The Mid-infrared Imager, Spectrometer Coronagraph (MISC) instrument studied for the Origins Space Telescope (OST) Mission Concept 1 is designed to observe at mid-infrared (MIR) wavelengths ranging from 5 to 38 microns for OST. In the OST Mission Concept 1 study, MISC consists of three separate optical modules providing imaging, spectroscopy, and coronagraph capabilities. The MISC Coronagraph module (MISC COR) employs Phase-Induced Amplitude Apodization (PIAA) coronagraph (Guyon et al. 2014) in which pupil apodization is modified by reflection on mirrors and central starlight is blocked by focal plane mask and Lyot mask. The performance target of MISC COR is to achieve 10-7 contrast at 0.5" from the central star with covering wavelength of 6-38 microns using 2 optical channels. MISC COR will be a powerful tool to bring a revolutionary progress in exoplanet sciences. In this paper, we present detailed design of its optics and optomechanics, and discuss expected performances for a variety of combination of focal plane mask and Lyot mask.