For recall, assuming that European options are traded at discrete strikes:
- the portfolio of vanilla options that minimally super-replicates an option $O$ is the portfolio of options that costs least but still pays out at least as much as $O$ in all states of nature.
- the portfolio of vanilla options that maximally sub-replicates an option $O$ is the portfolio of options that costs most but still pays out no more than the option $O$ in all states of nature.
How does those two portfolios relate to hedging the option $O$? I've seen that the super-replicating portfolio is preferred if one is short the option $O$ and the sub-replicating portfolio is preferred used when one is long the option $O$.