public static double m1_beta_black_b(double beta, double mu, double sigma, ref double beta_b, ref double mu_b, ref double sigma_b, double res_b)
{
var betaMinusOne = beta - 1.0;
var betaMinusOneSq = betaMinusOne * betaMinusOne;
var sigma2 = sigma * sigma;
var sqrt2 = Math.Sqrt(2.0);
var sqrtPi = Math.Sqrt(Math.PI);
var t = -mu / sigma;
var t2 = t * t;
var one_over_two_betaMinusOneSq = 0.5 / betaMinusOneSq;
var e4 = Math.Exp(-0.5 * t2);
var a1 = t + sigma / betaMinusOne;
var b1 = -beta * mu / betaMinusOneSq + (sigma2 + 2.0 * mu) * one_over_two_betaMinusOneSq;
var dmgroup1_db1 = 0.0;
var dmgroup1_da1 = 0.0;
var group1 = -SpecialFunction.exp_x_N_y_b(b1, a1, ref dmgroup1_db1, ref dmgroup1_da1, 1.0);
var a2 = sigma - t;
var muOneHalfS2 = mu + 0.5 * sigma2;
var dgroup2_dmuOneHalfS2 = 0.0;
var dgroup2_da2 = 0.0;
var group2 = SpecialFunction.exp_x_N_y_b(muOneHalfS2, a2, ref dgroup2_dmuOneHalfS2, ref dgroup2_da2, 1.0);
var f0 = -1.0 / (2.0 * sqrtPi);
var dnmt_dmt = 0.0;
var nmt = SpecialFunction.cdf_normal_b(-t, ref dnmt_dmt, 1.0);
var res = f0 * sqrt2 * e4 * beta * sigma
- betaMinusOneSq * group1
+ group2
- beta * (mu - beta + 2) * (nmt - 1.0);
var e4_b = res_b * f0 * sqrt2 * beta * sigma;
beta_b += res_b * f0 * sqrt2 * e4 * sigma;
beta_b += -res_b * (mu - 2.0 * beta + 2.0) * (nmt - 1.0);
sigma_b += res_b * f0 * sqrt2 * e4 * beta;
var betaMinusOneSq_b = -res_b * group1;
var group1_b = -res_b * betaMinusOneSq;
var group2_b = res_b;
mu_b += -res_b * beta * (nmt - 1.0);
var nmt_b = -res_b * beta * (mu - beta + 2);
var t_b = -nmt_b * dnmt_dmt;
var a2_b = group2_b * dgroup2_da2;
var muOneHalfS2_b = group2_b * dgroup2_dmuOneHalfS2;
mu_b += muOneHalfS2_b;
sigma_b += muOneHalfS2_b * sigma;
sigma_b += a2_b;
t_b += -a2_b;
var b1_b = -group1_b * dmgroup1_db1;
var a1_b = -group1_b * dmgroup1_da1;
beta_b += -b1_b * mu / betaMinusOneSq;
mu_b += b1_b * (-beta / betaMinusOneSq + 2.0 * one_over_two_betaMinusOneSq);
sigma_b += b1_b * 2.0 * sigma * one_over_two_betaMinusOneSq;
betaMinusOneSq_b += b1_b * beta * mu / betaMinusOneSq / betaMinusOneSq;
var one_over_two_betaMinusOneSq_b = b1_b * (sigma2 + 2.0 * mu);
t_b += a1_b;
var betaMinusOne_b = -a1_b * sigma / betaMinusOne / betaMinusOne;
sigma_b += a1_b / betaMinusOne;
var t2_b = -0.5 * e4_b * e4;
betaMinusOneSq_b += -one_over_two_betaMinusOneSq_b * 0.5 / betaMinusOneSq / betaMinusOneSq;
t_b += t2_b * 2.0 * t;
mu_b += -t_b / sigma;
sigma_b += t_b * mu / sigma / sigma;
betaMinusOne_b += betaMinusOneSq_b * 2.0 * betaMinusOne;
beta_b += betaMinusOne_b;
return(res);
}
public static double beta_black_trunc_b(double beta, double mu, double sigma, double x, ref double beta_b, ref double mu_b, ref double sigma_b, ref double x_b, double res_b)
{
var betaMinusOne = beta - 1.0;
var betaMinusOneSq = betaMinusOne * betaMinusOne;
var sigma2 = sigma * sigma;
var sqrt2 = Math.Sqrt(2.0);
var sqrtPi = Math.Sqrt(Math.PI);
var t = -mu / sigma;
var one_over_two_betaMinusOneSq = 0.5 / betaMinusOneSq;
var f0 = -1.0 / (2.0 * sqrtPi);
var res = 0.0;
var betaMinusOne_b = 0.0;
var betaMinusOneSq_b = 0.0;
var one_over_two_betaMinusOneSq_b = 0.0;
var x2_b = 0.0;
var t_b = 0.0;
if (x < t)
{
var x2 = x * x;
var ar2 = -0.5 * x2;
var e2 = Math.Exp(ar2);
var a4 = x + sigma / betaMinusOne;
var ar4 = -beta * mu / betaMinusOneSq + (sigma2 + 2.0 * mu) * one_over_two_betaMinusOneSq;
var dgroup4_dar4 = 0.0;
var dgroup4_da4 = 0.0;
var group4 = SpecialFunction.exp_x_N_y_b(ar4, a4, ref dgroup4_dar4, ref dgroup4_da4, 1.0);
