/// <summary>
/// Calculation of
/// b = Φ(h+t)·exp(h·t) - Φ(h-t)·exp(-h·t)
/// exp(-(h²+t²)/2)
/// = --------------- · [ Y(h+t) - Y(h-t) ]
/// √(2π)
/// with
/// Y(z) := Φ(z)/φ(z)
/// using an expansion of Y(h+t)-Y(h-t) for small t to twelvth order in t.
/// Theoretically accurate to (better than) precision ε = 2.23E-16 when h<=0 and t < τ with τ :=
/// 2·ε^(1/16) ≈ 0.21.
/// The main bottleneck for precision is the coefficient a:=1+h·Y(h) when |h|>1 .
/// Smalltexpansions the of normalised black call.
/// Y(h) := Φ(h)/φ(h) = √(π/2)·erfcx(-h/√2)
/// a := 1+h·Y(h) --- Note that due to h<0, and h·Y(h) -> -1 (from above) as h -> -∞, we also have that a>
/// 0 and a -> 0 as h -> -∞
/// w := t² , h2 := h²
/// </summary>
/// <param name="h">The h.</param>
/// <param name="t">The t.</param>
/// <returns></returns>
private double SmalltexpansionOfNormalisedBlackCall(double h, double t)
{
var a = 1 + h * (0.5 * SqrtTwoPi) * Erfc.ErfcxCody(-OneOverSqrtTwo * h);
var w = t * t;
var h2 = h * h;
var expansion = 2 * t * (a + w * ((-1 + 3 * a + a * h2) / 6 + w *
((-7 + 15 * a + h2 * (-1 + 10 * a + a * h2)) / 120 + w *
((-57 + 105 * a +
h2 * (-18 + 105 * a + h2 * (-1 + 21 * a + a * h2))) / 5040 +
w * ((-561 + 945 * a +
h2 * (-285 + 1260 * a +
h2 * (-33 + 378 * a + h2 * (-1 + 36 * a + a * h2)))
) / 362880 + w *
((-6555 + 10395 * a +
h2 * (-4680 + 17325 * a +
h2 * (-840 + 6930 * a +
h2 * (-52 + 990 * a +
h2 * (-1 + 55 * a + a * h2))))) /
39916800 + (-89055 + 135135 * a +
h2 * (-82845 + 270270 * a +
h2 * (-20370 + 135135 * a +
h2 * (-1926 + 25740 * a +
h2 * (-75 + 2145 * a +
h2 * (-1 + 78 * a +
a * h2)))))) *
w / 6227020800.0))))));
var b = OneOverSqrtTwoPi * Math.Exp(-0.5 * (h * h + t * t)) * expansion;
return(Math.Max(b, 0.0));
}
//
// Introduced on 2017-02-18
//
// b(x,s) = Φ(x/s+s/2)·exp(x/2) - Φ(x/s-s/2)·exp(-x/2)
// = Φ(h+t)·exp(x/2) - Φ(h-t)·exp(-x/2)
// = ½ · exp(-u²-v²) · [ erfcx(u-v) - erfcx(u+v) ]
// = ½ · [ exp(x/2)·erfc(u-v) - exp(-x/2)·erfc(u+v) ]
// = ½ · [ exp(x/2)·erfc(u-v) - exp(-u²-v²)·erfcx(u+v) ]
// = ½ · [ exp(-u²-v²)·erfcx(u-v) - exp(-x/2)·erfc(u+v) ]
// with
// h = x/s , t = s/2 ,
// and
// u = -h/√2 and v = t/√2 .
//
// Cody's erfc() and erfcx() functions each, for some values of their argument, involve the evaluation
// of the exponential function exp(). The normalised Black function requires additional evaluation(s)
// of the exponential function irrespective of which of the above formulations is used. However, the total
// number of exponential function evaluations can be minimised by a judicious choice of one of the above
// formulations depending on the input values and the branch logic in Cody's erfc() and erfcx().
//
private double NormalisedBlackCallWithOptimalUseOfCodysFunctions(double x, double s)
{
const double codysThreshold = 0.46875;
var h = x / s;
var t = 0.5 * s;
var q1 = -OneOverSqrtTwo * (h + t);
var q2 = -OneOverSqrtTwo * (h - t);
double twoB;
if (q1 < codysThreshold)
{
if (q2 < codysThreshold)
{
twoB = Math.Exp(0.5 * x) * Erfc.ErfcCody(q1) -
Math.Exp(-0.5 * x) * Erfc.ErfcCody(q2);
}
else
{
twoB = Math.Exp(0.5 * x) * Erfc.ErfcCody(q1) -
Math.Exp(-0.5 * (h * h + t * t)) * Erfc.ErfcCody(q2);
}
}
else
{
if (q2 < codysThreshold)
{
twoB = Math.Exp(-0.5 * (h * h + t * t)) * Erfc.ErfcxCody(q1) -
Math.Exp(-0.5 * x) * Erfc.ErfcCody(q2);
}
else
{
twoB = Math.Exp(-0.5 * (h * h + t * t)) *
(Erfc.ErfcxCody(q1) - Erfc.ErfcxCody(q2));
}
}
return(Math.Max(0.5 * twoB, 0.0));
}
