We present a search for optical bursts from the repeating fast radio burst FRB 121102 using simultaneous observations with the high-speed optical camera ULTRASPEC on the 2.4-m Thai National Telescope and radio observations with the 100-m Effelsberg Radio Telescope. A total of 13 radio bursts were detected, but we found no evidence for corresponding optical bursts in our 70.7-ms frames. The 5σ upper limit to the optical flux density during our observations is 0.33 mJy at 767 nm. This gives an upper limit for the optical burst fluence of 0.046 Jy ms, which constrains the broad-band spectral index of the burst emission to α ≤ -0.2. Two of the radio pulses are separated by just 34 ms, which may represent an upper limit on a possible underlying periodicity (a rotation period typical of pulsars), or these pulses may have come from a single emission window that is a small fraction of a possible period.