public static double m1_beta_black(double beta, double mu, double sigma)
{
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 group1 = -SpecialFunction.exp_x_N_y(-beta * mu / betaMinusOneSq + (sigma2 + 2.0 * mu) * one_over_two_betaMinusOneSq, a1);
var a2 = sigma - t;
var group2 = SpecialFunction.exp_x_N_y(mu + 0.5 * sigma2, a2);
var f0 = -1.0 / (2.0 * sqrtPi);
return(f0 * sqrt2 * e4 * beta * sigma
- betaMinusOneSq * group1
+ group2
- beta * (mu - beta + 2) * (SpecialFunction.cdf_normal(-t) - 1.0));
}
public static double beta_black_trunc(double beta, double mu, double sigma, double x)
{
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);
if (x < t)
{
var x2 = x * x;
var e2 = Math.Exp(-0.5 * x2);
var a4 = x + sigma / betaMinusOne;
var group4 = SpecialFunction.exp_x_N_y(-beta * mu / betaMinusOneSq + (sigma2 + 2.0 * mu) * one_over_two_betaMinusOneSq, a4);
return(f0 * sqrt2 * e2 * beta * sigma +
+betaMinusOneSq * group4
+ beta * SpecialFunction.cdf_normal(x) * (mu - beta + 2.0));
}
else
{
var t2 = t * t;
var e4 = Math.Exp(-0.5 * t2);
var a1 = t + sigma / betaMinusOne;
var group1 = -SpecialFunction.exp_x_N_y(-beta * mu / betaMinusOneSq + (sigma2 + 2.0 * mu) * one_over_two_betaMinusOneSq, a1);
var a2 = sigma - t;
var a3 = x - sigma;
var group2 = -SpecialFunction.exp_x_N_y(mu + 0.5 * sigma2, -a2);
var group3 = SpecialFunction.exp_x_N_y(mu + 0.5 * sigma2, a3);
return(f0 * sqrt2 * e4 * beta * sigma
- betaMinusOneSq * group1
+ (group2 + group3)
- beta * (mu - beta + 2) * (SpecialFunction.cdf_normal(-t) - 1.0));
}
}
