  We consider the pricing of derivatives in a setting with trading
restrictions, but without any probabilistic assumptions on the underlying
model, in discrete and continuous time. In particular, we assume that European
put or call options are traded at certain maturities, and the forward price
implied by these option prices may be strictly decreasing in time. In discrete
time, when call options are traded, the short-selling restrictions ensure no
arbitrage, and we show that classical duality holds between the smallest
super-replication price and the supremum over expectations of the payoff over
all supermartingale measures. More surprisingly in the case where the only
vanilla options are put options, we show that there is a duality gap. Embedding
the discrete time model into a continuous time setup, we make a connection with
(strict) local-martingale models, and derive framework and results often seen
in the literature on financial bubbles. This connection suggests a certain
natural interpretation of many existing results in the literature on financial
bubbles.
