Asphalt, a soft matter consisting of more than a thousand chemical species, is of vital importance for the transportation infrastructure, yet it poses significant challenges for microscopic theory and modeling approaches due to its multicomponent nature. Polymeric additives can potentially enhance the thermo-mechanical properties of asphalt, thus helping reduce the road repair costs; rational design of such systems requires knowledge of the molecular structure and dynamics of these systems. We have used molecular dynamics (MD) simulations to investigate the volumetric, structural, and dynamic properties of the neat asphalt as well as styrene-butadiene rubber (SBR) modified asphalt systems. The volume-temperature behavior of the asphalt systems exhibited a glass transition phenomenon, akin to that observed in experiments. The glass transition temperature, room temperature density, and coefficient of volume thermal expansion of the neat asphalt systems so evaluated were in agreement with experimental data when the effect of the high cooling rate used in simulations was accounted for. While the volumetric properties of SBR modified asphalt were found to be insensitive to the presence of the SBR additive, the addition of SBR led to an increase in the aggregation of asphaltene molecules. Furthermore, addition of SBR caused a reduction in the mobility of the constituent molecules of asphalt, with the reduction being more significant for the larger constituent molecules. Similar to other glass forming liquids, the reciprocal of the diffusion coefficient of the selected molecules was observed to follow the Vogel-Fulcher-Tammann (VFT) behavior as a function of temperature. These results suggest the potential for using polymeric additives for enhancing the dynamic mechanical properties of asphalt without affecting its volumetric properties.