  We show that when the price process $S$ represents a fully incomplete market,
the optimal super-replication of any Markovian claim $g(S_T)$ with $g(\cdot)$
being nonnegative and lower semicontinuous is of buy-and-hold type. Since both
(unbounded) stochastic volatility models and rough volatility models are
examples of fully incomplete markets, one can interpret the buy-and-hold
property when super-replicating Markovian claims as a natural phenomenon in
incomplete markets.
