Option Pricing with Time-Changed Fractional Brownian Motion: A Fractional Variance Gamma Model

Jayen Tan; Robert Jarrow

REVIEW 4 major objections 4 minor 205 references

Option Pricing with Time-Changed Fractional Brownian Motion: A Fractional Variance Gamma Model

T0 review · 4 major / 4 minor · reviewed 2026-08-05 · deepseek-v4-flash

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Pith's one-line read This paper claims that evaluating fractional Brownian motion at a gamma-distributed activity clock makes it a semimartingale, opening classical no-arbitrage option pricing to fractional dynamics.

desk verdict The fVG construction is plausible and the semimartingale claim is likely true, but a central proof uses a false identity for pure-jump compensators, leaving the true-martingale result unproven as written. read the letter →

arxiv 2608.03925 v1 pith:X4ZP5IWO submitted 2026-08-04 q-fin.MF q-fin.PR

Robert Jarrow , Jayen Tan This is my paper · ORCID

classification q-fin.MFq-fin.PR MSC 60G2260G5160G5591G20

keywords fractionalBrownianmotiontime-changedprocessgammaactivitytimesemimartingalevariancemodeloptionpricingGMMestimationHurstexponent

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

Fractional Brownian motion (fBm) has the features stock returns seem to show — long-range dependence, rough paths, anomalous diffusion — but it is not a semimartingale, and under continuous trading that creates arbitrage, ruling it out of the standard option-pricing framework. The paper claims the obstacle disappears if fBm is evaluated not at calendar time but at a gamma-distributed stochastic clock called activity time: the composition X = B_H(γ) is a pure-jump semimartingale, while retaining fBm's defining statistical properties. On this construction the paper builds the fractional Variance Gamma (fVG) model, derives the compensator that acts as the process's predictable drift, obtains a Girsanov change of measure for risk-neutral pricing, and proposes a GMM estimation procedure. Fitting the model to S&P 500 index returns yields an estimated Hurst exponent of about 0.45, mildly below the Brownian value 0.5, which the authors read as mildly sublinear scaling of return moments. If the semimartingale claim is right, fractional persistence and roughness become compatible with classical arbitrage-free valuation without altering trading restrictions or the definition of arbitrage.

What carries the argument

The load-bearing object is the time-changed process X = B_H(γ), fBm run on a gamma 'activity-time' clock. Because the gamma process is an infinite-activity pure-jump subordinator of bounded variation and fBm has continuous paths, the composition is a pure-jump process of bounded variation — the property that makes it a semimartingale. The argument then runs through the marked point process compensator ν_X(dt, dx) = ψ_X(t,x) dx dt, whose density is a convolution of the conditional Gaussian density of fBm increments with the gamma Lévy density; integrating x against ψ_X gives the explicit predictable compensator A_X that plays the role of the drift, and a Girsanov kernel λ(t,x) yields the equi

What would settle it

Simulate the fVG model with known parameters and run the proposed GMM on the observed price path alone: if the moment conditions leave a flat ridge in (H, v) or the estimates do not converge to the true values as the sample grows, the latent activity time cannot be recovered from prices, and the Proposition 5.2 drift condition cannot be implemented in the natural filtration — the compatibility claim fails at the observable-market level.

Watch

Extended reading notes

Core claim

The central discovery is Theorem 4.2: X(t) = B_H(γ(t)), fBm evaluated at an independent gamma process, is a semimartingale. Because γ is a pure-jump process of bounded variation and B_H is continuous, X is itself a pure-jump bounded-variation process, hence a special semimartingale with a unique predictable compensator A_X. The paper computes A_X as a Gaussian–Gamma mixture, derives a Girsanov density yielding equivalent martingale measures, and builds the fVG stock-price model, whose log returns split into risk-free accrual, jump-risk premium, dependence-induced endogenous drift, and martingale innovation. Fitting the model to S&P 500 returns by GMM gives H ≈ 0.45, read as mildly sublinear

Load-bearing premise

The load-bearing premise is that the no-arbitrage and risk-neutral results hold in the enlarged filtration where the gamma activity clock is observable; from observed prices alone the compensator driving the drift condition cannot be computed, because the activity time is latent — a gap the paper states in Section 4.3 and leaves open.

