Tradable mobility credit (TMC) schemes are an approach to travel demand management that have received significant attention in the transportation domain in recent years as a promising means to mitigate the adverse environmental, economic and social effects of urban traffic congestion. In TMC schemes, a regulator provides an initial endowment of mobility credits (or tokens) to all potential travelers. In order to use the transportation system, travelers need to spend a certain amount of tokens (tariff) that could vary with their choice of mode, route, departure time etc. The tokens can be bought and sold in a market that is managed by and operated by a regulator at a price that is dynamically determined by the demand and supply of tokens. This paper proposes and analyzes alternative market models for a TMC system (focusing on market design aspects such as allocation/expiration of credits, rules governing trading, transaction costs, regulator intervention, price dynamics), and develops a methodology to explicitly model the disaggregate behavior of individuals within the market. Extensive simulation experiments are conducted within a departure time context for the morning commute problem to compare the performance of the alternative designs relative to congestion pricing and a no control scenario. The simulation experiments employ a day to day assignment framework wherein transportation demand is modeled using a logit-mixture model and supply is modeled using a standard bottleneck model. The paper addresses a growing and imminent need to develop methodologies to realistically model TMCs that are suited for real-world deployments and can help us better understand the performance of these systems and the impact in particular, of market dynamics.