Full-year sabbatical is the norm, but the home institution does not necessarily offer salary for that full year, so it's up to the professor to find funding for the rest of it from the host or some grant. According to their web site[1], MIT pays salary for half of the sabbatical year, leaving Lloyd to raise the other half himself. 125k sounds to me like a believable half-year salary for a full professor at MIT.

1: https://vpf.mit.edu/300-sabbaticals