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Knock-out options start as plain vanilla options and are 'knocked-out', ie cease to exist, when the barrier is hit. Knock-in options start lifeless and 'kick-in', ie come into play as plain vanilla options when the barrier is hit. Therefore statement I is not correct.

Barrier options are certainly cheaper than equivalent vanilla options because vanilla options have a larger range of prices over which they pay out. Therefore statement II is correct.

Statement III is not correct. Average price options, also called Asian options, are less attractive to a buyer of options and therefore they are cheaper and not more expensive than vanilla options. This is because average prices are less volatile, and also when compared to a strip of equivalent vanilla options, some of the individual vanilla options in the strip may pay out whereas on the average nothing may pay out.

Digital options have a very high gamma close to the strike price as the option payout becomes uncertain and fluctuates sharply between 0 and 1. Therefore statement IV is correct.