Robust pricing and hedging under trading restrictions and the emergence of local martingale models

Abstract

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 the inclusion of robust modelling assumptions by specifying a set of feasible paths on which (super)hedging arguments are required to work. 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.