_
Real	theta0_
Array	SABR_Prob_Vec_
Array	SABR_Cum_Prob_Vec_

Detailed Description

Definition at line 36 of file kienitzlawsonswaynesabrpdedensity.hpp.

Constructor & Destructor Documentation

KienitzLawsonSwayneSabrPdeDensity()

KienitzLawsonSwayneSabrPdeDensity	(	const Real	alpha,
		const Real	beta,
		const Real	nu,
		const Real	rho,
		const Real	forward,
		const Real	expiryTime,
		const Real	displacement,
		const Size	zSteps,
		const Size	tSteps,
		const Real	nStdDev
	)

Definition at line 25 of file kienitzlawsonswaynesabrpdedensity.cpp.

    : alpha_(alpha), beta_(beta), nu_(nu), rho_(rho), forward_(forward), expiryTime_(expiryTime),
      displacement_(displacement), zSteps_(zSteps), tSteps_(tSteps), nStdDev_(nStdDev) {
 
    QL_REQUIRE(alpha > 0.0, "KienitzLawsonSwayneSabrPdeDensity: alpha (" << alpha << ") must be positive");
    QL_REQUIRE(beta >= 0.0 && beta < 1.0, "KienitzLawsonSwayneSabrPdeDensity: beta (" << beta << ") must be in [0,1)");
    QL_REQUIRE(nu > 0.0, "KienitzLawsonSwayneSabrPdeDensity: nu (" << nu << ") must be positive");
    QL_REQUIRE(rho > -1.0 && rho < 1.0, "KienitzLawsonSwayneSabrPdeDensity: rho (" << rho << ") must be in (-1,1)");
    QL_REQUIRE(expiryTime > 0.0,
               "KienitzLawsonSwayneSabrPdeDensity: expiryTime (" << expiryTime << ") must be positve");
    QL_REQUIRE(zSteps > 1, "KienitzLawsonSwayneSabrPdeDensity: zSteps (" << zSteps << ") must be >1");
    QL_REQUIRE(tSteps > 0, "KienitzLawsonSwayneSabrPdeDensity: tSteps (" << tSteps << ") must be positive");
    QL_REQUIRE(nStdDev > 0.0, "KienitzLawsonSwayneSabrPdeDensity: nStdDev (" << nStdDev << ") must be positive");
 
    calculate();
}

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Member Function Documentation

alpha()

Real alpha

(

)

const

Definition at line 43 of file kienitzlawsonswaynesabrpdedensity.hpp.

{ return alpha_; }

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beta()

Real beta

(

)

const

Definition at line 44 of file kienitzlawsonswaynesabrpdedensity.hpp.

{ return beta_; }

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nu()

Real nu

(

)

const

Definition at line 45 of file kienitzlawsonswaynesabrpdedensity.hpp.

{ return nu_; }

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rho()

Real rho

(

)

const

Definition at line 46 of file kienitzlawsonswaynesabrpdedensity.hpp.

{ return rho_; }

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forward()

Real forward

(

)

const

Definition at line 47 of file kienitzlawsonswaynesabrpdedensity.hpp.

{ return forward_; }

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expiryTime()

Real expiryTime

(

)

const

Definition at line 48 of file kienitzlawsonswaynesabrpdedensity.hpp.

{ return expiryTime_; }

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displacement()

Real displacement

(

)

const

Definition at line 49 of file kienitzlawsonswaynesabrpdedensity.hpp.

{ return displacement_; }

zSteps()

Size zSteps

(

)

const

Definition at line 50 of file kienitzlawsonswaynesabrpdedensity.hpp.

{ return zSteps_; }

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tSteps()

Size tSteps

(

)

const

Definition at line 51 of file kienitzlawsonswaynesabrpdedensity.hpp.

{ return tSteps_; }

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nStdDev()

Real nStdDev

(

)

const

Definition at line 52 of file kienitzlawsonswaynesabrpdedensity.hpp.

{ return nStdDev_; }

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callPrices()

std::vector< Real > callPrices

(

const std::vector< Real > &

strikes

)

const

Definition at line 242 of file kienitzlawsonswaynesabrpdedensity.cpp.

