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We derive upper and lower bounds on the expectation of $f(\mathbf{S})$ underdependence uncertainty, i.e. when the marginal distributions of the randomvector $\mathbf{S}=(S_1,\dots,S_d)$ are known but their dependence structure ispartially unknown. We solve the problem by providing improved \FH bounds on thecopula of $\mathbf{S}$ that account for additional information. In particular,we derive bounds when the values of the copula are given on a compact subset of$[0,1]^d$, the value of a functional of the copula is prescribed or differenttypes of information are available on the lower dimensional marginals of thecopula. We then show that, in contrast to the two-dimensional case, the boundsare quasi-copulas but fail to be copulas if $d>2$. Thus, in order to translatethe improved \FH bounds into bounds on the expectation of $f(\mathbf{S})$, wedevelop an alternative representation of multivariate integrals with respect tocopulas that admits also quasi-copulas as integrators. By means of thisrepresentation, we provide an integral characterization of orthant orders onthe set of quasi-copulas which relates the improved \FH bounds to bounds on theexpectation of $f(\mathbf{S})$. Finally, we apply these results to computemodel-free bounds on the prices of multi-asset options that take partialinformation on the dependence structure into account, such as correlations ormarket prices of other traded derivatives. The numerical results show that theadditional information leads to a significant improvement of the option pricebounds compared to the situation where only the marginal distributions areknown.

Improved Fr\'echet$-$Hoeffding bounds on $d$-copulas and applications in model-free finance