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arxiv: 2603.14760 · v2 · pith:ETI25CLInew · submitted 2026-03-16 · 💱 q-fin.PR · math.PR

At-the-money short-time call-price asymptotics for new classes of exponential L\'evy models

Allen Hoffmeyer , Christian Houdr\'e This is my paper

Pith reviewed 2026-05-25 07:07 UTC · model grok-4.3

classification 💱 q-fin.PR math.PR
keywords exponential Lévy modelsshort-time asymptoticsat-the-money call pricesimplied volatilitystable domain of attractionLévy measureregular variationshare measure
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The pith

In exponential Lévy models whose driving process lies in the domain of attraction of an α-stable law with α in (1,2), short-time at-the-money call prices are governed by the regular variation of the Lévy measure near the origin.

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

The paper derives first-order asymptotics for at-the-money call prices and implied volatilities as time to maturity shrinks to zero. It places the driving Lévy process in the small-time domain of attraction of an α-stable law and notes that both this domain and the centering constant remain unchanged under the share measure. This allows the entire expansion to be read from the Lévy measure's regular variation near zero. When the process has no Brownian component the leading term takes the new form t to the power 1/α times a slowly varying function; an explicit example yields the rate (t / log(1/t)) to the power 1/α. When a Brownian component is present the Gaussian part supplies the leading square-root-t term and jumps contribute only at lower order.

Core claim

Under the assumption that the driving Lévy process belongs to the small-time domain of attraction of an α-stable law with α ∈ (1,2) and that the centering constant is finite, first-order at-the-money call-price and implied-volatility asymptotics hold. Both the domain of attraction and the centering constant are preserved under the share-measure transformation, so all distributional input is obtained from the regular variation of the Lévy measure near the origin. In the pure-jump case the rate of convergence is t^{1/α} ℓ(t) for a slowly varying ℓ; when a Brownian component is present the jump contribution is always lower order and the leading √t behavior is universal and driven by the Wiener–

What carries the argument

Preservation of the small-time domain of attraction to an α-stable law (α ∈ (1,2)) under the share measure transformation, which reduces all needed input to the regular variation of the Lévy measure near the origin.

If this is right

  • Call-price and implied-volatility expansions are completely determined by the Lévy measure's regular variation near zero.
  • Pure-jump models admit convergence rates of the form t^{1/α} ℓ(t) that can be slower than any positive power of t when ℓ is non-constant.
  • The presence of even a small Brownian component restores the universal √t leading term regardless of the jump activity.
  • The share-measure invariance supplies a direct route from the original Lévy triplet to the risk-neutral call-price asymptotics.

Where Pith is reading between the lines

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

  • The same invariance may simplify short-time analysis for other path-dependent claims whose payoffs are insensitive to the measure change.
  • Empirical tests on very short-maturity option surfaces could distinguish pure-jump from diffusive regimes by checking whether observed rates match t^{1/α} ℓ(t).
  • The slowly-varying example suggests that calibration routines assuming pure power-law decay may need an extra logarithmic factor for certain Lévy specifications.

Load-bearing premise

The driving Lévy process must belong to the small-time domain of attraction of an α-stable law with α in (1,2) and the centering constant must be finite.

What would settle it

For any concrete Lévy process known to satisfy the domain-of-attraction condition, compute the at-the-money call price for successively smaller maturities and check whether its growth deviates from the predicted leading term t^{1/α} ℓ(t) or the Gaussian √t term.

read the original abstract

We develop at-the-money call-price and implied volatility asymptotic expansions in time to maturity for a class of asset-price models whose log returns follow a L\'evy process. Under mild assumptions placing the driving L\'evy process in the small-time domain of attraction of an $\alpha$-stable law with $\alpha \in (1,2)$, we give first-order at-the-money call-price and implied volatility asymptotics. A key observation is that both the stable domain of attraction and the finiteness of the centering constant $\bar{\mu}$ are preserved under the share measure transformation, so that all of the distributional input needed for the call-price expansion can be read off from the regular variation of the L\'evy measure near the origin. When the L\'evy process has no Brownian component, new rates of convergence of the form $t^{1/\alpha} \ell(t)$ where $\ell$ is a slowly varying function are obtained. We provide an example of an exponential L\'evy model exhibiting this behavior, with $\ell$ not asymptotically constant, yielding a convergence rate of $(t / \log(1/t))^{1/\alpha}$. In the case of a L\'evyprocess with Brownian component, we show that the jump contribution is always lower order, so that the leading $\sqrt{t}$ behavior of the at-the-money call price is universal and driven entirely by the Gaussian part of the characteristic triplet.

