Compared to the predictions of complete market models, actual exchange rates are puzzlingly smooth and only weakly correlated with macro-economic fundamentals, suggesting that market incompleteness plays a key role in exchange rate dynamics. Incompleteness in international financial markets introduces a stochastic wedge between the growth rates of marginal utility at home and abroad, and the change in the exchange rate. We derive a preference-free upper bound on the effects of the FX wedges. Even if domestic agents can invest only in the foreign risk-free asset, incomplete spanning fails to simultaneously match the exchange rate volatility, cyclicality and the FX risk premia in the data.