In this paper we show that many problems of multivariable stability margin optimization can be reduced to a spectral tangential interpolation problem of Nevanlinna-Pick type, where the interpolating transfer matrices are bounded not in infinity norm but in spectral radius. This equivalence is established via conformal mappings. Necessary and sufficient conditions for the existence of solutions to the spectral interpolation problem are given in terms of the positivity of an associated Nevanlinna-Pick matrix. We conclude by applying this approach to solve a multivariable gain margin problem.