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arxiv: 2606.28238 · v1 · pith:PA34TOJEnew · submitted 2026-06-26 · 🧮 math.DS

Affine Option Pricing with Hawkes-Type Endogenous Jump Activity

Ziyang Fang , Shenglan Yuan This is my paper

Pith reviewed 2026-06-29 01:53 UTC · model grok-4.3

classification 🧮 math.DS
keywords affine processesoption pricingendogenous jumpsHawkes feedbackRiccati systemFourier cosine methodtempered stable jumpsmartingale condition
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The pith

The two-dimensional state process with endogenous jump activity admits an affine transform representation.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper builds an option pricing model in which jump activity is driven endogenously by past realized price jumps through a bounded feedback function. It proves that the joint dynamics of log-price and activity level still admit an affine structure, so the characteristic function is obtained by solving a system of ordinary differential equations. This structure supports direct Fourier-cosine pricing that uses only real-axis values. The construction relies on state-dependent thinning of a Poisson measure and a mean-subcriticality condition to guarantee existence, uniqueness, and the martingale property under the risk-neutral measure. If the representation holds, option prices can be computed without simulating paths even when jump intensity reacts to its own history.

Core claim

The coupled log-price and activity-state dynamics, constructed via state-dependent thinning of a Poisson random measure with normalized asymmetric tempered-stable jump sizes and bounded excitation g(y)=1-e^{-a y^2}, admit an affine transform representation. The associated generalized Riccati system is derived and shown to be well-posed on the real axis with forward invariance of the relevant complex half-plane. European options are then priced by a Fourier-cosine method that requires only the real-axis transform, under the risk-neutral drift restriction and a sufficient true-martingale condition obtained from the mean-subcriticality assumption.

What carries the argument

The affine transform representation of the two-dimensional state process (log-price and activity scale), which yields the generalized Riccati system.

If this is right

  • Pathwise existence and uniqueness of the coupled dynamics hold under the mean-subcriticality condition.
  • A risk-neutral drift restriction together with a sufficient martingale condition is obtained explicitly.
  • European option prices are computed via the Fourier-cosine method using only the real-axis transform.
  • Numerical experiments produce the model-implied volatility surface and show current activity shifting near-term volatility levels.
  • Endogenous feedback controls the persistence of jump-induced skew across different maturities.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same affine closure might be preserved if the excitation function is replaced by other bounded maps that keep total excitation finite.
  • The real-axis well-posedness could simplify calibration routines that avoid complex-plane contour choices.
  • The framework suggests a route for adding self-exciting jumps to other affine models without losing the transform representation.
  • Market data on maturity-dependent skew could be used to test whether the estimated feedback strength matches the model's mean-subcriticality bound.

Load-bearing premise

The bounded excitation function together with the mean-subcriticality condition keeps average excitation finite and guarantees existence of the risk-neutral measure and the martingale property.

What would settle it

Monte Carlo paths of the price and activity process whose empirical characteristic function deviates from the solution of the Riccati system on the real axis, or simulated discounted prices that fail to be martingales.

Figures

Figures reproduced from arXiv: 2606.28238 by Shenglan Yuan, Ziyang Fang.

Figure 1
Figure 1. Figure 1: shows the pronounced short-maturity curvature and the gradual flat￾tening of the implied-volatility profile as maturity increases. In addition, Fig￾ure 2 reports the model-implied volatility smiles, which display a pronounced short-maturity curvature. For T = 1/12, the implied volatilities at k = −0.30, k = 0, and k = 0.30 are 0.516470, 0.281031, 0.456148, respectively. The left wing is higher than the rig… view at source ↗
Figure 2
Figure 2. Figure 2: Model-implied volatility smiles. Firstly, [PITH_FULL_IMAGE:figures/full_fig_p024_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Initial-activity channel. 1/12 1/4 1/2 1 Maturity T 0.02 0.03 0.04 0.05 0.06 L eft--rig ht B S IV spre a d σIV(− 0.3 0, T) − σIV(0.3 0, T) η = 0 η = 0.5 η = 1 [PITH_FULL_IMAGE:figures/full_fig_p025_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Endogenous-feedback channel. 5.6 Carr–Madan Check We compare COS prices with prices computed from the Carr–Madan inversion formula. For the Carr–Madan calculation, we use the damping parameter δ = 25 [PITH_FULL_IMAGE:figures/full_fig_p025_4.png] view at source ↗
read the original 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.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 3 minor

Summary. The paper develops a risk-neutral option-pricing model in which the activity scale of an infinite-activity jump process is endogenously driven by the asset's own realized price jumps. Jump sizes follow a normalized asymmetric tempered-stable Lévy shape, activity is updated via the bounded excitation function g(y)=1-e^{-a y^2}, and the coupled log-price/activity dynamics are constructed by state-dependent thinning of a Poisson random measure. The authors prove pathwise existence and uniqueness, derive the mean-subcriticality condition together with risk-neutral drift and true-martingale restrictions, establish the affine transform representation, 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 via the Fourier-cosine method and benchmarked against damped Carr-Madan inversion, with numerical illustrations of the implied volatility surface and the effects of current activity and endogenous feedback.

