Hearing aids are expected to improve speech intelligibility for listeners with hearing impairment. An appropriate amplification fitting tuned for the listener's hearing disability is critical for good performance. The developments of most prescriptive fittings are based on data collected in subjective listening experiments, which are usually expensive and time-consuming. In this paper, we explore an alternative approach to finding the optimal fitting by introducing a hearing aid speech processing framework, in which the fitting is optimised in an automated way using an intelligibility objective function based on the HASPI physiological auditory model. The framework is fully differentiable, thus can employ the back-propagation algorithm for efficient, data-driven optimisation. Our initial objective experiments show promising results for noise-free speech amplification, where the automatically optimised processors outperform one of the well recognised hearing aid prescriptions.