                                                                                                  {
    std::vector<Real> result;
    for (auto const& strike : strikes) {
        Real zstrike = zy(yf(strike));
        if (zstrike <= zMin())
            result.push_back(forward() - strike);
        else if (zstrike >= zMax())
            result.push_back(0.0);
        else {
            Real fmax = fy(yz(zMax()));
            Real p = (fmax - strike) * pR(); // rightmost value
            // unused ???
            // Real k0 = std::ceil((zstrike - zMin()) / hh());
            // Real ftilde = fy(yz(zMin() + k0 * hh));
            // Real term1 = ftilde - strike;
            // in between points
            Real ft1;
            Size k = zSteps() - 2;
            for (; k >= 1; --k) {
                Real zm1 = zMin() + (static_cast<Real>(k) - 0.5) * hh();
                ft1 = fy(yz(zm1));
                if (ft1 > strike)
                    p += (ft1 - strike) * sabrProb()[k] * hh();
                else
                    break;
            }
            // k=k+1 ???
            // now k is the value where the payoff is zero and at k+1 the payoff is positive
            // calculating the value for subgridscale
            Real zmK = zMin() + static_cast<Real>(k) * hh();       // last admissable value ft > strike
            Real zmKm1 = zMin() + static_cast<Real>(k - 1) * hh(); // first value with ft < strike
            Real ftK = fy(yz(zmK));
            Real ftKm1 = fy(yz(zmKm1));
            Real diff = ftK - ftKm1;
            Real b = (2.0 * ft1 - ftKm1 - ftK) / diff;
            Real subgridadjust = 0.5 * hh() * sabrProb()[k] * (ftK - strike) * (ftK - strike) / diff *
                                 (1.0 + b * (ftK + 2.0 * strike - 3.0 * ftKm1) / diff);
            p += subgridadjust;
            result.push_back(p);
        }
    }
    return result;
} // callPrices

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putPrices()

std::vector< Real > putPrices

(

const std::vector< Real > &

strikes

)

const

Definition at line 286 of file kienitzlawsonswaynesabrpdedensity.cpp.

                                                                                                 {
    std::vector<Real> result;
    for (auto const& strike : strikes) {
        Real zstrike = zy(yf(strike));
        if (zstrike <= zMin())
            result.push_back(0.0);
        else if (zstrike >= zMax())
            result.push_back(strike - forward()); // wrong in VBA ???
        else {
            Real fmin = fy(yz(zMin()));
            Real p = (strike - fmin) * pL(); // leftmost value
            // in between points
            Real ft1;
            Size k = 1;
            for (; k <= zSteps() - 2; ++k) {
                Real zm1 = zMin() + (static_cast<Real>(k) - 0.5) * hh();
                ft1 = fy(yz(zm1));
                if (strike >= ft1)
                    p += (strike - ft1) * sabrProb()[k] * hh();
                else
                    break;
            }
            // ???
            if (k > 1)
                --k;
            Real zmK = zMin() + static_cast<Real>(k) * hh();       // last admissable value ft > strike
            Real zmKm1 = zMin() + static_cast<Real>(k - 1) * hh(); // first value with ft < strike
            Real ftK = fy(yz(zmK));
            Real ftKm1 = fy(yz(zmKm1));
            Real diff = ftK - ftKm1;
            Real b = (2.0 * ft1 - ftKm1 - ftK) / diff;
            Real subgridadjust = 0.5 * hh() * sabrProb()[k] * (strike - ftKm1) * (strike - ftKm1) / diff *
                                 (1.0 - b * (3.0 * ftK - 2.0 * strike - ftKm1) / diff);
            p += subgridadjust;
            result.push_back(p);
        }
    }
    return result;
} // putPrices

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inverseCumulative()

Real inverseCumulative

(

const Real

p

)

const

Definition at line 227 of file kienitzlawsonswaynesabrpdedensity.cpp.

                                                                            {
    int idx = std::upper_bound(SABR_Cum_Prob_Vec_.begin(), SABR_Cum_Prob_Vec_.end(), std::max(std::min(p, 1.0), 0.0)) -
              SABR_Cum_Prob_Vec_.begin();
    if (idx == 0)
        return fy(yz(zMin()));
    else if (idx == SABR_Cum_Prob_Vec_.size())
        return fy(yz(zMax()));
    Real zl = zMin() + static_cast<Real>(idx - 1) * hh();
    Real zr = zMin() + static_cast<Real>(idx) * hh();
    Real cl = SABR_Cum_Prob_Vec_[idx - 1];
    Real cr = SABR_Cum_Prob_Vec_[idx];
    Real alpha = (cr - p) / (cr - cl);
    return fy(yz(alpha * zl + (1.0 - alpha) * zr));
}

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sabrProb()

const Array & sabrProb

(

)

const

Definition at line 62 of file kienitzlawsonswaynesabrpdedensity.hpp.

{ return SABR_Prob_Vec_; }

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zMin()

Real zMin

(

)

const

Definition at line 63 of file kienitzlawsonswaynesabrpdedensity.hpp.

