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)));
}
public static double beta_black_strike_to_quick_delta(double beta, double F, double T, double sigma, double strike)
{
var qd = SpecialFunction.cdf_normal((inv_g(beta, F) - inv_g(beta, strike)) / (sigma * Math.Sqrt(T)));
if (qd <= -0.5)
{
return(1.0 + qd);
}
if (qd > 0.5)
{
return(qd - 1.0);
}
return(qd);
}
public static double black76(OptionType optType, double F, double K, double sdev)
{
var num = optType.ToEpsilon();
if (sdev < 0.0)
{
throw FS.FailWith($"sdev must be positive. Got {sdev}");
}
if (sdev == 0.0 || K <= 0.0)
{
return(Math.Max(num * (F - K), 0.0));
}
else
{
var num2 = (Math.Log(F / K) + 0.5 * sdev * sdev) / sdev;
var num3 = num2 - sdev;
return(num * (F * SpecialFunction.cdf_normal(num * num2) -
K * SpecialFunction.cdf_normal(num * num3)));
}
}
public static double m1_beta_black(double beta, double mu, double sigma)
{
var betaMinusOne = beta - 1.0;
var betaMinusOneSq = betaMinusOne * betaMinusOne;
var sigma2 = sigma * sigma;
var sqrt2 = Math.Sqrt(2.0);
var sqrtPi = Math.Sqrt(Math.PI);
var t = -mu / sigma;
var t2 = t * t;
var one_over_two_betaMinusOneSq = 0.5 / betaMinusOneSq;
var e4 = Math.Exp(-0.5 * t2);
var a1 = t + sigma / betaMinusOne;
var group1 = -SpecialFunction.exp_x_N_y(-beta * mu / betaMinusOneSq + (sigma2 + 2.0 * mu) * one_over_two_betaMinusOneSq, a1);
var a2 = sigma - t;
var group2 = SpecialFunction.exp_x_N_y(mu + 0.5 * sigma2, a2);
var f0 = -1.0 / (2.0 * sqrtPi);
return(f0 * sqrt2 * e4 * beta * sigma
- betaMinusOneSq * group1
+ group2
- beta * (mu - beta + 2) * (SpecialFunction.cdf_normal(-t) - 1.0));
}
public static double beta_black(double beta, OptionType optType, double F, double K, double T, double sigma)
{
if (T < 0.0)
{
throw FS.FailWith($"T < 0 not allowed, {T}");
}
if (sigma < 0.0)
{
throw FS.FailWith($"sigma < 0 not allowed {sigma}");
}
var stdev = sigma * Math.Sqrt(T);
if (stdev == 0.0)
{
return(Math.Max(optType.ToEpsilon() * (F - K), 0.0));
}
var mu = inv_m1_beta_black(beta, F, stdev);
var d = (inv_g(beta, K) - mu) / stdev;
var t1 = K * SpecialFunction.cdf_normal(d);
var t2 = beta_black_trunc(beta, mu, stdev, d);
var put = t1 - t2;
var intrinsic = Math.Max(optType.ToEpsilon() * (F - K), 0.0);
if (optType == OptionType.Put)
{
return(Math.Max(put, intrinsic));
}
else if (optType == OptionType.Call)
{
return(Math.Max(put + F - K, intrinsic));
}
else
{
throw FS.E_CASE(optType);
}
}
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));
}
}
public static double beta_black_cdf(double beta, double mu, double sigma, double x)
{
return(SpecialFunction.cdf_normal((inv_g(beta, x) - mu) / sigma));
}
