A convex parimutuel call auction implemented at a central market organizer computer includes receiving orders from market participants, calculating a quantity of accepted bids for each of the orders, and communicating to the participants the calculated quantity of accepted bids for each of the orders. Each order includes a specification by a participant of contingent claims on outcomes of a future event, a limit bid price, and a limit bid quantity. The calculation involves maximizing an objective function given by an approximate profit to the market organizer plus a weighted logarithmic penalty function. Because the formulation is convex, the solution may be computed in polynomial time using standard techniques, such as a path-following algorithm.