Replicating European Payoffs and Path-Dependent Gap Risk

Summary

The document asks which underlying positions cannot be replicated using vanilla options when strikes and expirations are continuously available and liquid. It states that a non-path-dependent European payoff can be replicated with calls and puts through the Carr-Madan formula, provided the payoff is twice differentiable in the distribution sense. This is a model-independent replication result for terminal payoffs.

The response contrasts this with gap risk contracts, described as daily-restriking put spreads that pay and cancel only when a sufficiently large drop occurs relative to the previous close. Because their outcome depends on the path of daily prices and a conditional cancellation feature, the answer says Europeans cannot replicate them. The discussion is brief: it does not derive the replication formula or examine approximation, market frictions, or other path-dependent instruments.

Key ideas

• Vanilla calls and puts can replicate suitable non-path-dependent European terminal payoffs.
• The stated replication condition is twice differentiability in the distribution sense.
• Daily-restriking gap risk contracts depend on price movements across time.
• The response says European options alone cannot replicate the described path-dependent contract.

Tags

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


# 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.

## Answer by Antoine Conze (score 6, accepted)

https://quant.stackexchange.com/a/36791

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.

## Answer by James Spencer-Lavan (score 3)

https://quant.stackexchange.com/a/36790

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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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.