Electricity prices determined from economic dispatch without considering fixed costs may cause high uplift payments. With fixed costs, however, a price is not a monotonic function of demand, affecting market transparency. To overcome these, convex hull (CH) pricing has recently been introduced for unit commitment with fixed costs. Several CH pricing methods were presented, and a feasible cost was used to quantify the CH price quality. The associated difficulties are 1. high computational effort required to obtain a feasible cost and 2. the associated duality gap may not be tight enough to provide accurate measure. In this paper, a novel measure to quantify the quality of CH prices is presented by establishing an upper bound to the optimal dual value approaching it from above. Near-optimal CH prices are efficiently obtained by using Surrogate Lagrangian Relaxation (SLR), meanwhile, the upper bound decreases fast due to convergence of SLR. Testing results on the IEEE 118-bus system without transmission capacities indicate that the novel quality measure reaches a value of less than 0.1% in seconds – much more accurate and faster than the measure provided by a feasible cost – demonstrating the high quality of the upper bound and the efficiency of SLR.