Variance-Hawkes Process and its Application to Energy Markets

Paper: arXiv 2410.08420

Authors: Joshua McGillivray, Anatoliy Swishchuk

Abstract

We define a new model using a Hawkes process as a subordinator in a standard Brownian motion. We demonstrate that this Hawkes subordinated Brownian motion or more succinctly, variance-Hawkes process can be fit to 2018 and 2019 natural gas and crude oil front-month futures log returns. This variance-Hawkes process allows financial models to easily have clustering effects encoded into their behaviour in a simple and tractable way. We also compare the simulations of a square of a variance Hawkes process with its Ito formula. We simulate both processes and compare their distributions, trajectories, and percent errors across multiple runs. We derive the generator relating to this Hawkes subordinated Brownian motion, calculate several moments, and conjecture its distribution. We also provide explicit solutions to the second moments of the Hawkes process and its intensity as well as the cross moment between the Hawkes process and its intensity in the case of an exponential kernel.

Complexity vs Empirical Score

• Math Complexity: 7.0/10
• Empirical Rigor: 5.5/10
• Quadrant: Holy Grail — high math complexity, high empirical rigor

Why this score: The paper introduces a novel Variance-Hawkes process by using a Hawkes process as a discontinuous subordinator in Brownian motion to capture volatility clustering in energy markets. It features rigorous mathematical derivations including moment calculations, generators, and Ito’s lemma, accompanied by empirical fitting to crude oil and natural gas futures.

Research Flowchart

  flowchart TD
    A["Research Goal:<br>Model Volatility Clustering via<br>Variance-Hawkes Process (VHP)"] --> B["Data Inputs:<br>2018-2019 Natural Gas & Crude Oil<br>Front-Month Futures Log Returns"]
    
    B --> C["Model Formulation:<br>Subordinate Brownian Motion<br>with Hawkes Process"]
    
    C --> D["Mathematical Derivations:<br>Generators, Moments &<br>Itô Formula for Squared VHP"]
    
    D --> E["Simulation & Validation:<br>Compare Process Trajectories,<br>Distributions & Percent Errors"]
    
    E --> F["Key Outcomes:<br>Tractable Model for Energy Markets &<br>Explicit Moment Solutions"]

Cited References (13)

1. An elementary derivation of moments of H awkes processes (2020) — link
2. Compound H awkes processes in limit order books (2019) — link
3. Electricity price modeling with stochastic time change (2017) — link
4. The E uropean intraday electricity market: a modeling based on the H awkes process (2019) — link
5. Hawkes processes in finance (2015) — link
6. Risk model based on compound H awkes process (2018) — link
7. Variance and volatility swaps in energy markets (2010) — link
8. S& P D ow J ones Indices Announces 2021 S & P GSCI Weights (2020) — link

0 of 13 cited works are in the corpus; resolved via arXiv source bundles, Semantic Scholar and OpenAlex. Corpus-wide: 2518 linked papers.