Option Pricing with a Compound CARMA(p,q)-Hawkes

TL;DR

This paper introduces a novel asset price model based on a compound CARMA(p,q)-Hawkes process, capable of capturing complex market dynamics and reproducing observed volatility smiles in option pricing.

Contribution

It extends the Hawkes process by integrating a CARMA(p,q) intensity, offering greater flexibility and realism in modeling financial market data and option prices.

Findings

Model reproduces market volatility smile

Higher order parameters improve pricing accuracy

Empirical calibration demonstrates model's effectiveness

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.