SAX J2103.5+4545 is the Be/X-ray binary (BeX) with the shortest orbital period. It shows extended bright and faint X-ray states that last for a few hundred days. The main objective of this work is to investigate the relationship between the X-ray and optical variability and to characterize the spectral and timing properties of the bright and faint states. We have found a correlation between the spectral and temporal parameters that fit the energy and power spectra. Softer energy spectra correspond to softer power spectra. That is to say, when the energy spectrum is soft, the power at high frequencies is suppressed. We also present the results of our monitoring of the Hα line of the optical counterpart since its discovery in 2003. There is a correlation between the strength and shape of the Hα line, originated in the circumstellar envelope of the massive companion and the X-ray emission from the vicinity of the neutron star. Hα emission, indicative of an equatorial disc around the B-type star, is detected whenever the source is bright in X-rays. When the disc is absent, the X-ray emission decreases significantly. The long-term variability of SAX J2103.5+4545 is characterized by fast episodes of disc loss and subsequent reformation. The time-scales for the loss and reformation of the disc (about 2 yr) are the fastest among BeXs.