The aim of this paper is to compute upper and lower bounds for convex value functions of derivative contracts. Laprise et al. (2006) compute bounds for American-style vanilla options by selected portfolios of call options. We provide an alternative interpretation of their numerical procedure as a stochastic dynamic program for which the Bellman value function is approximated by selected piecewise linear interpolations at each decision date. The stochastic dynamic program does not (directly) depend on portfolios of call options, but rather on a key ingredient: some transition parameters of the underlying asset. More in line with the literature on dynamic programming, our procedure is contract free and is well designed to accomodate all one-dimensional convex value functions of derivative contracts. In support of this, we revisit the numerical investigation of Laprise et al. (2006) and enlarge their findings to include options embedded in bonds under affine term-structure models of interest rates.