Pointwise Arbitrage Pricing Theory in Discrete Time

We pursue robust approach to pricing and hedging in mathematical
finance. We develop a general discrete time setting in which some
underlying assets and options are available for dynamic trading and a
further set of European options, possibly with varying maturities, is
available for static trading. We include in our setup modelling beliefs by
allowing to specify a set of paths to be considered, e.g.
super-replication of a contingent claim is required only for paths falling
in the given set. Our framework thus interpolates between
model-independent and model-specific settings and allows to quantify the
impact of making assumptions. We establish suitable FTAP and
Pricing-Hedging duality results which include as special cases previous
results of Acciaio et al. (2013), Burzoni et al. (2016) as well the
Dalang-Morton-Willinger theorem. Finally, we explain how to treat further
problems, such as insider trading (information quantification) or American
options pricing.
Based on joint works with Burzoni, Frittelli, Hou, Maggis; Aksamit, Deng and Tan.