{ return zMin_; }

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zMax()

Real zMax

(

)

const

Definition at line 64 of file kienitzlawsonswaynesabrpdedensity.hpp.

{ return zMax_; }

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hh()

Real hh

(

)

const

Definition at line 65 of file kienitzlawsonswaynesabrpdedensity.hpp.

{ return hh_; }

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pL()

Real pL

(

)

const

Definition at line 66 of file kienitzlawsonswaynesabrpdedensity.hpp.

{ return pL_; }

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pR()

Real pR

(

)

const

Definition at line 67 of file kienitzlawsonswaynesabrpdedensity.hpp.

{ return pR_; }

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theta0()

Real theta0

(

)

const

Definition at line 68 of file kienitzlawsonswaynesabrpdedensity.hpp.

{ return theta0_; }

yf()

Real yf

(

const Real

f

)

const

Definition at line 46 of file kienitzlawsonswaynesabrpdedensity.cpp.

                                                             {
    Real betam = 1.0 - beta_;
    return (std::pow(f + displacement_, betam) - std::pow(forward_ + displacement_, betam)) / betam;
}

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fy()

Real fy

(

const Real

y

)

const

Definition at line 51 of file kienitzlawsonswaynesabrpdedensity.cpp.

                                                             {
    Real betam = 1.0 - beta_;
    Real par = std::pow(forward_ + displacement_, betam) + betam * y;
    if (beta_ > 0) {
        par = std::max(par, 0.0);
        return std::pow(par, 1.0 / betam) - displacement_;
    } else {
        return par - displacement_;
    }
}

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yz()

Real yz

(

const Real

z

)

const

Definition at line 62 of file kienitzlawsonswaynesabrpdedensity.cpp.

                                                             {
    return alpha_ / nu_ * (std::sinh(nu_ * z) + rho_ * (std::cosh(nu_ * z) - 1.0));
}

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zy()

Real zy

(

const Real

y

)

const

Definition at line 66 of file kienitzlawsonswaynesabrpdedensity.cpp.

                                                             {
    Real tmp = rho_ + nu_ * y / alpha_;
    return -1.0 / nu_ *
           std::log((std::sqrt(1.0 - rho_ * rho_ + tmp * tmp) - rho_ - nu_ * y / alpha_) / (1.0 - rho_)); // ???
}

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calculate()

void calculate ( )

private

Definition at line 166 of file kienitzlawsonswaynesabrpdedensity.cpp.

                                                  {
 
    Real betam = 1.0 - beta_;
    zMin_ = -nStdDev_ * std::sqrt(expiryTime_);
    zMax_ = -zMin_;
 
    if (beta_ > 0.0) {
        zMin_ = std::max(zMin_, zy(yf(-displacement_)));
    }
 
    Size j = zSteps_ - 2;
    Real h0 = (zMax_ - zMin_) / static_cast<Real>(j);
    Size j0 = static_cast<int>(-zMin_ / h0 + 0.5);
    hh_ = -zMin_ / (static_cast<Real>(j0) - 0.5);
 
    Array zm(zSteps_), ym(zSteps_), fm(zSteps_), CCm(zSteps_), Gamma_vec(zSteps_);
 
    for (Size i = 0; i < zSteps_; ++i) {
        zm[i] = static_cast<Real>(i) * hh_ + zMin_ - 0.5 * hh_;
        ym[i] = yz(zm[i]);
        fm[i] = fy(ym[i]);
    }
 
    zMax_ = static_cast<Real>(zSteps_ - 1) * hh_ + zMin_;
    Real ymax = yz(zMax_);
    Real ymin = yz(zMin_);
    Real fmax = fy(ymax);
    Real fmin = fy(ymin);
 
    fm[0] = 2 * fmin - fm[1];
    fm[zSteps_ - 1] = 2 * fmax - fm[zSteps_ - 2];
 
    for (Size i = 1; i < zSteps_ - 1; ++i) {
        CCm[i] = std::sqrt(alpha_ * alpha_ + 2.0 * rho_ * alpha_ * nu_ * ym[i] + nu_ * nu_ * ym[i] * ym[i]) *
                 std::pow(fm[i] + displacement_, beta_);
        if (i != j0)
            Gamma_vec[i] = (std::pow(fm[i] + displacement_, beta_) - std::pow(forward_ + displacement_, beta_)) /
                           (fm[i] - forward_);
    }
    CCm[0] = CCm[1];
    CCm[zSteps_ - 1] = CCm[zSteps_ - 2];
    Gamma_vec[0] = 0.0;           // need to init this ?
    Gamma_vec[zSteps_ - 1] = 0.0; // need to init this ?
 