Significance. If the stated existence, uniqueness, martingale, and Riccati well-posedness results hold, the work supplies a tractable affine framework that incorporates Hawkes-type endogenous jump feedback while preserving the ability to price options by Fourier methods. The bounded excitation function together with the mean-subcriticality condition is a key technical device that keeps average excitation finite; the real-axis well-posedness result directly enables the COS pricing routine. These features distinguish the model from standard affine jump-diffusions and allow examination of how jump-induced skew persists across maturities.

major comments (2)
  1. [dynamics construction / generator derivation] The central claim that the two-dimensional state process is affine rests on the generator being affine after the state-dependent thinning construction. The manuscript should explicitly verify (in the section deriving the infinitesimal generator) that no non-affine remainder terms arise from the interaction between the bounded g(y) and the mark distribution of the tempered-stable measure.
  2. [mean-subcriticality and martingale conditions] The mean-subcriticality condition is invoked both for the true-martingale property and for the risk-neutral drift restriction. The paper should state the precise inequality (in terms of the parameters a and the tempering coefficients) and show that it is sufficient to close the integrability argument for the compensator of the thinned random measure.
minor comments (3)
  1. [model specification] The abstract refers to 'tempering parameters of the asymmetric tempered-stable Lévy measure' without listing their symbols; these should be introduced with explicit notation when the Lévy measure is first defined.
  2. [numerical experiments] In the numerical section the COS versus damped Carr-Madan comparison would benefit from tabulated relative pricing errors or convergence rates rather than qualitative statements alone.
  3. [Riccati well-posedness] A short remark clarifying why the forward invariance of the complex half-plane guarantees that the real-axis Riccati solution suffices for the COS method would improve readability for readers unfamiliar with the complex-plane analysis.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the positive evaluation and the precise suggestions. Both major comments concern points of explicitness in the derivations rather than errors in the results; we will incorporate the requested verifications and statements in the revised manuscript.

read point-by-point responses

  1. Referee: [dynamics construction / generator derivation] The central claim that the two-dimensional state process is affine rests on the generator being affine after the state-dependent thinning construction. The manuscript should explicitly verify (in the section deriving the infinitesimal generator) that no non-affine remainder terms arise from the interaction between the bounded g(y) and the mark distribution of the tempered-stable measure.

    Authors: We agree that an explicit verification improves clarity. The state-dependent thinning is constructed so that the compensator of the thinned measure is exactly the activity coordinate multiplied by the fixed integral ∫ g(y) u(dy), where u is the normalized tempered-stable mark measure; because g(y) depends only on the mark and not on the state, and the mark measure is state-independent, the resulting generator applied to exponential-affine functions remains affine in the two state variables with no remainder. In the revision we will add a short paragraph immediately after the generator derivation that writes out this calculation and confirms the absence of non-affine terms. revision: yes

  2. Referee: [mean-subcriticality and martingale conditions] The mean-subcriticality condition is invoked both for the true-martingale property and for the risk-neutral drift restriction. The paper should state the precise inequality (in terms of the parameters a and the tempering coefficients) and show that it is sufficient to close the integrability argument for the compensator of the thinned random measure.

    Authors: The mean-subcriticality condition appears in Theorem 2.3 as the requirement that the expected excitation under the normalized mark measure stays below the reciprocal of the maximal tempering-adjusted moment; we will expand it explicitly to the inequality a ∫ y^{2} u_{\alpha,\beta}(dy) < 1 (with u_{\alpha,\beta} the normalized asymmetric tempered-stable measure) and add a short integrability lemma showing that this bound, together with the linear growth control on the compensator, guarantees that the integrated intensity remains finite almost surely. This will be inserted in Section 2.3 of the revision. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation self-contained

full rationale

The paper constructs the two-dimensional state process explicitly from state-dependent thinning of a Poisson random measure whose mark distribution is independent of state and whose activity update uses the bounded function g(y). The generator is therefore affine in the state variables by the explicit form of the dynamics, so the characteristic function satisfies the generalized Riccati system by the standard affine-process argument; this is not a reduction of the claim to its own inputs. No self-citations appear as load-bearing premises, no parameters are fitted then renamed as predictions, and the mean-subcriticality condition is derived from the model to guarantee integrability rather than assumed to force the affine representation. The well-posedness proof on the real axis likewise proceeds from the boundedness and subcriticality assumptions without circular redefinition.

Axiom & Free-Parameter Ledger

2 free parameters · 2 axioms · 0 invented entities

The model introduces several parameters whose values are not derived from first principles and relies on domain assumptions about the Lévy measure and excitation function.

free parameters (2)
  • a
    Controls the strength of endogenous feedback in g(y)=1-e^{-ay^2}; chosen to keep average excitation finite.
  • tempering parameters of the asymmetric tempered-stable Lévy measure
    Shape and scale parameters of the jump-size distribution; required to normalize the activity scale to the local quadratic-variation rate.
axioms (2)
  • domain assumption Mean-subcriticality condition
    Invoked to guarantee finite moments and the existence of the risk-neutral measure.
  • standard math Pathwise existence and uniqueness of the state-dependent thinned Poisson random measure
    Standard result from stochastic calculus assumed to hold under the chosen excitation function.

pith-pipeline@v0.9.1-grok · 5776 in / 1353 out tokens · 25998 ms · 2026-06-29T01:53:19.452356+00:00 · methodology

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