Affine Option Pricing with Hawkes-Type Endogenous Jump Activity

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 Lévy 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.