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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 $O$.

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They idea is that these provide portfolios that you can use to statically replicate an option and have no state in the world where you lose money. From this it follows that they provide bounds on option prices.

To illustrate: suppose you bought and option and you want to hedge it statically. To do that, you would want to sell a replicating portfolio that will at most pay out what your option pays out (because if it pays out more, then you'll have a net loss). In addition, you don't want to pay for this portfolio more than you did for the option, because then there are still states of the world where you end up with a net loss. So you'd want to sell the maximally subreplicating portfolio if it's price is below that of your option (or if you're willing to lock in a loss). If your switch the argument to selling an option, you get that you need the minimal super-replicating portfolio.

From the above, it follows directly that to admit no arbitrage, an option price should lie between the maximally sub-replicating portfolio and the minimally super-replicating one.

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