We present a high-performance laser frequency stabilization method using modulation transfer spectroscopy (MTS) on the rubidium 87D2 transition line. A substantial improvement of the laser frequency stability was achieved by searching for the optimal diameter and intensity settings of the probe and pump beam. The frequency instability measured from the beat frequency of two locked external cavity diode lasers (ECDLs) reached a short-term stability of 4.5×10-14/τ and did not exceed 2 × 10-12 until 105 s, which is the best performance reported thus far with a D2 transition. The long-term stability is limited by the offset fluctuations of the baseline induced by the residual amplitude modulation (RAM), which can be further improved by reducing the current temperature variation of about 0.2 K by means of temperature stabilization or through a further reduction of the RAM.