We present full pseudopotential band-structure calculations of the frequency dependence of the second-order susceptibility due to transitions between valence minibands in p-type asymmetric stepped-well superlattices. These theoretical predictions are supported by experimental observations of second-harmonic generation. From an analysis of the theoretical results the microscopic processes that contribute to the response are identified and the features of the response are related to the superlattice band structure over a range of temperatures. We demonstrate that simple particle-in-a-box models are unable to correctly describe the positions of the peaks, or the mechanisms involved, in the second-order susceptibility. Experimental results confirming the detection of second-harmonic generation in these structures are presented and compared with the theoretical predictions. An enhancement over the response in bulk of approximately 8 times is observed. The angular dependence observed is consistent with the directional properties of the second-order susceptibility predicted by the theoretical calculations.