Superhedging and Dynamic Risk Measures under Volatility Uncertainty

TL;DR

This paper develops a framework for superhedging and dynamic risk measures under volatility uncertainty, introducing a nonlinear martingale approach and characterizing it via second order backward SDEs.

Contribution

It introduces a novel nonlinear martingale representation for superhedging under volatility uncertainty and characterizes it through second order backward SDEs.

Findings

Derived a c d ag nonlinear martingale as superhedging value process

Proved an optional sampling theorem for the nonlinear martingale

Characterized the nonlinear martingale as a solution to a second order backward SDE

Abstract

We consider dynamic sublinear expectations (i.e., time-consistent coherent risk measures) whose scenario sets consist of singular measures corresponding to a general form of volatility uncertainty. We derive a c\`adl\`ag nonlinear martingale which is also the value process of a superhedging problem. The superhedging strategy is obtained from a representation similar to the optional decomposition. Furthermore, we prove an optional sampling theorem for the nonlinear martingale and characterize it as the solution of a second order backward SDE. The uniqueness of dynamic extensions of static sublinear expectations is also studied.