Class PlackettCopula

Namespace

NanoQL.Numerics

Assembly

NanoQL.dll

The Plackett copula, C(x, y) = (s − √(s² − 4xyθ(θ − 1)))/(2(θ − 1)), s = 1 + (θ − 1)(x + y).

public sealed class PlackettCopula

Inheritance

object

PlackettCopula

Inherited Members

object.GetType()

object.ToString()

object.Equals(object)

object.Equals(object, object)

object.ReferenceEquals(object, object)

object.GetHashCode()

Remarks

Immutable. On the unit square's edges it returns the boundary values, C(u, 0) = C(0, v) = 0, C(u, 1) = u and C(1, v) = v, before its formula, as every copula here does. QuantLib's formula cancels wherever the root nears s: near the edges at every θ (1e-7 relative at θ 0.5 and 2 on points 1e-8 from them) and everywhere as θ nears 1 (9.5e-5). nanoql takes the stable root, 2xyθ/(s + √…) where s is positive, QuantLib's form elsewhere, and above θ 1 the radicand as 1 + 2(θ − 1)(x(1 − y) + y(1 − x)) + ((θ − 1)(x − y))², whose terms are all non-negative, where QuantLib's cancels as θ grows (1e-8 at θ 1e8): within 4e-16 relative of a high-precision evaluation of QuantLib's formula on points within 1e-12 of the edges, from θ 0.1 to 1e12; toward θ 0 the copula nears max(x + y − 1, 0), whose cancellation near x + y = 1 no form avoids.

Constructors

PlackettCopula(double)

Initializes a new instance of the PlackettCopula class.

public PlackettCopula(double theta)

Parameters

theta double

The parameter θ, at least 0, not 1.

Exceptions

ArgumentException

theta is outside its range.

Methods

Evaluate(double, double)

The copula's value.

public double Evaluate(double x, double y)

Parameters

x double

The first argument, in [0, 1].

y double

The second argument, in [0, 1].

Returns

double

C(x, y).

Exceptions

ArgumentException

x or y is outside [0, 1].