Affine Option Pricing with Hawkes-Type Endogenous Jump Activity

Ziyang Fang; Shenglan Yuan

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arXiv · 2606.28238

Affine Option Pricing with Hawkes-Type Endogenous Jump Activity

Ziyang Fang·Shenglan Yuan

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Abstract

We develop a risk-neutral option-pricing model where the activity scale of an infinite-activity jump process is endogenously driven by the asset's own realized price jumps. Jump sizes are governed by a normalized asymmetric tempered-stable L\'evy shape, while a predictable activity scale controls the overall jump intensity and is normalized to coincide with the local jump-induced quadratic-variation rate. Endogenous feedback is introduced through the bounded excitation function $g(y)=1-e^{-ay^2}$, so that small realized jumps excite future activity approximately in proportion to squared jump size while the total average excitation remains finite. We construct the coupled log-price and activity-state dynamics by state-dependent thinning of a Poisson random measure, prove pathwise existence and uniqueness, derive the mean-subcriticality condition, and obtain both the risk-neutral drift restriction and a sufficient true-martingale condition. The resulting two-dimensional state process admits an affine transform representation. We derive the associated generalized Riccati system and prove real-axis well-posedness with forward invariance of the relevant complex half-plane. European options are priced by a Fourier-cosine (COS) method, which requires only the real-axis transform, and are benchmarked against a damped Carr--Madan (CM) inversion. Numerical experiments illustrate the model-implied volatility surface and show how current activity shifts near-term volatility levels, while endogenous feedback affects the persistence of jump-induced skew across maturities.

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BibTeX RIS

Ziyang Fang, Shenglan Yuan. 2026-06-26. Affine Option Pricing with Hawkes-Type Endogenous Jump Activity. https://arxiv.org/abs/2606.28238

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