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 modelled by a suitably integrable, strictly positive, càdlàg process S over [0, T ]. Every option price is defined as the conditional expectation under a given equivalent (true) martingale measure P, the same for all options. We show that every positive contingent claim on S can be approximately replicated (in L-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 counter-example to static hedging with European call options of all strike prices and all maturities t ≤ T .