Editorial extensions

If this is right

fBm's long-range dependence (H > 1/2), roughness (small H), and anomalous diffusion can be embedded in a stock price process that still satisfies the fundamental theorem of asset pricing — no restricted trading strategies, transaction costs, or redefined arbitrage needed.
Under the fVG model, log returns decompose into risk-free accrual, a jump-risk premium, an endogenous drift from the dependence structure, and a martingale innovation; persistence therefore affects expected returns, with H > 1/2 implying positively autocorrelated returns and H < 1/2 anti-persistent dynamics.
Risk-neutral option prices can be computed as expectations under the statistical measure using the simulated conditional distribution of X together with the Girsanov likelihood factor; if the market price of jump risk is independent of jump size, the pricing measure is fixed by the estimated physical dynamics alone.
Estimated on daily S&P 500 returns, the full fVG specification gives H ≈ 0.45, implying that the term structure of return moments scales mildly sublinearly; the authors argue that apparent long memory in simpler models may partly reflect unmodeled heavy tails.
The fVG and time-changed fractional geometric Brownian motion specifications are equivalent up to the jump-amplitude transformation x ↦ ln(1+x), so the valuation framework covers both formulations.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

The construction is portable: any continuous Gaussian process run on a finite-variation subordinator becomes a pure-jump semimartingale with the same compensator machinery, so the result plausibly extends to multifractional or fractional Ornstein–Uhlenbeck drivers, not just fBm.
The paper's own Section 4.3 concedes that the compensator A_X is not adapted to the natural filtration of observed prices, because the activity time is latent; closing that gap would require a filtering or nonparametric method that recovers γ (or its conditional law) from the path of X — without it, the EMM drift condition is not directly implementable from market data.
A testable reading of H ≈ 0.45: compute separate tail-index estimates on the same S&P 500 returns and check whether correcting for heavy tails genuinely moves the Hurst estimate toward the Brownian benchmark — currently that attribution rests on in-sample model comparison.
Because the paper leaves option pricing to future research, the nearest decisive test is whether the fVG's implied volatility surface — smile persistence across maturity generated by H < 1/2 and mixture kurtosis — is observed; a mismatch would localize which of the five parameters must absorb the misspecification.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

What is genuinely new here is the packaging: taking the already-known time-changed fBm B_H(gamma), proving it is a semimartingale via bounded variation, and then building the whole apparatus—compensator, Girsanov, EMM drift conditions, marked-point representation, and a GMM estimation recipe—around it. That financial wrapper is absent from the statistical papers the authors cite, so the extension is real. The moment formulas in Section 4 are clean and potentially useful, and the paper is honest about what it does not do: no option price validation, no standard errors on the GMM estimates, and a Hurst exponent that wanders between 0.35 and 0.48 depending on specification. That level of candor earns credit.

The soft spot is more serious than the paper lets on. In the proof of Proposition 4.3 (Appendix A.3), the authors assert that because A is the predictable compensator of X, A(t) = E[X(t)|F(t-)] on pure-jump processes. That is false for any process with predictable jump times, and even for a Poisson process the identity fails—the left side is deterministic, the right side is N(t-) plus an infinitesimal. The square-integrability bound on E[A^2] therefore does not go through, and X-A is only established as a local martingale, not a true martingale. The Girsanov corollary, the sufficient conditions in Proposition 5.3, and the drift decomposition in (25) all lean on the true-martingale status. This is a load-bearing gap, but it looks repairable: the finite-variation structure may allow a direct L^2 bound, or one can cite existing results on compensators of subordinated processes.

Separately, Section 4.3 concedes that the compensator is not adapted to the natural filtration of X, so the EMM conditions are not directly implementable from observed prices. That is a known limitation of latent-activity-time models, but it undercuts the claim that the framework is operationally compatible with classical observable-market pricing. The empirical exercise is illustrative only, which the authors themselves state.

Bottom line: the semimartingale construction is sound, the model is economically motivated, and the authors flag the right caveats. But the central martingale proof has a concrete error, and the non-adaptivity issue needs a cleaner resolution before the arbitrage-free machinery is fully trustworthy. Worth sending to referees—they can push on the proof and the operational gap. I would not cite it as-is, but I would put it on a reading list for fractional-model people.