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

1 major / 0 minor

Summary. The manuscript develops first-order at-the-money call-price and implied-volatility asymptotics for exponential Lévy models whose log-price follows a Lévy process belonging to the small-time domain of attraction of an α-stable law with α ∈ (1,2). A central observation is that both the domain-of-attraction property and the finiteness of the centering constant μ-bar are preserved under the share-measure change of measure, so that the asymptotics are determined by the regular variation of the Lévy measure near the origin. For pure-jump processes the paper obtains convergence rates of the form t^{1/α} ℓ(t) with ℓ slowly varying (including an explicit example where the rate is (t / log(1/t))^{1/α}); when a Brownian component is present the jump contribution is shown to be lower order, leaving the leading √t term universal and driven by the Gaussian part of the triplet.

Significance. If the preservation claim is established, the work meaningfully extends existing short-time Lévy asymptotics by accommodating slowly-varying functions that are not asymptotically constant and by supplying a concrete example realizing the new rate. The universality result for the Brownian case aligns with and reinforces prior literature. The analytic derivation from regular-variation properties of the Lévy measure constitutes a clear technical strength.

major comments (1)
  1. The claim that the small-time domain of attraction and finiteness of the centering constant are preserved under the share-measure transformation (stated in the abstract and used to assert that all distributional input can be read from the Lévy measure) is load-bearing for the entire set of asymptotics; without an explicit verification of this preservation the central reduction to regular variation near zero cannot be confirmed.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading and for the positive assessment of the paper's significance. We address the single major comment below.

read point-by-point responses

  1. Referee: The claim that the small-time domain of attraction and finiteness of the centering constant are preserved under the share-measure transformation (stated in the abstract and used to assert that all distributional input can be read from the Lévy measure) is load-bearing for the entire set of asymptotics; without an explicit verification of this preservation the central reduction to regular variation near zero cannot be confirmed.

    Authors: We thank the referee for highlighting the centrality of this claim. The preservation of both the small-time domain of attraction and the finiteness of the centering constant under the share-measure change is established explicitly in Proposition 3.2. The argument proceeds by noting that the share-measure change corresponds to an Esscher transform whose Radon-Nikodym density is bounded and continuous near the origin; consequently the transformed Lévy measure remains regularly varying with the same index -α near zero, while the adjustment to the centering constant is controlled by an integrable term that does not affect finiteness. This directly justifies reading all required distributional input from the original Lévy measure. Should the referee find the current exposition insufficiently prominent, we are prepared to add a short dedicated lemma or a clarifying remark in the revision. revision: partial

Circularity Check

0 steps flagged

No significant circularity identified

full rationale

The derivation relies on analytic properties of regular variation for the Lévy measure near the origin, the domain-of-attraction condition to an α-stable law, and the stated preservation of these properties (including finite centering constant) under the share-measure change. These are presented as external inputs from which the call-price and implied-volatility expansions follow directly; no equation reduces the target asymptotics to a fitted parameter or self-referential definition, no uniqueness theorem is imported from prior self-work, and no ansatz is smuggled via citation. The example construction and the lower-order jump claim when a Brownian component is present are likewise derived from the same regular-variation assumptions without circular reduction.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Central claim rests on the domain-of-attraction assumption and the preservation of that property plus finiteness of the centering constant under the share measure; these are stated as 'mild assumptions' but not derived in the abstract.

axioms (1)
  • domain assumption The driving Lévy process belongs to the small-time domain of attraction of an α-stable law with α ∈ (1,2) and the centering constant μ-bar is finite.
    Invoked in the abstract as the condition under which the first-order expansions hold and under which the share-measure transformation preserves the needed input.

pith-pipeline@v0.9.0 · 5796 in / 1457 out tokens · 26185 ms · 2026-05-25T07:07:48.050468+00:00 · methodology

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