  We propose a distributional formulation of the spanning problem of a
multi-asset payoff by vanilla basket options. This problem is shown to have a
unique solution if and only if the payoff function is even and absolutely
homogeneous, and we establish a Fourier-based formula to calculate the
solution. Financial payoffs are typically piecewise linear, resulting in a
solution that may be derived explicitly, yet may also be hard to numerically
exploit. One-hidden-layer feedforward neural networks instead provide a natural
and efficient numerical alternative for discrete spanning. We test this
approach for a selection of archetypal payoffs and obtain better hedging
results with vanilla basket options compared to industry-favored approaches
based on single-asset vanilla hedges.
