# What Positions on an Underlier CANNOT be Hedged with Vanillas?

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.

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.
• 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/… – Antoine Conze Nov 8 '17 at 15:24