Kappa logarithm: private API

kappa_log(u::Number, kappa::Number)

Evaluate the Kaniadakis logarithm.

The relativistic risk measures, their JuMP constraint layer and the entropy pooling views of the relativistic value at risk all scale a dual variable by this quantity, so the library states it once here. The function checks neither argument, because every caller holds kappa in a field that its constructor has already passed to assert_unit_interval, and every caller passes the reciprocal of a tail mass, which is positive.

Mathematical definition

\[\begin{align} \ln_{\kappa}(u) &= \dfrac{u^{\kappa} - u^{-\kappa}}{2 \kappa}\,. \end{align}\]

Where:

• $\ln_{\kappa}(u) = \dfrac{u^{\kappa} - u^{-\kappa}}{2 \kappa}$: Kaniadakis logarithm.
• $u > 0$: Argument of the logarithm.
• $\kappa \in (0, 1)$: Deformation parameter.

The value is negative for every $u < 1$, and $\ln_{\kappa}(u) \to \ln(u)$ as $\kappa \to 0$.

Arguments

• u: Argument of the logarithm. Must be positive.
• kappa: Relativistic deformation parameter. The bound is Num_DefCal, so the slot takes the parameter itself, an AbstractDeformationCalibrationAlgorithm that computes it from the prior result, or a plain function of the same five arguments. Must lie in $(0, 1)$.

Returns

• lnk::Number: Value of the Kaniadakis logarithm.

Examples

julia> PortfolioOptimisers.kappa_log(2, 0.3)0.6981533616478014julia> PortfolioOptimisers.kappa_log(0.5, 0.3)-0.6981533616478014

Related

• RRM: every relativistic risk measure reaches the logarithm through it.
• RelativisticValueatRisk
• ConicRelativisticValueatRiskView
• GridRelativisticValueatRiskView
• assert_unit_interval: the check that every caller's constructor runs on kappa.

References

• [105] D. Cajas. Portfolio Optimization of Relativistic Value at Risk. Available at SSRN 4378498 (2023).