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Improved Fréchet-Hoeffding bounds on \(d\)-copulas and applications in model-free finance. (English) Zbl 1390.62091

The paper deals with the partial point-wise ordering of certain families of Lipschitz functions defined of the unit cube \([0,1]^d\), \(d \geq 2\), and taking values in the unit interval \([0,1]\), namely \(d\)-dimensional copulas and their generalizations quasi-copulas. The goal is to determine the point-wise bounds for constrained sets of such functions.
Between others the authors provided the formulas for the upper and lower bounds for sets of quasi-copulas and copulas with fixed values on some closed subset of the unit cube. As a possible application of the obtained results they derive the bounds on prices of certain European style exotic options written on several basic assets, under the dependence uncertainty.

MSC:

62H05 Characterization and structure theory for multivariate probability distributions; copulas
60E15 Inequalities; stochastic orderings
62P05 Applications of statistics to actuarial sciences and financial mathematics
91G20 Derivative securities (option pricing, hedging, etc.)
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References:

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