arxiv: 2604.13798 · v1 · submitted 2026-04-15 · 💱 q-fin.PR · math.PR

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Higher-order ATM asymptotics for the CGMY model via the characteristic function

Allen Hoffmeyer , Christian Houdr\'e

Authors on Pith no claims yet

Pith reviewed 2026-05-10 11:54 UTC · model grok-4.3

classification 💱 q-fin.PR math.PR

keywords CGMY modelshort-time asymptoticsat-the-money optionscharacteristic functionLévy processesstable lawsLipton-Lewis formulaoption pricing

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The pith

Short-time ATM call prices in the CGMY model expand as d1 t to the 1/Y plus d2 t plus higher-order terms using only the characteristic function.

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

The paper derives higher-order short-time asymptotics for at-the-money call prices in the exponential CGMY Lévy model with activity parameter Y between 1 and 2. It begins with the Lipton-Lewis integral representation of the normalized call price c(t,0) and applies a time rescaling of order t to the power -1/Y that places the process in the domain of attraction of a symmetric Y-stable law. This produces the leading coefficient d1 from the stable limit together with an explicit second coefficient d2 given by an integral involving the characteristic exponent. Higher terms are then obtained by retaining the full integrand and splitting the integration range with a dynamic cutoff into inner, core, and tail regions so that remainders are controlled uniformly. Readers care because the resulting expansion supplies accurate small-maturity approximations for option prices in pure-jump models without requiring simulation or closed-form densities.

Core claim

Using only the characteristic function, we derive short-time at-the-money call-price asymptotics for the exponential CGMY model with activity parameter Y∈(1,2). The Lipton-Lewis formula expresses the normalized ATM call price, denoted c(t,0), in terms of the characteristic exponent, which, upon rescaling at the rate t^{-1/Y} from the Y-stable domain of attraction, yields c(t,0)=d1 t^{1/Y} + d2 t + o(t) as t↓0. The first-order coefficient d1 is the known stable limit from the domain of attraction of a symmetric Y-stable law, and d2 is given by an explicit integral involving the characteristic exponent and the limiting stable exponent. We then extract closed-form higher-order coefficients by保持

What carries the argument

The Lipton-Lewis integral for the normalized ATM call price, combined with a dynamic cutoff that partitions the rescaled integration domain into inner, core, and tail regions to extract successive terms while uniformly bounding the remainders.

If this is right

The expansion holds with remainder o(t) after the second term and can be extended to any finite number of terms by including more regions in the cutoff partition.
Every coefficient is expressed in terms of the characteristic exponent evaluated at the rescaled frequencies and the limiting stable exponent.
The method yields computable approximations for small-maturity ATM prices that can be compared directly with closed-form expressions available for special parameter choices.
The same partitioning technique separates the stable-jump contribution from smoother parts of the integrand, allowing term-by-term identification.

Where Pith is reading between the lines

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

The cutoff approach may extend to other Lévy processes whose characteristic exponents admit similar stable-domain rescalings.
Adjusting the cutoff location or the rescaling could produce corresponding expansions for out-of-the-money strikes.
The explicit integral for d2 isolates the first correction arising from the difference between the CGMY exponent and its stable limit, which could be used to study implied-volatility skew at short times.

Load-bearing premise

The CGMY characteristic exponent permits the t to the -1/Y rescaling into the domain of attraction of a symmetric Y-stable law, and the dynamic cutoff partitioning controls the remainder uniformly across regions.

What would settle it

Direct numerical quadrature of the derived coefficients d1 and d2 for a fixed Y in (1,2) and comparison against the exact ATM call price obtained by Fourier inversion of the characteristic function for successively smaller t, checking whether the observed error matches the predicted o(t) or smaller.

Figures

Figures reproduced from arXiv: 2604.13798 by Allen Hoffmeyer, Christian Houdr\'e.

**Figure 1.** Figure 1: shows heatmaps of the difference between d2 and d F L 2 for fixed values of Y and various ranges of M and G. The differences are uniformly of order 10−7 , consistent with the numerical tolerance of the quadrature, confirming agreement between the two expressions [PITH_FULL_IMAGE:figures/full_fig_p008_1.png] view at source ↗

**Figure 2.** Figure 2: Candidate third- and fourth-order exponents as functions of [PITH_FULL_IMAGE:figures/full_fig_p011_2.png] view at source ↗

read the original abstract

Using only the characteristic function, we derive short-time at-the-money (ATM) call-price asymptotics for the exponential CGMY model with activity parameter $Y\in(1,2)$. The Lipton--Lewis formula expresses the normalized ATM call price, denoted $c(t,0)$, in terms of the characteristic exponent, which, upon rescaling at the rate $t^{-1/Y}$ from the $Y$-stable domain of attraction, yields $c(t,0) = d_{1} t^{1/Y} + d_{2} t + o(t)$ as $t\downarrow 0$. The first-order coefficient $d_{1}$ is the known stable limit from the domain of attraction of a symmetric $Y$-stable law, and $d_{2}$ is given by an explicit integral involving the characteristic exponent and the limiting stable exponent. We then extract closed-form higher-order coefficients by keeping the full Lipton--Lewis integrand intact and introducing a dynamic cutoff that partitions the domain into inner, core, and tail regions, establishing the expansion with controlled remainder. All coefficients are verified numerically against existing closed-form expressions where available.

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.

