© 2017 Elsevier B.V. This paper provides new identification results for the bid–ask spread and the nonparametric distribution of the latent fundamental price increments (ε t ) from the observed transaction prices alone. The results are established via the characteristic function approach, and hence allow for discrete or continuous ε t and the observed price increments do not need to have any finite moments. Constructive identification (and overidentification) results are established first in the basic Roll (1984) model, and then in various extended Roll models, including general unbalanced order flow, serially dependent latent trade direction indicators, adverse selection, random spread and a multivariate Roll model.