Hedging under arbitrage

TL;DR

This paper demonstrates that delta hedging remains optimal for replicating payoffs in markets lacking an equivalent local martingale measure, by constructing a new probability measure and establishing conditions for hedge existence.

Contribution

It introduces a new probability measure to extend delta hedging applicability in non-martingale market models with market bubbles.

Findings

Delta hedging minimizes initial capital in non-martingale markets.

A new probability measure replaces the equivalent local martingale measure.

Conditions for differentiability of expectations ensure hedge existence.

Abstract

It is shown that delta hedging provides the optimal trading strategy in terms of minimal required initial capital to replicate a given terminal payoff in a continuous-time Markovian context. This holds true in market models where no equivalent local martingale measure exists but only a square-integrable market price of risk. A new probability measure is constructed, which takes the place of an equivalent local martingale measure. In order to ensure the existence of the delta hedge, sufficient conditions are derived for the necessary differentiability of expectations indexed over the initial market configuration. The recently often discussed phenomenon of "bubbles" is a special case of the setting in this paper. Several examples at the end illustrate the techniques described in this work.