Bounding a 33-strike put from a 35-strike put

A stock trades at $32$, and a European put struck at $35$ costs $4.5$. Using only this information, give the best estimate (the tightest super-replication bound) for the price of a put on the same stock and expiry struck at $33$, to two decimals.

About $4.24$. For a non-negative stock, $\tfrac{33}{35}(35-S_T)^+=(33-\tfrac{33}{35}S_T)^+\ge(33-S_T)^+$, so $\tfrac{33}{35}$ of a $35$ put super-replicates the $33$ put. That bounds its price by $\tfrac{33}{35}\times4.5\approx4.24$, tighter than the naive bound $4.5$.