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For a single asset we can infer (in theory) the exact distribution of future outcomes via option prices: Either via option butterflies or the Breeden-Litzenberger formula.

Is there a name for (one of) the classes of multi-options that determine (i.e. can be used to compute) the joint distribution of several assets?

For two assets $X,Y$, for example, "pair-options" with payoff of the form $$\mathbf 1\{X\geq K_1\}(Y-K_2)^+,$$ with $K_1,K_2\in\mathbb R$ and the roles of $X,Y$ switched, should be one such family. This is because it allows us to construct butterflies for (and hence get the distribution of) $Y$, conditional on $X\geq K_1$, etc.

(Is there a publicly available historical dataset consisting of "pair-options" or suitable alternatives as above?)

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