Abstract
We unify and establish equivalence between the pathwise and the quasi-sure approaches to robust modelling of financial markets in finite discrete time. In particular, we prove a fundamental theorem of asset pricing and a superhedging theorem which encompass the formulations of Bouchard and Nutz [12] and Burzoni et al. [13]. In bringing the two streams of literature together, we examine and compare their many different notions of arbitrage. We also clarify the relation between robust and classical ℙ-specific results. Furthermore, we prove when a superhedging property with respect to the set of martingale measures supported on a set Omega of paths may be extended to a pathwise superhedging on Omega without changing the superhedging price.
Highlights
Mathematical models of financial markets are of great significance in economics and finance and have played a key role in the theory of pricing and hedging derivativesWe gratefully acknowledge funding received from the European Research Council under the European Union’s Seventh Framework Programme (FP7/2007-2013)/ERC grant agreement no. 335421) and from St
We show that under mild technical assumptions, the pathwise and quasi-sure fundamental theorems of asset pricing and superhedging dualities can be inferred from one another and are equivalent
One of the principal aims of this paper is to show that both model approaches are equivalent in terms of corresponding FTAPs and superhedging prices
Summary
We gratefully acknowledge funding received from the European Research Council under the European Union’s Seventh Framework Programme (FP7/2007-2013)/ERC grant agreement no. 335421) and from St. The quasi-sure approach introduces a set P of priors representing possible market scenarios These priors can be very different, and P typically contains measures which are mutually singular. The quasi-sure and the pathwise, allow to interpolate between the two ends of the modelling spectrum, as identified by Merton [30]: the model-independent and the model-specific settings (see Fig. 1) In doing so, they allow capturing how their outputs change in terms of adding or removing modelling assumptions, allowing us to quantify the impact and risk that a given set of assumptions bear on the problem at hand; see Cont [19].
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