Time-changed \levy processes and option pricing: a critical comment

Hasan Fallahgoul; Kihun Nam

arxiv: 1907.00149 · v1 · pith:6T5IWRIHnew · submitted 2019-06-29 · 💱 q-fin.MF

Time-changed levy processes and option pricing: a critical comment

Hasan Fallahgoul , Kihun Nam This is my paper

Pith reviewed 2026-05-25 13:09 UTC · model grok-4.3

classification 💱 q-fin.MF

keywords time-changed Lévy processesoption pricingstopping timesmeasurabilityfiltrationCarr-Wu frameworksubordinators

0 comments

The pith

Time changes proposed by Carr and Wu for option pricing models fail the required stopping-time measurability condition.

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

Carr and Wu (2004) built a framework meant to cover nearly all continuous-time option pricing models by using time-changed Lévy processes. The framework depends on the time changes being stopping times, which requires them to be measurable with respect to the underlying filtration. The paper checks the specific time-change constructions listed in that work and concludes that every one of them violates the measurability condition. A reader would care because the result directly questions whether the framework can actually support the models it claims to include. The analysis rests on standard filtration arguments rather than on any new pricing formula.

Core claim

Carr and Wu (2004), henceforth CW, developed a framework that encompasses almost all of the continuous-time models proposed in the option pricing literature. Their framework hinges on the stopping time property of the time changes. By analyzing the measurability of the time changes with respect to the underlying filtration, we show that all models CW proposed for the time changes fail to satisfy this assumption.

What carries the argument

The stopping-time property of time changes, defined as measurability with respect to the underlying filtration.

If this is right

The time-changed Lévy models listed by Carr and Wu cannot be justified inside their own framework.
The claim that the framework covers almost all continuous-time option pricing models does not hold for the proposed time changes.
Any pricing results derived from those specific time changes lack the stopping-time justification given in the framework.

Where Pith is reading between the lines

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

New time-change constructions that are adapted to the filtration would be needed to restore the framework's coverage.
Papers that rely on the Carr-Wu setup without separate measurability checks may inherit the same gap.
The same filtration argument could be applied to later time-change proposals in the literature to test whether they satisfy the property.

Load-bearing premise

The stopping-time property is both necessary for the Carr-Wu framework and that the specific constructions listed in their 2004 paper are the only ones that need to be examined.

What would settle it

An explicit construction showing that at least one of the time-change processes listed in Carr and Wu (2004) is measurable with respect to the filtration at every time would falsify the central claim.

read the original abstract

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

The paper flags that Carr-Wu time changes fail measurability for stopping times but never shows why that property is required for their pricing results to hold.

read the letter

The main point is that this short comment claims every time-change construction listed in Carr and Wu (2004) is not a stopping time because it is not measurable with respect to the underlying filtration. If that is accurate, it would be a technical problem for those specific setups. The paper does apply the stopping-time definition directly to the models CW proposed, which is a straightforward check that had not been reported before. That is the actual new content. It also keeps the focus on the cited work without adding extra machinery. The citation pattern is narrow and relevant. The soft spot is the missing link the stress-test note already flagged. The abstract states that the CW framework hinges on the stopping-time property, yet the text supplies no derivation or citation showing that non-measurable time changes would break the martingale property, the characteristic function, or the pricing formulas. Without that step, the measurability observation stays a side note rather than a refutation. The abstract also gives no explicit filtration definitions or verification that the listed models are represented exactly as CW wrote them. This makes the central claim hard to assess from the given material. The paper is aimed at people who already work on time-changed Lévy models in mathematical finance and want to check the foundations of the 2004 constructions. A reader looking for a quick technical alert might find it useful, but anyone wanting a complete argument would have to fill in the necessity gap themselves. I would not bring it to reading group. I would not cite it. It does not look ready for peer review until the necessity of the stopping-time property is shown explicitly.

Referee Report

1 major / 0 minor

Summary. The manuscript argues that the Carr-Wu (2004) framework for option pricing via time-changed Lévy processes requires the time changes to be stopping times with respect to the underlying filtration, and that all time-change constructions proposed in CW fail this measurability requirement.

Significance. If the stopping-time property is required for the martingale property and characteristic-function derivations in the CW framework to hold, the result would undermine a broad class of continuous-time option pricing models. The manuscript supplies no derivation establishing this necessity, so the significance cannot be assessed from the provided text.

major comments (1)

[Abstract] Abstract: the claim that the CW framework 'hinges on' the stopping time property is asserted without any derivation, citation, or explicit verification that non-measurable time changes would invalidate the martingale property, the characteristic function, or the pricing formulas. This link is load-bearing for the central claim that the measurability failures constitute a refutation.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the detailed report and the opportunity to address the central concern regarding the necessity of the stopping-time property. We respond to the major comment below and commit to revisions that strengthen the manuscript.

read point-by-point responses

Referee: [Abstract] Abstract: the claim that the CW framework 'hinges on' the stopping time property is asserted without any derivation, citation, or explicit verification that non-measurable time changes would invalidate the martingale property, the characteristic function, or the pricing formulas. This link is load-bearing for the central claim that the measurability failures constitute a refutation.

Authors: We agree that the manuscript asserts the dependence on the stopping-time property without supplying an explicit derivation of why this property is required for the martingale property, characteristic function, or pricing formulas to hold. This is a substantive gap. In the revised version we will add a dedicated subsection (likely in Section 2) that derives the necessity: we will show step-by-step that the time-change process must be a stopping time with respect to the underlying filtration for the time-changed Lévy process to remain a martingale under the risk-neutral measure, for the stochastic integral representation to be well-defined, and for the Fourier inversion used by CW to recover the option price to be justified. The derivation will rely on standard results from stochastic calculus (e.g., the optional sampling theorem and the definition of the stochastic exponential). We will also note any relevant citations from the Lévy-process literature that make this assumption explicit. This revision directly addresses the load-bearing link identified by the referee. revision: yes

Circularity Check

0 steps flagged

No significant circularity; direct application of stopping-time definition to external models

full rationale

The paper's argument consists of checking the measurability of time-change processes (proposed in the external Carr-Wu 2004 reference) against the underlying filtration and concluding they are not stopping times. This is a straightforward definitional application with no fitted parameters renamed as predictions, no self-citation load-bearing the central claim, and no reduction of the result to the authors' own prior work or ansatz. The necessity of the stopping-time property is attributed to the CW framework rather than derived internally, but this does not create circularity within the present derivation chain. The paper is self-contained against external benchmarks and receives the default non-finding.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The paper relies on standard definitions of stopping times and adapted processes from stochastic calculus; no free parameters or invented entities are introduced.

axioms (2)

domain assumption A time change must be a stopping time with respect to the underlying filtration for the Carr-Wu framework to apply.
Stated in the abstract as the hinge of the CW framework.
standard math Standard definitions of measurability and stopping times from stochastic processes apply without modification.
Invoked when analyzing the time changes with respect to the filtration.

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