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The variability data for the BL Lacertae object TXS 1902+556 in the optical and |$\gamma$| -ray wavebands were obtained from the 0.76-m Katzman Automatic Imaging Telescope and the Fermi Large Area Telescope (Fermi -LAT), covering periods of 14.4 and 14.7 yr, respectively. The variability properties were systematically analysed, with particular emphasis on the first comprehensive investigation of radiation variation in the optical waveband. Four well-established techniques were employed for this purpose: the Lomb–Scargle periodogram, REDFIT program, Jurkevich method, and discrete correlation function (DCF) approach. The optical waveband exhibits quasi-periodic oscillations (QPO) with a time-scale of |$P_{\rm O}=276.8\pm 6.1$|  d at a significance level |$3.87\sigma$|⁠ , while the |$\gamma$| -ray waveband does not exhibit any significant periodicity. However, it should be noted that the QPO time-scale is consistent with the Sun-gaps in the optical light curve within 2 |$\sigma$| uncertainties. The optical QPO behaviour is most likely attributed to the helical motion of the jet driven by the orbital motion in a supermassive black hole binary system. Moreover, we have provided an explanation for the absence of QPO in the |$\gamma$| -ray light curves. Furthermore, utilizing the DCF method, a weak correlation between the variability in the optical and |$\gamma$| -ray wavebands was observed, suggesting that the emission of TXS 1902+556 may be generated through a combination of synchrotron self-Compton (SSC) and external Compton (EC) processes, or a leptonic–hadronic hybrid process. [ABSTRACT FROM AUTHOR]