The Bertino Family of Copulas

By Gregory A. Fredricks, Roger B. Nelsen. 2002

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Improved Fréchet Bounds and Model-Free Pricing of Multi-Asset Options (2010, methodology): “[10] computed the improved bounds when a measure of association, such as Kendall’s τ or Spearman’s ρ, is given, and the Bertino’s family of copulas [2] yields best-possible bounds when the values of the copula on the main diagonal are known.”
Best-possible bounds on sets of bivariate distribution functions (2004, background): “We prove only (b), since the proof of (a) is immediate and the proof of (c) can be found in [5], where one can also find a statistical characterization of random variables with copula Bd: For the boundary conditions (1.”
On the construction of copulas and quasi-copulas with given diagonal sections (2008, methodology): “In the following theorem we review three different ways to construct copulas and quasi-copulas with given diagonal sections, and some of their known properties (Fredricks and Nelsen, 1997, 2002; Nelsen and Fredricks, 1997; Nelsen et al., 2004; Úbeda-Flores, 2001).”
Kendall distribution functions (2003, unknown): “Let : I → I be a function such that (1) = 1, (t)6 t, and 06 (t2)− (t1)6 2(t2 − t1) for t, t1 and t2 in I with t1¡t2; and for u; v in I set B (u; v) = min(u; v)−min(s− (s)|min(u; v)6 s6max(u; v)): (3) Each B is a copula; for t in I, B (t; t) = (t); if (t) = t, B = M ; and if (t) = max(2t − 1; 0), B =”