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Hedging of exotic options in Hawkes jump-diffusion models by Malliavin calculus

Ayub Ahmadi, Mahdieh Tahmasebi

TL;DR

This paper tackles delta hedging for exotic options when the underlying follows a Hawkes jump-diffusion with stochastic intensity. It develops Malliavin-calculus tools on a Wiener–Poisson space to derive explicit Wiener–Poisson weights for European and Asian deltas, leveraging a duality between Skorokhod integrals and Malliavin derivatives. A careful analysis of the Hawkes SDEs provides moment bounds and weak derivatives, enabling a robust numerical scheme with proven convergence. Numerical experiments show that the Wiener–Poisson weight (WP) method yields highly accurate deltas compared to pure Wiener or Poisson weights and finite differences, validating the approach for practical hedging in jump-clustered markets.

Abstract

In financial mathematics, the calculation of the Greeks, especially the delta, is emphasized due to its role in risk management. In this article, we employ Malliavin calculus to determine the delta of European and Asian options, where the underlying asset evolves according to a Hawkes jump-diffusion process. A central feature is that the Hawkes jump intensity is stochastic, which substantially affects the delta representation.

Hedging of exotic options in Hawkes jump-diffusion models by Malliavin calculus

TL;DR

This paper tackles delta hedging for exotic options when the underlying follows a Hawkes jump-diffusion with stochastic intensity. It develops Malliavin-calculus tools on a Wiener–Poisson space to derive explicit Wiener–Poisson weights for European and Asian deltas, leveraging a duality between Skorokhod integrals and Malliavin derivatives. A careful analysis of the Hawkes SDEs provides moment bounds and weak derivatives, enabling a robust numerical scheme with proven convergence. Numerical experiments show that the Wiener–Poisson weight (WP) method yields highly accurate deltas compared to pure Wiener or Poisson weights and finite differences, validating the approach for practical hedging in jump-clustered markets.

Abstract

In financial mathematics, the calculation of the Greeks, especially the delta, is emphasized due to its role in risk management. In this article, we employ Malliavin calculus to determine the delta of European and Asian options, where the underlying asset evolves according to a Hawkes jump-diffusion process. A central feature is that the Hawkes jump intensity is stochastic, which substantially affects the delta representation.

Paper Structure

This paper contains 15 sections, 16 theorems, 116 equations, 3 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Hawkes processes and properties of stochastic intensities
  3. Hawkes stochastic differential equations
  4. Properties of the solution
  5. Weak derivatives of the solution
  6. Delta of options derived by the solution of Hawkes SDEs
  7. Delta Greek for European options
  8. Delta Greek for Simple Asian options
  9. The convergence of the numerical scheme
  10. Numerical Example
  11. Conclusions
  12. Proof of Lemma \ref{['yvaroon']}
  13. Proof of Lemma \ref{['deltadomain']}
  14. Proof of Lemma \ref{['deltadomainY']}
  15. Proof of Lemma \ref{['dlambdaexp']}

Key Result

Lemma 2.1

For every $p \geq 2$,

Figures (3)

  • Figure 1: Pricing Asian and European call options.
  • Figure 2: The delta of the European call option under Hawkes process dynamics.
  • Figure 3: The delta of the simple Asian call option under Hawkes process dynamics.

Theorems & Definitions (27)

  • Lemma 2.1
  • proof
  • Lemma 3.1
  • proof
  • Lemma 3.2
  • proof
  • Lemma 3.3
  • proof
  • Lemma 3.4
  • proof
  • ...and 17 more