On a Heath-Jarrow-Morton approach for stock options

TL;DR

This paper extends the Heath-Jarrow-Morton framework to model stock options across all strikes and maturities using time-inhomogeneous Lévy processes, providing conditions for arbitrage-free models and their construction.

Contribution

It introduces a novel HJM-inspired approach for stock options using Lévy processes, differing from previous local volatility models, and establishes existence and uniqueness of arbitrage-free models.

Findings

Derived necessary and sufficient conditions for no arbitrage.

Proved existence and uniqueness of arbitrage-free models.

Provided a framework for constructing such models.

Abstract

This paper aims at transferring the philosophy behind Heath-Jarrow-Morton to the modelling of call options with all strikes and maturities. Contrary to the approach by Carmona and Nadtochiy (2009) and related to the recent contribution Carmona and Nadtochiy (2012) by the same authors, the key parametrisation of our approach involves time-inhomogeneous L\'evy processes instead of local volatility models. We provide necessary and sufficient conditions for absence of arbitrage. Moreover we discuss the construction of arbitrage-free models. Specifically, we prove their existence and uniqueness given basic building blocks.