Multi-period static hedging of European options

TL;DR

This paper extends static hedging techniques for European options to multiple short maturities within a Markovian framework, using Gaussian Quadrature for practical implementation and comparing Black-Scholes and Merton models.

Contribution

It generalizes Carr and Wu's approach to include multiple short-term options and demonstrates a practical finite-set hedging method with numerical experiments.

Findings

Effective multi-maturity hedging strategy developed

Gaussian Quadrature accurately estimates hedging error

Comparative analysis of Black-Scholes and Merton models

Abstract

We consider the hedging of European options when the price of the underlying asset follows a single-factor Markovian framework. By working in such a setting, Carr and Wu \cite{carr2014static} derived a spanning relation between a given option and a continuum of shorter-term options written on the same asset. In this paper, we have extended their approach to simultaneously include options over multiple short maturities. We then show a practical implementation of this with a finite set of shorter-term options to determine the hedging error using a Gaussian Quadrature method. We perform a wide range of experiments for both the \textit{Black-Scholes} and \textit{Merton Jump Diffusion} models, illustrating the comparative performance of the two methods.