This paper studies international stochastic discount factors (SDFs) under different degrees of market segmentation in incomplete markets. In complete markets, the real exchange rate is equal to the ratio of foreign and domestic SDFs. Incomplete markets give rise to a stochastic wedge and SDFs are not necessarily invariant to the numeraire. We first posit model-free SDFs that are (i) numeraire invariant and (ii) can be decomposed into permanent and transitory components. We then derive novel measures of similarity of SDFs to quantify the amount of risk-sharing in international financial markets. In the data, we find that large permanent components are necessary to address well-known exchange rate puzzles, however, at the cost of highly volatile SDFs. Market segmentation in equity and/or bond markets helps avoid implausibly large SDF dispersions. Moreover, we find the ensuing SDFs are also well aligned with SDF similarity recovered from FX option data. Hence, economies featuring some form of mild market segmentation and large martingale components can match salient features of exchange rates, bond, equity, and FX option markets.