Using Super- and Sub-Replicating Portfolios to Bound Option Prices
Summary
The document explains how static portfolios of vanilla options can bound the value and payoff of another option when European options are available at discrete strikes. A minimal super-replicating portfolio costs as little as possible while paying at least as much as the target in every state. A maximal sub-replicating portfolio costs as much as possible while never paying more than the target.
For a long position in the target option, selling a sub-replicating portfolio can provide a hedge whose payout does not exceed the option’s payoff; for a short position, a super-replicating portfolio can cover the obligation in every state. The cost and payoff comparisons matter together, since a hedge that costs more than the option premium can still leave a loss. These constructions imply no-arbitrage price bounds between the sub- and super-replicating portfolio costs. The explanation is conceptual and does not give a numerical example or a procedure for finding the portfolios.
Key ideas
- A minimal super-replicating portfolio costs least while matching or exceeding the target option payoff in every state.
- A maximal sub-replicating portfolio costs most while staying at or below the target payoff in every state.
- A long option position can be hedged with a sub-replicating portfolio when its cost and payout fit the intended hedge.
- A short option position can be covered with a super-replicating portfolio.
- The two portfolio costs provide lower and upper no-arbitrage bounds for the option price.
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Full text
# Super-replicating and sub-replicating portfolios and hedging # Super-replicating and sub-replicating portfolios and hedging 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 $O$. ## Answer by Bram (score 1) https://quant.stackexchange.com/a/46334 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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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.