Perturbation analysis of sub/super hedging problems

TL;DR

This paper explores the theoretical foundations of no-arbitrage conditions and duality in hedging problems, providing a perturbation analysis that reveals how smile extrapolation affects exotic option bounds.

Contribution

It establishes the Fundamental Theorem of Asset Pricing in a general setting and analyzes the stability of hedging bounds under perturbations, including numerical insights.

Findings

Duality is preserved when reducing infinite-dimensional problems to finite dimensions.

Perturbation analysis reveals the impact of smile extrapolation on option bounds.

Theoretical links between no-arbitrage conditions and pricing functionals are clarified.

Abstract

We investigate the links between various no-arbitrage conditions and the existence of pricing functionals in general markets, and prove the Fundamental Theorem of Asset Pricing therein. No-arbitrage conditions, either in this abstract setting or in the case of a market consisting of European Call options, give rise to duality properties of infinite-dimensional sub- and super-hedging problems. With a view towards applications, we show how duality is preserved when reducing these problems over finite-dimensional bases. We finally perform a rigorous perturbation analysis of those linear programming problems, and highlight numerically the influence of smile extrapolation on the bounds of exotic options.