We introduce ambiguity sets based on the nested distance for stochastic processes. We show how these sets can be constructed as nonparametric confidence sets from statistical observations with a prescribed confidence level. For the problem of finding the acceptability price for a contingent claim under model ambiguity, we derive a duality result and show the relationship between size of the ambiguity set and the distributionally robust price. Since we consider both, the optimal ask and the optimal bid price, we can relate the bid-ask spread to the size of the ambiguity set and indirectly to the sample size of the observations. It turns out that lowering the acceptance level decreases the bid-ask spread, while increasing the model uncertainty increases this spread. We do not assume a complete market situation. We also may report on similar results for an ambiguity model for the multiperiod optimal management problem of hydrostorage electricity production.