We establish the existence of initial data for Einstein's constraint equations that describe the exterior of a black hole. The black hole itself does not belong to the 3-dimensional initial data manifold, since the physical fields diverge there. The initial data manifold has a compact set removed, hence containing an inner boundary. We use the well-known Lichnerowicz-York conformal rescaling method to transform the Einstein constraint equations into a boundary value problem for a non-linear elliptic system in a scalar and a vector-valued variables. Boundary conditions are chosen at the inner boundary of the initial data manifold having two main properties: First, the coupled non-linear elliptic system has a solution; second, the inner boundary is a trapped surface. The latter condition implies that the solution of this elliptic system describes the exterior of a black hole. A trapped surface is a 2-dimensional subset of a 4-dimensional spacetime with the property that light leaving the surface in any direction always contracts, never expands. It has been shown that such unusual surface must be located in a neighborhood of a black hole. Although a trapped surface is defined in a 4-dimensional spacetime, it can be characterized only in terms of initial data fields in a 3-dimensional manifold. This last property makes trapped surfaces a useful 3-dimensional characterization of black hole initial data.