Functional Calculus for Pathwise Hedging

Uncertainty about the underlying probabilistic dynamics is a key problem in finance. Indeed, in this context, every model is at best a useful but rough approximation of reality. It is therefore crucial to discern which results depend delicately on the chosen model assumptions, and which ones are robust, in that they can be deduced from broad qualitative properties.

Accordingly, the analysis of model uncertainty and how to take it into account via the robust pricing and hedging of financial derivatives are key directions of current research. From a mathematical perspective, this naturally leads to deep questions about what parts of stochastic calculus can be developed in a purely pathwise manner. The present research project will make profound contributions at this intersection of stochastic analysis and its financial applications.

The key tool in this context is "functional calculus", which describes the actions of functionals on general, path-dependent random systems. In a financial context, this allows to link "superhedging strategies" (that completely hedge a given financial risk) to path-dependent optimality equations. These in turn lead to explicit solutions in some concrete examples and generally open the door to the deployment of efficient numerical methods.

The present project explores this approach in a number of practically important settings, e.g., the case where a complex financial derivative is not only hedged with the underlying asset that determines its payoff, but also by continuously readjusting a position in simpler derivatives. Such risk management strategies are routinely used in practice, but the underlying theory is not well understood - a gap in the literature that will be filled in this project using functional calculus.

In addition to these financial applications, the proposed research will also further develop the general theory of functional calculus in a number of fundamental ways, e.g., to functionals that do not evolve continuously in time.