Class ClaytonCopula

Namespace

NanoQL.Numerics

Assembly

NanoQL.dll

The Clayton copula, C(x, y) = max(x^−θ + y^−θ − 1, 0)^(−1/θ).

public sealed class ClaytonCopula

Inheritance

object

ClaytonCopula

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.

• Below θ 0 the base is taken at 0 before the power, the copula's definition, where QuantLib takes the maximum after it, so a negative base gives NaN or, at an even exponent, a positive value: at θ −0.5, C(0.2, 0.2) 0 for QuantLib's 0.0111.
• Near independence, |θ| below 0.1, the base is computed as (x^−θ − 1) + (y^−θ − 1) by expm1 and the power by log1p, where QuantLib's cancels (4e-8 relative at θ 1e-8, on points within 1e-8 of the edges); above θ 0.1 QuantLib's formula, but where x^−θ or y^−θ would overflow (θ·(−ln min(x, y)) above 709, where QuantLib's gives 0): there m(1 + (m/M)^θ − m^θ)^(−1/θ), with m and M the smaller and larger argument. Each form is within 5e-15 relative of a 50-digit evaluation of QuantLib's formula on points within 1e-12 of the edges; at θ −1 the copula is max(x + y − 1, 0), whose cancellation near x + y = 1 no form avoids.

Constructors

ClaytonCopula(double)

Initializes a new instance of the ClaytonCopula class.

public ClaytonCopula(double theta)

Parameters

theta double

The parameter θ, at least −1, not 0.

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].