On robust pricing-hedging duality in continuous time

TL;DR

This paper develops a unified robust framework for pricing and hedging in continuous-time finance, incorporating model uncertainty and beliefs, and extends duality results to multi-dimensional, multi-maturity settings.

Contribution

It introduces a general duality framework that interpolates between model-independent and model-specific approaches, extending martingale optimal transport to multiple dimensions and maturities.

Findings

Established a duality between superhedging prices and calibrated martingale measures.

Extended martingale optimal transport duality to multi-dimensional, multi-maturity cases.

Included the impact of beliefs and model uncertainty in pricing-hedging duality.

Abstract

We pursue robust approach to pricing and hedging in mathematical finance. We consider a continuous time setting in which some underlying assets and options, with continuous paths, are available for dynamic trading and a further set of European options, possibly with varying maturities, is available for static trading. Motivated by the notion of prediction set in Mykland (2003), 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 or gaining information. We obtain a general pricing-hedging duality result: the infimum over superhedging prices is equal to supremum over calibrated martingale measures. In presence of non-trivial beliefs, the equality is between limiting values of perturbed problems. In particular, our results include the martingale optimal transport duality of Dolinsky and Soner (2013) and extend it to multiple dimensions and multiple maturities.