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Piezo-optical and elasto-optical properties of lead molibdate (PbMoO4) crystals are investigated, as determined by the quantum-mechanical ab initio calculation of the full set of elasto-optic, piezo-optic and elastic tensor components. Indicative surfaces are built, which describe the anisotropy of these stress- and strain-optical effects and allow for its rigorous analysis. Maximum values of these properties are determined as well as the geometries of acousto-optic interaction displaying the maximum efficiency. Lead molibdate crystals are found to be characterized by a large piezo-optic efficiency, the maximum change of the optical path (per unit of mechanical stress and specimen length) being calculated to 24.9 Br, which is a much larger value than previously reported for other well-known piezo-optic materials such as LiNbO3, CaWO4 and GaP. It is also shown that a significant rotation (up to several tens of degrees) of the optical indicatrix about the X3 optical axis is induced by uniaxial pressure or deformation, depending only on pressure (or deformation) direction, which has relevant implications in the design of acousto-optic cells.