Clark-Ocone formula for the maximum of processes with the stochastic intensity and its application

TL;DR

This paper generalizes the Clark-Ocone formula for the maximum of certain Lévy processes with stochastic intensity, facilitating the pricing of lookback options driven by complex stochastic models.

Contribution

It introduces a new explicit representation of the Clark-Ocone formula for maxima of Cox and Hawkes processes with stochastic intensities, extending previous results.

Findings

Derived explicit Clark-Ocone formulas for Cox processes with CIR intensities.

Extended the formula to Hawkes processes with stochastic intensity.

Facilitated more accurate pricing of lookback options in complex stochastic models.

Abstract

Pricing of the lookback options using the Clark-Ocone formula for the underlying assets driven by stochastic L\'evy processes requires computing the Malliavin derivatives of their maximum or minimum on the Wiener-Poisson space and their distributions. In this work, we will find a generalization of the explicit representation of the Clark-Ocone formula on the maximum of two types of L\'evy processes with stochastic intensity: Cox processes with CIR-modeled intensities, and the Hawkes processes.