Paper: arXiv 2412.15172

Authors: Lorenzo Mercuri, Andrea Perchiazzo, Edit Rroji

Abstract

A self-exciting point process with a continuous-time autoregressive moving average intensity process, named CARMA(p,q)-Hawkes model, has recently been introduced. The model generalizes the Hawkes process by substituting the Ornstein-Uhlenbeck intensity with a CARMA(p,q) model where the associated state process is driven by the counting process itself. The proposed model preserves the same degree of tractability as the Hawkes process, but it can reproduce more complex time-dependent structures observed in several market data. The paper presents a new model of asset price dynamics based on the CARMA(p,q) Hawkes model. It is constructed using a compound version of it with a random jump size that is independent of both the counting and the intensity processes and can be employed as the main block for pure jump and (stochastic volatility) jump-diffusion processes. The numerical results for pricing European options illustrate that the new model can replicate the volatility smile observed in financial markets. Through an empirical analysis, which is presented as a calibration exercise, we highlight the role of higher order autoregressive and moving average parameters in pricing options.

Complexity vs Empirical Score

  • Math Complexity: 7.5/10
  • Empirical Rigor: 6.5/10
  • Quadrant: Holy Grail — high math complexity, high empirical rigor

Why this score: The paper introduces an innovative compound CARMA(p,q)-Hawkes process for option pricing that effectively captures complex memory structures and volatility smiles. It combines rigorous mathematical modeling (log-affine characteristic functions, measure transformations, Gauss-Laguerre quadrature) with empirical calibration on option market data.

Research Flowchart

  flowchart TD
    A[Research Goal: Develop Option Pricing Model with CARMA p,q-Hawkes Asset Dynamics] --> B[Model Formulation: Construct Compound CARMA p,q-Hawkes Jump Process]
    B --> C[Framework Integration: Build Pure Jump & Stochastic Volatility Jump-Diffusion Models]
    D[Inputs: Market Data & European Option Prices] --> E[Computational Process: Model Calibration & Numerical Option Pricing]
    C --> E
    E --> F[Outcomes: Replicates Volatility Smile & Shows Value of Higher-Order p,q Parameters]

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