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

Mahdieh Tahmasebi

TL;DR

The paper extends the Clark-Ocone formula to the maximum of processes driven by Lévy processes with stochastic intensity, focusing on Cox (CIR-type) and Hawkes models. It derives explicit Malliavin derivatives of the running maximum $M_T=\max_{0\le t\le T} X_t$ and provides concrete Clark-Ocone representations with coefficients expressed via conditional tail probabilities. The authors develop Laplace-inversion techniques to obtain the distributions of the supremum under these stochastic-intensity dynamics, enabling lookback-option pricing and hedging in models with jump clustering. The results offer a principled, semi-analytical framework for pricing and risk management of path-dependent options in markets with stochastic intensities and jumps.

Abstract

Pricing of the lookback options using the Clark-Ocone formula for the underlying assets driven by stochastic Lévy 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 Lévy processes with stochastic intensity: Cox processes with CIR-modeled intensities, and the Hawkes processes.

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

TL;DR

The paper extends the Clark-Ocone formula to the maximum of processes driven by Lévy processes with stochastic intensity, focusing on Cox (CIR-type) and Hawkes models. It derives explicit Malliavin derivatives of the running maximum and provides concrete Clark-Ocone representations with coefficients expressed via conditional tail probabilities. The authors develop Laplace-inversion techniques to obtain the distributions of the supremum under these stochastic-intensity dynamics, enabling lookback-option pricing and hedging in models with jump clustering. The results offer a principled, semi-analytical framework for pricing and risk management of path-dependent options in markets with stochastic intensities and jumps.

Abstract

Pricing of the lookback options using the Clark-Ocone formula for the underlying assets driven by stochastic Lévy 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 Lévy processes with stochastic intensity: Cox processes with CIR-modeled intensities, and the Hawkes processes.

Paper Structure

This paper contains 5 sections, 7 theorems, 74 equations.

Table of Contents

  1. Malliavin calculus on Wiener-Poisson Space
  2. Cox processes
  3. Hawkes process
  4. Distribution of the supremum running of Hawkes processes
  5. Appendix

Key Result

Proposition 1

Let $F \in L^{2}(\Omega, \mathcal{F}, \mathbb{P})$. There exists a unique Borel predictable process $\psi \in L^{2}([0, T] \times \mathbb{R} \times \Omega)$ and a unique predictable process $\phi \in L^{2}([0, T] \times \Omega)$ such that

Theorems & Definitions (7)

  • Proposition 1
  • Proposition 2
  • Proposition 1.1
  • Proposition 1.2
  • Theorem 2.1
  • Lemma 2.2
  • Theorem 3.1