This note is an extended version of Arai (2016), in which convex risk measures describing the upper and lower bounds of a good deal bound are studied for the case where the set of 0-attainable claims is convex as an extension of Arai and Fukasawa (2014). Here a good deal bound is defined as a subinterval of a no-arbitrage pricing bound. An outline of good deal bounds is given firstly for the readers who are not familiar with good deal bounds. In addition, many examples of convex markets are also introduced; and precise proofs for all mathematical results are provided.