Hedging under arbitrage

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.

Minor changes, accepted for publication in Journal of Mathematical Finance

References in corpus (3)

• Analysis of continuous strict local martingales via h-transforms
• On optimal arbitrage
• Bubbles, convexity and the Black--Scholes equation

Cited by in corpus (11)

• Strict local martingales and bubbles
• Weak and strong no-arbitrage conditions for continuous financial markets
• Supermartingales as Radon-Nikodym densities and related measure extensions
• Pathwise stochastic integrals for model free finance
• Optimal arbitrage under model uncertainty
• Local times for typical price paths and pathwise Tanaka formulas
• A new proof for the conditions of Novikov and Kazamaki
• The fundamental theorem of asset pricing, the hedging problem and maximal claims in financial markets with short sales prohibitions
• Negative Call Prices
• Viscosity Characterization of the Explosion Time Distribution for Diffusions
• Exploiting arbitrage requires short selling