A note on market completeness with American put options

We consider a non necessarily complete financial market with one bond and one risky asset, whose price process is modeled by a suitably integrable, strictly positive, càdlàg process S on [0,T]. Every option price is defined as the conditional expectation under a given equivalent (true) martingale measure, the same for all options. We show that every positive contingent claim on S can be approximately replicated in L 2-sense by investing dynamically in the underlying and statically in all American put options (of every strike price k and with the same maturity T). We also provide a counterexample to static hedging with European call options of all strike prices and all maturities t≤T.