Functional Expansions

TL;DR

This paper explores various functional expansions, including static and dynamic types, to better understand path-dependent problems across disciplines, with applications in finance for pricing and hedging exotic options.

Contribution

It introduces the intrinsic value expansion (IVE) and revisits the functional Taylor expansion (FTE), connecting functional calculus with practical financial applications.

Findings

FTE effectively separates functional effects from future trajectories.

Extensions of analyticity concepts to path space are developed.

Financial applications demonstrate the utility of FTE in pricing and hedging.

Abstract

Path dependence is omnipresent in many disciplines such as engineering, system theory and finance. It reflects the influence of the past on the future, often expressed through functionals. However, non-Markovian problems are often infinite-dimensional, thus challenging from a conceptual and computational perspective. In this work, we shed light on expansions of functionals. First, we treat static expansions made around paths of fixed length and propose a generalization of the Wiener series$-$the intrinsic value expansion (IVE). In the dynamic case, we revisit the functional Taylor expansion (FTE). The latter connects the functional It\^o calculus with the signature to quantify the effect in a functional when a "perturbation" path is concatenated with the source path. In particular, the FTE elegantly separates the functional from future trajectories. The notions of real analyticity and radius of convergence are also extended to the path space. We discuss other dynamic expansions arising from Hilbert projections and the Wiener chaos, and finally show financial applications of the FTE to the pricing and hedging of exotic contingent claims.