public static double beta_black_pdf(double beta, double mu, double sigma, double x)
{
var gm1 = inv_g(beta, x);
var den = dg(beta, gm1);
return(SpecialFunction.pdf_normal((gm1 - mu) / sigma) / den / sigma);
}
public static double ibachelier(OptionType optType, double F, double K, double T, double premium, bool safe = false)
{
if (T <= 0.0)
{
throw FS.FailWith($"T must be postive, got {T}");
}
var epsilon = optType.ToEpsilon();
var intrinsic = Math.Max(epsilon * (F - K), 0.0);
if (premium <= intrinsic)
{
if (DoubleUtils.ApproximatelyEqual(premium, intrinsic, DoubleUtils.DoubleEpsilon))
{
return(0.0);
}
if (safe)
{
return(0.0);
}
throw FS.FailWith($"premium must be larger than intrinsic. Got premium = {premium}, vs intrinsic = {intrinsic}"); //!
}
if (F == K)
{
return(premium * Math.Sqrt(2.0 * Math.PI / T));
}
var sqrtT = Math.Sqrt(T);
var phiStar = -Math.Abs((premium - intrinsic) / (K - F));
var xBar = 0.0;
if (phiStar < -0.001882039271)
{
var g = 1.0 / (phiStar - 0.5);
var g2 = g * g;
var zeta = (0.032114372355 - g2 * (0.016969777977 - g2 * (2.6207332461e-3 - 9.6066952861e-5 * g2)))
/ (1.0 - g2 * (0.6635646938 - g2 * (0.14528712196 - 0.010472855461 * g2)));
xBar = g * (1.0 / Math.Sqrt(2.0 * Math.PI) + zeta * g2);
}
else
{
var h = Math.Sqrt(-Math.Log(-phiStar));
xBar = (9.4883409779 - h * (9.6320903635 - h * (0.58556997323 + 2.1464093351 * h)))
/ (1 - h * (0.65174820867 + h * (1.5120247828 + 6.6437847132e-5 * h)));
}
var xBar2 = xBar * xBar;
var nxBar = SpecialFunction.pdf_normal(xBar);
var phiTildaXBar = SpecialFunction.cdf_normal(xBar) + nxBar / xBar;
var q = (phiTildaXBar - phiStar) / nxBar;
var xStar = xBar + (3.0 * q * xBar2 * (2.0 - q * xBar * (2.0 + xBar2)))
/ (6.0 + q * xBar * (-12.0 + xBar * (6.0 * q + xBar * (-6.0 + q * xBar * (3.0 + xBar2)))));
return(Math.Abs((K - F) / (xStar * sqrtT)));
}
