We examine the light curves of two quasars, motivated by recent suggestions that a supermassive black hole binary (SMBHB) can exhibit sharp lensing spikes. We model the variability of each light curve as due to a combination of two relativistic effects: the orbital relativistic Doppler boost and gravitational binary self-lensing. In order to model each system, we extend previous Doppler plus self-lensing models to include eccentricity. The first quasar is identified in optical data as a binary candidate with a 20-yr period (Ark 120), and shows a prominent spike. For this source, we rule out the lensing hypothesis and disfavour the Doppler-boost hypothesis due to discrepancies in the measured versus recovered values of the binary mass and optical spectral slope. The second source, which we nickname Spikey, is the rare case of an active galactic nucleus identified in Kepler's high-quality, high-cadence photometric data. For this source, we find a model, consisting of a combination of Doppler modulation and a narrow symmetric lensing spike, consistent with an eccentric SMBHB with a mass of $M_{\text{tot}} = 3\times 10^{7} {\, \mathrm{M}_{\odot }}$ , rest- frame orbital period T = 418 d, eccentricity e = 0.5, and seen at an inclination of 8○ from edge-on. This interpretation can be tested by monitoring Spikey for periodic behaviour and recurring flares in the next few years. In preparation for such monitoring, we present the first X-ray observations of this object taken by the Neil Gehrels Swift Observatory.
