We present time-resolved Gran Telescopio Canarias optical spectroscopy and William Herschel Telescope i-band photometry of the X-ray transient SWIFT J1753.5–0127 in quiescence. The i-band light curve is dominated by flickering with an amplitude of ∼0.5 mag and shows no evidence of the ellipsoidal modulation of the companion star. The telluric-corrected average spectrum, on the other hand, reveals the presence of weak (strongly veiled) TiO bands at 7055 Å and 7589 Å. We used them for a spectral classification, finding an M4-5 V companion star. However, as velocity shifts are not clearly detected in the individual spectra, we turned the analysis to the double-peaked Hα emission line from the accretion disc. By exploiting the empirical correlations established for quiescent X-ray transients between the line morphology and fundamental binary parameters, we estimated the radial velocity semi-amplitude of the companion K2 = 820 ± 36 km s‑1, a mass ratio q = 0.023 ± 0.006 and an inclination i = 79 ± 5 deg. Moreover, an orbital period of 3.26 ± 0.02 h was measured from the modulation of the centroid velocities and the double-peak trough depth of the Hα profile. These quantities yielded a mass function f(M1) = 7.8 ± 1.0 M⊙ and black hole and companion star masses of M1 = 8.8 ± 1.3 M⊙ and M2 = 0.20 ± 0.06 M⊙, respectively. The companion star mass is in line with the spectral classification obtained from the relative depth of the TiO bands. Based on the mean quiescent magnitude (i = 21.4 ± 0.1), orbital period, and interstellar extinction, we estimate the distance to the source to be 3.9 ± 0.7 kpc and a Galactic plane elevation of 0.8 ± 0.2 kpc, supporting the case for a large natal kick.