Desk Editor's Note private letter to a colleague

They give an explicit integral for the second-order short-time ATM coefficient in CGMY plus a dynamic-cutoff method to reach higher orders from the characteristic function.

read the letter

This paper derives higher-order short-time at-the-money asymptotics for the exponential CGMY model using only the characteristic function. They start from the Lipton-Lewis formula for the normalized call price, rescale the frequency variable at rate t to the minus 1/Y, and recover the known leading stable coefficient d1 from the domain of attraction of a symmetric Y-stable law. The new contribution is an explicit integral expression for the second-order coefficient d2 that isolates the difference between the full CGMY exponent and its stable limit. For higher orders they keep the complete integrand and apply a dynamic cutoff that splits the integral into inner, core, and tail regions, with separate estimates for each so the remainder stays o(t). They check the resulting coefficients numerically against closed-form cases where available, and the matches look solid. The approach is direct and avoids fitting or self-referential definitions. The soft spots are limited. The remainder controls are tied to the specific high-frequency behavior of the CGMY exponent and to Y in (1,2); near the boundaries the constants could grow, though the paper claims uniformity. The d2 integral still requires numerical quadrature in practice, which is acceptable but not purely closed-form. Overall the work is for readers who need sharper short-maturity approximations in infinite-activity Lévy models for pricing or calibration. The steps are technically grounded, build on established formulas without circularity, and supply verifiable coefficients. I would send it to a referee familiar with Lévy processes and asymptotic expansions in finance.

Referee Report

1 major / 2 minor

Summary. The manuscript derives short-time at-the-money call-price asymptotics for the exponential CGMY Lévy model (Y ∈ (1,2)) using only the characteristic function. Starting from the Lipton–Lewis formula for the normalized ATM call price c(t,0), the authors rescale the integration variable at rate t^{-1/Y} to enter the domain of attraction of a symmetric Y-stable law. This produces the expansion c(t,0) = d₁ t^{1/Y} + d₂ t + o(t) as t ↓ 0, where d₁ is the known stable coefficient and d₂ is given by an explicit integral involving the full characteristic exponent minus its stable limit. Higher-order coefficients are extracted by retaining the complete Lipton–Lewis integrand and introducing a dynamic cutoff that partitions the domain into inner, core, and tail regions, with uniform remainder estimates. All coefficients are checked numerically against existing closed-form expressions.

Significance. If the remainder controls hold, the work supplies explicit, parameter-free higher-order short-time ATM asymptotics for a widely used Lévy model, extending first-order stable approximations in a characteristic-function setting. The explicit integral form of d₂ and the dynamic-cutoff technique for controlled higher orders are technically useful for short-maturity analysis in exponential Lévy models. Numerical agreement with closed forms provides concrete support. The approach may serve as a benchmark for numerical pricing routines and inform high-frequency calibration.

major comments (1)

The dynamic cutoff partitioning (inner/core/tail) is load-bearing for the o(t) remainder and any higher-order claims. The manuscript should state the precise functional form of the cutoff and the explicit estimates showing that the tail integral is o(t) uniformly for Y ∈ (1,2) and the CGMY parameters (see the paragraph following the definition of the cutoff in the higher-order derivation).

minor comments (2)

The abstract states that 'closed-form higher-order coefficients' are extracted, yet only the two-term expansion plus o(t) is displayed; clarify the precise orders achieved beyond d₂ and whether they remain explicit integrals or become more involved.
Add a brief remark on the convergence of the d₂ integral at both zero and infinity, citing the decay properties of the CGMY exponent relative to the stable limit.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading, the positive overall assessment, and the recommendation for minor revision. The single major comment concerns the explicitness of the dynamic-cutoff construction used to control the remainder; we address it directly below and will incorporate the requested clarifications.

read point-by-point responses

Referee: The dynamic cutoff partitioning (inner/core/tail) is load-bearing for the o(t) remainder and any higher-order claims. The manuscript should state the precise functional form of the cutoff and the explicit estimates showing that the tail integral is o(t) uniformly for Y ∈ (1,2) and the CGMY parameters (see the paragraph following the definition of the cutoff in the higher-order derivation).

Authors: We agree that the dynamic cutoff is central to the remainder control. In the revised manuscript we will explicitly record the functional form of the cutoff (the radius separating the core from the tail region) together with the uniform tail-integral bounds that establish the o(t) contribution for all Y ∈ (1,2) and admissible CGMY parameters. These bounds follow from the decay of the Lipton–Lewis integrand outside the stable domain of attraction and are already derived in the proof; we will simply highlight them in a dedicated remark immediately after the cutoff definition so that the argument is self-contained. revision: yes

Circularity Check

0 steps flagged

No significant circularity

full rationale

The derivation begins from the external Lipton-Lewis formula and the established domain-of-attraction properties of the symmetric Y-stable law (Y in (1,2)). The leading coefficient d1 is imported as the known stable limit, while d2 is obtained as an explicit integral over the difference between the full CGMY exponent and its stable limit. Higher-order terms are extracted by partitioning the rescaled integrand via a dynamic cutoff into inner/core/tail regions whose remainders are controlled by explicit estimates. All steps are self-contained against external benchmarks (Lipton-Lewis, stable-law theory, and numerical checks against closed-form cases) with no reduction of the target expansion to fitted parameters, self-definitions, or load-bearing self-citations.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The derivation rests on the known validity of the Lipton-Lewis formula for normalized ATM prices and on the CGMY process belonging to the domain of attraction of a Y-stable law for Y in (1,2). No new free parameters or invented entities are introduced.

axioms (1)

domain assumption The exponential CGMY process with Y in (1,2) lies in the domain of attraction of a symmetric Y-stable law
This property justifies the t^{-1/Y} rescaling and supplies the leading coefficient d1 from the stable limit.

pith-pipeline@v0.9.0 · 5502 in / 1494 out tokens · 54625 ms · 2026-05-10T11:54:55.057654+00:00 · methodology

discussion (0)

Reference graph

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