We pursue the robust approach to pricing and hedging in which no probability measure is fixed but call or put options with different maturities and strikes can be traded initially at their market prices. We allow for inclusion of robust modelling assumptions by specifying the set of feasible paths. In a discrete time setup with no short selling, we characterise absence of arbitrage and show that if call options are traded then the usual pricing-hedging duality is preserved. In contrast, if only put options are traded a duality gap may appear. Embedding the results into a continuous time framework, we show that the duality gap may be interpreted as a financial bubble and link it to strict local martingales. This provides an intrinsic justification of strict local martingales as models for financial bubbles arising from a combination of trading restrictions and current market prices.