    SABR_Prob_Vec_ = Array(zSteps_, 0.0);
 
    Gamma_vec[j0] = beta_ / std::pow(forward_ + displacement_, betam);
    SABR_Prob_Vec_[j0] = 1.0 / hh_;
 
    PDE_method(fm, CCm, Gamma_vec, expiryTime_ / static_cast<Real>(tSteps_), zSteps_);
    theta0_ = SABR_Prob_Vec_[j0];
 
    // cumulative probability
    SABR_Cum_Prob_Vec_ = Array(SABR_Prob_Vec_.size());
    SABR_Cum_Prob_Vec_[0] = pL();
    for (Size k = 1; k < SABR_Cum_Prob_Vec_.size() - 1; ++k)
        SABR_Cum_Prob_Vec_[k] = SABR_Cum_Prob_Vec_[k - 1] + SABR_Prob_Vec_[k] * hh_;
    SABR_Cum_Prob_Vec_.back() = 1.0;
 
} // calculate

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PDE_method()

void PDE_method ( const Array &  fm, const Array &  Ccm, const Array &  Gamma_Vec, const Real  dt, const Size  Nj  )

private

Definition at line 125 of file kienitzlawsonswaynesabrpdedensity.cpp.

                                                                                 {
 
    Array Emdt1_vec(Nj), Emdt2_vec(Nj), PP1(Nj), PP2(Nj), Em(Nj, 1.0), EmO(Nj);
    constexpr Real b = 1.0 - M_SQRT_2;
    Real dt1 = dt * b;
    Real dt2 = dt * (1.0 - 2.0 * b);
    for (Size i = 1; i < Nj - 1; ++i) {
        Emdt1_vec[i] = std::exp(rho_ * nu_ * alpha_ * Gamma_Vec[i] * dt1);
        Emdt2_vec[i] = std::exp(rho_ * nu_ * alpha_ * Gamma_Vec[i] * dt2);
    }
    Emdt1_vec[0] = Emdt1_vec[1];
    Emdt1_vec[Nj - 1] = Emdt1_vec[Nj - 2];
    Emdt2_vec[0] = Emdt2_vec[1];
    Emdt2_vec[Nj - 1] = Emdt2_vec[Nj - 2];
    pL_ = pR_ = 0.0;
 
    // we need two time steps
 
    Real PL1, PR1, PL2, PR2;
    for (Size it = 0; it < tSteps_; ++it) {
        for (Size j = 1; j < Nj - 1; ++j) {
            Em[j] *= Emdt1_vec[j];
        }
        solveTimeStep_LS(fm, Ccm, Em, dt1, SABR_Prob_Vec_, pL_, pR_, Nj, PP1, PL1, PR1);
        for (Size j = 1; j < Nj - 1; ++j) {
            Em[j] *= Emdt1_vec[j];
        }
        solveTimeStep_LS(fm, Ccm, Em, dt1, PP1, PL1, PR1, Nj, PP2, PL2, PR2);
        for (Size j = 1; j < Nj - 1; ++j) {
            SABR_Prob_Vec_[j] = (M_SQRT2 + 1.0) * PP2[j] - M_SQRT2 * PP1[j];
            Em[j] *= Emdt2_vec[j];
        }
        SABR_Prob_Vec_[0] = -SABR_Prob_Vec_[1];
        SABR_Prob_Vec_[Nj - 1] = -SABR_Prob_Vec_[Nj - 2];
        pL_ = (M_SQRT2 + 1.0) * PL2 - M_SQRT2 * PL1;
        pR_ = (M_SQRT2 + 1.0) * PR2 - M_SQRT2 * PR1;
    }
 
} // PDE_method

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solveTimeStep_LS()

void solveTimeStep_LS ( const Array &  fm, const Array &  cm, const Array &  Em, const Real  dt, const Array &  PP_in, const Real  PL_in, const Real  PR_in, const Size  n, Array &  PP_out, Real &  PL_out, Real &  PR_out  )

private

Definition at line 97 of file kienitzlawsonswaynesabrpdedensity.cpp.

