Abstract Market-makers serve an important role as providers of liquidity and order in financial markets, particularly during periods of high volatility. Optimal market-makers solve a sequential decision making problem, where they face an exploration versus exploitation dilemma at each time step. A belief state MDP based solution was presented by Das and Magdon-Ismail [2008]. This solution however, was closely tied to the choice of a Gaussian belief state prior and did not take asset inventory into consideration when calculating an optimal policy. In this work we introduce a novel continuous state POMDP framework which is the first to solve, exactly and in closed-form, the optimal market-making problem with inventory, fixed asset value, arbitrary belief state priors, trader models and reward functions via symbolic dynamic programming. We use this novel model and solution to show that sequentially optimal policies are heavily inventory-dependent and calculate policies that operate with bounded loss guarantees under a variety of market models and conditions.