Before revision: The Carr-Madan replication formula ensures that knowledge of the prices of options at every strike is equivalent to knowing the entire pricing distribution . In many situations, the available market data is insufficient to determine this distribution precisely,  and the question arises: what are the bounds for the option price at a specified strike, given the market-implied constraints?
After revision: The Carr-Madan replication formula ensures that knowledge of the prices of options at every strike is equivalent to knowing the entire pricing distribution for a security, as derivatives contingent on the security can be replicated using options. The available data may be insufficient to determine this distribution precisely, and the question arises: what are the bounds for the option price at a specified strike, given the market-implied constraints?