We present here an atomistic theory of the electronic and optical properties of hexagonal InAsP quantum dots in InP nanowires in the wurtzite phase. These self-assembled quantum dots are unique in that their heights, shapes, and diameters are well known. Using a combined valence-force-field, tight-binding, and configuration-interaction approach we perform atomistic calculations of single-particle states and excitonic, biexcitonic and trion complexes as well as emission spectra as a function of the quantum dot height, diameter and As versus P concentration. The atomistic tight-binding parameters for InAs and InP in the wurtzite crystal phase were obtained by ab initio methods corrected by empirical band gaps. The low energy electron and hole states form electronic shells similar to parabolic or cylindrical quantum confinement, only weakly affected by hexagonal symmetry and As fluctuations. The relative alignment of the emission lines from excitons, trions and biexcitons agrees with that for InAs/InP dots in the zincblende phase in that biexcitons and positive trions are only weakly bound. The random distribution of As atoms leads to dot-to-dot fluctuations of a few meV for the single-particle states and the spectral lines. Due to the high symmetry of hexagonal InAsP nanowire quantum dots the exciton fine structure splitting is found to be small, of the order a few $mu$eV with significant random fluctuations in accordance with experiments.
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