Hedging strategies and minimal variance portfolios for European and exotic options in a Levy market

TL;DR

This paper develops and analyzes hedging strategies for European and exotic options in Levy markets, utilizing Taylor's Theorem and minimal variance portfolios to achieve effective risk management.

Contribution

It introduces a novel approach to hedge higher moments in Levy markets using Taylor expansion and minimal variance portfolios, including practical algorithms and performance analysis.

Findings

Perfect hedging of European options is achievable.

Higher moment hedging can be done with minimal variance portfolios.

Numerical results demonstrate practical effectiveness of the strategies.

Abstract

This paper presents hedging strategies for European and exotic options in a Levy market. By applying Taylor's Theorem, dynamic hedging portfolios are con- structed under different market assumptions, such as the existence of power jump assets or moment swaps. In the case of European options or baskets of European options, static hedging is implemented. It is shown that perfect hedging can be achieved. Delta and gamma hedging strategies are extended to higher moment hedging by investing in other traded derivatives depending on the same underlying asset. This development is of practical importance as such other derivatives might be readily available. Moment swaps or power jump assets are not typically liquidly traded. It is shown how minimal variance portfolios can be used to hedge the higher order terms in a Taylor expansion of the pricing function, investing only in a risk-free bank account, the underlying asset and potentially variance swaps. The numerical algorithms and performance of the hedging strategies are presented, showing the practical utility of the derived results.