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We prove existence, uniqueness and optimal regularity of solutions to thestationary obstacle problem defined by the fractional Laplacian operator withdrift, in the subcritical regime. We localize our problem by considering asuitable extension operator introduced by L. Caffarelli and L. Silvestre. Thestructure of the extension equation is different from the one considered by L.Caffarelli, S. Salsa and L. Silvestre in their study of the obstacle problemfor the fractional Laplacian without drift, in that the obstacle function hasless regularity, and exhibits some singularities. To take into account the newfeatures of the problem, we prove a new Almgren-type monotonicity formula,which we then use to establish the optimal regularity of solutions.

Optimal regularity of solutions to the obstacle problem for the fractional Laplacian with drift