                                                                       {
 
    Array bb(n), aa(n), cc(n);
 
    Real frac = dt / (2.0 * hh_);
 
    for (Size i = 1; i < n - 1; ++i) {
        aa[i] = -frac * cm[i - 1] * Em[i - 1] / (fm[i] - fm[i - 1]); // lower diagonal
        bb[i] = 1.0 + frac * (cm[i] * Em[i] * (1.0 / (fm[i + 1] - fm[i]) + 1.0 / (fm[i] - fm[i - 1]))); // diagonal
        cc[i] = -frac * cm[i + 1] * Em[i + 1] / (fm[i + 1] - fm[i]); // upper diagonal
    }
 
    bb[0] = cm[0] / (fm[1] - fm[0]) * Em[0];
    bb[n - 1] = cm[n - 1] / (fm[n - 1] - fm[n - 2]) * Em[n - 1];
    aa[0] = 0.0; // need to init this ?
    aa[n - 1] = cm[n - 2] / (fm[n - 1] - fm[n - 2]) * Em[n - 2];
    cc[0] = cm[1] / (fm[1] - fm[0]) * Em[1];
    cc[n - 1] = 0.0; // need to init this ?
 
    tridag(aa, bb, cc, PP_in, PP_out, n, true);
 
    PL_out = PL_in + dt * cm[1] / (fm[1] - fm[0]) * Em[1] * PP_out[1];
    PR_out = PR_in + dt * cm[n - 2] / (fm[n - 1] - fm[n - 2]) * Em[n - 2] * PP_out[n - 2];
} // solveTimeStep_LS

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tridag()

void tridag ( const Array &  a, const Array &  b, const Array &  c, const Array &  R, Array &  u, const Size  n, const bool  firstLastRZero  )

private

Definition at line 72 of file kienitzlawsonswaynesabrpdedensity.cpp.

                                                                                  {
    Array gam(n);
    if (QuantLib::close_enough(b[0], 0.0)) {
        std::fill(u.begin(), u.end(), 0.0); // ???
        return;
    }
    Real bet = b[0];
    u[0] = firstLastRZero ? 0.0 : (R[0] / bet);
    for (Size j = 1; j < n; ++j) {
        gam[j] = c[j - 1] / bet;
        bet = b[j] - a[j] * gam[j];
        if (QuantLib::close_enough(bet, 0.0)) {
            QL_FAIL("KienitzLawsonSwayneSabrPdeDensity: tridag failed");
        }
        if (j < n - 1 || !firstLastRZero)
            u[j] = (R[j] - a[j] * u[j - 1]) / bet;
        else
            u[j] = (-a[j] * u[j - 1]) / bet;
    }
    for (Size j = n - 1; j >= 1; --j) {
        u[j - 1] -= gam[j] * u[j];
    }
} // tridag

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Member Data Documentation

alpha_

const Real alpha_

private

Definition at line 85 of file kienitzlawsonswaynesabrpdedensity.hpp.

beta_

const Real beta_

private

Definition at line 85 of file kienitzlawsonswaynesabrpdedensity.hpp.

nu_

const Real nu_

private

Definition at line 85 of file kienitzlawsonswaynesabrpdedensity.hpp.

rho_

const Real rho_

private

Definition at line 85 of file kienitzlawsonswaynesabrpdedensity.hpp.

forward_

const Real forward_

private

Definition at line 85 of file kienitzlawsonswaynesabrpdedensity.hpp.

expiryTime_

const Real expiryTime_

private

Definition at line 85 of file kienitzlawsonswaynesabrpdedensity.hpp.

displacement_

const Real displacement_

private

Definition at line 85 of file kienitzlawsonswaynesabrpdedensity.hpp.

zSteps_

const Size zSteps_

private

Definition at line 86 of file kienitzlawsonswaynesabrpdedensity.hpp.

tSteps_

const Size tSteps_

private

Definition at line 86 of file kienitzlawsonswaynesabrpdedensity.hpp.

nStdDev_

const Real nStdDev_

private

Definition at line 87 of file kienitzlawsonswaynesabrpdedensity.hpp.

zMin_

Real zMin_

private

Definition at line 89 of file kienitzlawsonswaynesabrpdedensity.hpp.

zMax_

Real zMax_

private

Definition at line 89 of file kienitzlawsonswaynesabrpdedensity.hpp.

hh_

Real hh_

private

Definition at line 89 of file kienitzlawsonswaynesabrpdedensity.hpp.

pL_

Real pL_

private

Definition at line 89 of file kienitzlawsonswaynesabrpdedensity.hpp.

pR_

Real pR_

private

Definition at line 89 of file kienitzlawsonswaynesabrpdedensity.hpp.

theta0_

Real theta0_

private

Definition at line 89 of file kienitzlawsonswaynesabrpdedensity.hpp.

SABR_Prob_Vec_

Array SABR_Prob_Vec_

private

Definition at line 90 of file kienitzlawsonswaynesabrpdedensity.hpp.

SABR_Cum_Prob_Vec_

Array SABR_Cum_Prob_Vec_

private

Definition at line 90 of file kienitzlawsonswaynesabrpdedensity.hpp.