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Say I have infinite precision of strikes $K$ (continuous world $dk$) and expirations $T$ (continuous $dT$) all with liquidity (so no practical limitations). What positions in an underlying can't be replicated? I'm under the impression I can replicate any European payoff, so if I had infinite expirations I could replicate really any position.

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Any non path dependent European type payoff $f(S_T)$ can be replicated in a model independent way with vanilla calls and puts provided $f$ is twice differentiable (in the distribution sense). This is a consequence of the Carr-Madan formula.

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  • $\begingroup$ I'm familiar with the replication formula... so is the distinction very clearly "path dependent payoffs cannot be replicated" with vanillas? Are there replication methods (separate from Carr-Madan) for path-dependent payoffs? $\endgroup$ – Jared Nov 8 '17 at 14:38
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    $\begingroup$ To clarify, any non path dependent European payoff $f(S_T)$ can be replicated with vanilla options in a model independent way, that is (as follows from the Carr-Madan formula) the replication weights depend only on the payoff function $f$, not on the model for the stochastic dynamics of $S_t$. For path dependent options in some cases it is possible to build static hedges but the weights depend on the model for $S_t$. See for instance static replication of barrier options and the paper by Andersen and al citeseerx.ist.psu.edu/viewdoc/… $\endgroup$ – Antoine Conze Nov 8 '17 at 15:24
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Gap risk contracts.

These are daily-restriking putspreads that pay & cancel only if the underlying drops more than (say) 20% as measured vs yesterday's closing level.

Contracts can range from as short as 6 months to 10 years.

Cannot replicate that using Europeans.

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