var dNx_dx = 0.0;
var Nx = SpecialFunction.cdf_normal_b(x, ref dNx_dx, 1.0);
res = f0 * sqrt2 * e2 * beta * sigma
+ betaMinusOneSq * group4
+ beta * Nx * (mu - beta + 2.0);
var e2_b = res_b * f0 * sqrt2 * beta * sigma;
beta_b += res_b * (f0 * sqrt2 * e2 * sigma + Nx * (mu - 2.0 * beta + 2.0));
sigma_b += res_b * f0 * sqrt2 * e2 * beta;
betaMinusOneSq_b += res_b * group4;
var group4_b = res_b * betaMinusOneSq;
var Nx_b = res_b * beta * (mu - beta + 2.0);
mu_b += res_b * beta * Nx;
x_b += Nx_b * dNx_dx;
var ar4_b = group4_b * dgroup4_dar4;
var a4_b = group4_b * dgroup4_da4;
beta_b += ar4_b * (-mu / betaMinusOneSq);
mu_b += ar4_b * (-beta / betaMinusOneSq + 2.0 * one_over_two_betaMinusOneSq);
betaMinusOneSq_b += ar4_b * (beta * mu / betaMinusOneSq / betaMinusOneSq);
sigma_b += ar4_b * 2.0 * sigma * one_over_two_betaMinusOneSq;
one_over_two_betaMinusOneSq_b += ar4_b * (sigma2 + 2.0 * mu);
x_b += a4_b;
sigma_b += a4_b / betaMinusOne;
betaMinusOne_b += a4_b * (-sigma / betaMinusOne / betaMinusOne);
var ar2_b = e2_b * e2;
x2_b += -0.5 * ar2_b;
x_b += x2_b * 2.0 * x;
}
else
{
var t2 = t * t;
var ar4 = -0.5 * t2;
var e4 = Math.Exp(ar4);
var a1 = t + sigma / betaMinusOne;
var ar1 = -beta * mu / betaMinusOneSq + (sigma2 + 2.0 * mu) * one_over_two_betaMinusOneSq;
var dmgroup1_dar1 = 0.0;
var dmgroup1_da1 = 0.0;
var group1 = -SpecialFunction.exp_x_N_y_b(ar1, a1, ref dmgroup1_dar1, ref dmgroup1_da1, 1.0);
var a2 = t - sigma;
var a3 = x - sigma;
var ar2 = mu + 0.5 * sigma2;
var dmgroup2_dar2 = 0.0;
var dmgroup2_da2 = 0.0;
var group2 = -SpecialFunction.exp_x_N_y_b(ar2, a2, ref dmgroup2_dar2, ref dmgroup2_da2, 1.0);
var dgroup3_dar2 = 0.0;
var dgroup3_da3 = 0.0;
var group3 = SpecialFunction.exp_x_N_y_b(ar2, a3, ref dgroup3_dar2, ref dgroup3_da3, 1.0);
var dnmt_dmt = 0.0;
var nmt = SpecialFunction.cdf_normal_b(-t, ref dnmt_dmt, 1.0);
res = f0 * sqrt2 * e4 * beta * sigma
- betaMinusOneSq * group1
+ group2 + group3
- beta * (mu - beta + 2) * (nmt - 1.0);
var e4_b = res_b * f0 * sqrt2 * beta * sigma;
beta_b += res_b * (f0 * sqrt2 * e4 * sigma - (mu - 2.0 * beta + 2.0) * (nmt - 1.0));
sigma_b += res_b * f0 * sqrt2 * e4 * beta;
betaMinusOneSq_b += -res_b * group1;
var group1_b = -res_b * betaMinusOneSq;
var group2_b = res_b;
var group3_b = res_b;
mu_b += -res_b * beta * (nmt - 1.0);
var nmt_b = res_b * -beta * (mu - beta + 2);
t_b += -nmt_b * dnmt_dmt;
var ar2_b = group3_b * dgroup3_dar2;
var a3_b = group3_b * dgroup3_da3;
var a2_b = -group2_b * dmgroup2_da2;
ar2_b += -group2_b * dmgroup2_dar2;
mu_b += ar2_b;
sigma_b += ar2_b * sigma;
x_b += a3_b;
sigma_b += -a3_b;
sigma_b += -a2_b;
t_b += a2_b;
var ar1_b = -group1_b * dmgroup1_dar1;
var a1_b = -group1_b * dmgroup1_da1;
beta_b += ar1_b * (-mu / betaMinusOneSq);
mu_b += ar1_b * (-beta / betaMinusOneSq + 2.0 * one_over_two_betaMinusOneSq);
betaMinusOneSq_b += ar1_b * (beta * mu / betaMinusOneSq / betaMinusOneSq);
sigma_b += ar1_b * 2.0 * sigma * one_over_two_betaMinusOneSq;
one_over_two_betaMinusOneSq_b += ar1_b * (sigma2 + 2.0 * mu);
t_b += a1_b;
sigma_b += a1_b / betaMinusOne;
betaMinusOne_b += -a1_b * sigma / betaMinusOne / betaMinusOne;
var ar4_b = e4_b * e4;
var t2_b = -0.5 * ar4_b;
t_b += t2_b * 2.0 * t;
}
betaMinusOneSq_b += -one_over_two_betaMinusOneSq_b * 0.5 / betaMinusOneSq / betaMinusOneSq;
mu_b += -t_b / sigma;
sigma_b += t_b * mu / sigma / sigma;
betaMinusOne_b += betaMinusOneSq_b * 2.0 * betaMinusOne;
beta_b += betaMinusOne_b;
return(res);
}
