| Summary: | <p>This thesis explores the question of model-free trading and hedging in markets where traded asset prices are continuous and where one may trade continuously in time with no transaction costs. In particular, we make no assumptions on the volatility of traded asset prices. The contributions of the thesis are as follows. First, we propose a framework of model-independent replication of financial derivatives based on solutions to systems of PDEs evaluated at market-observed inputs. This provides a model-independent extension of the paradigm of dynamic hedging to general markets with continuous prices. We then relate these replication strategies to local martingales of a certain closed form and characterise the latter for several specifications of markets. The markets we consider are: (1) a market with no traded claims, (2) a market with an underlying asset and a convex claim, (3) a market with an underlying asset and a set of co-maturing call options. The auxiliary results for the latter two markets may be of interest outside of the local martingale characterisation results. Thirdly, we propose a definition of integration with continuous paths that justifies a probability-free version of the hedging results outlined earlier. Finally, we present a number of smaller contributions related to model-free hedging and to probability-free integration with continuous paths.</p>
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