Referee Report

4 major / 4 minor

Summary. The paper introduces a time-changed fractional Brownian motion X(t)=B_H(γ(t)), where γ is an independent gamma activity time, and claims that this process is a semimartingale while retaining fBm-like roughness, long memory, and anomalous diffusion. It develops a marked-point-process representation, derives an explicit compensator A_X, states a Girsanov theorem, and uses these to formulate no-arbitrage drift conditions and an EMM existence criterion. On this basis it constructs the fractional Variance Gamma (fVG) model S(t)=S(0)exp{Ξ(t)+θγ(t)+σB_H(γ(t))}, obtains closed-form unconditional moment conditions, proposes a two-step feasible GMM estimator, and illustrates the procedure on ten years of daily S&P 500 returns, reporting H≈0.45.

Significance. If the technical gaps are repaired, the paper would be a useful contribution: it offers a way to keep fractional features in a tradable semimartingale price process, extends the VG model with a Hurst exponent, and provides closed-form moment conditions plus a simulation-based estimation and valuation strategy. Theorem 4.2 is plausible and rests on a bounded-variation argument for the composed process, and the moment formulas (Propositions 4.1 and 4.5, Corollary 4.6) are concrete and potentially useful. The paper is also honest that the conditional law and the pricing-performance evaluation are not yet fully developed. However, several load-bearing parts of the theoretical apparatus are not correctly established as written, and the acknowledged non-adaptivity of the compensator to the observable price filtration leaves the central claim of compatibility with the classical arbitrage-free pricing framework incomplete.

major comments (4)

[Section 4.2, Eq. (14) and definitions of m_B, σ_B] The compensator density is defined as ψ_X(t,x)=∫ φ(x;γ(t),g)ψ_γ(g)dg, with m_B(g;γ(t))=E[B_H(γ(t)+g)−B_H(γ(t))|F(t−)]=∫_0^{γ(t)} K_H(γ(t)+g,u)dB(u). For a jump of the gamma activity time at t, the pre-jump level is γ(t−), and the jump size is B_H(γ(t−)+g)−B_H(γ(t−)). The conditional distribution given F(t−) must be based on γ(t−), not on γ(t), and the integral for m_B should run to γ(t−) with kernel K_H(γ(t−)+g,u). As written, the integrand is not F(t−)-measurable and the displayed object is not a predictable compensator density. This affects Proposition 4.3 and everything downstream that uses ψ_X. Please correct the notation and derivation, or explain why γ(t) is intended to mean γ(t−) throughout.
[Appendix A.3 / Proposition 4.3] The proof that X−A is a true P-martingale uses the assertion 'Since A is the predictable compensator of X, A(t)=E[X(t)|F(t−)]'. This identity is false for pure-jump processes: for a Poisson process N with compensator βt, E[N(t)|F(t−)]=N(t−), not βt. The compensator is a dual predictable projection, not an optional projection. Consequently the Jensen bound on E[A(t)^2] does not establish square-integrability of A, and the argument elevating the local martingale to a true martingale is invalid. The true-martingale property is load-bearing for Corollary 4.4, Proposition 5.2/5.3, and the decomposition in Eq. (25). The statement may be salvageable by instead proving E[∫ x^2 ν_X(ds,dx)]<∞, using the gamma Lévy density and the Hölder continuity of fBm paths, but that proof is not what is in the manuscript.
[Section 4.3, final paragraph; Propositions 5.2–5.3] The paper acknowledges that A_X is not adapted to the natural filtration F^X generated by X, because it depends on the latent activity time γ and on conditional moments m_B(g;γ(s)). Yet the EMM construction and drift conditions are then stated for this A_X. If the filtration is enlarged to include γ, the discounted stock price is a semimartingale relative to F, but admissible trading strategies in the classical no-arbitrage framework are predictable with respect to the observable price filtration; the drift in Eq. (22) cannot be evaluated from observed prices, and Eq. (25) is not operational. The central claim of compatibility with classical observable-market arbitrage-free pricing therefore remains incomplete. The manuscript should either compute the compensator under F^X, or explicitly present the framework as a latent-factor/partially observed model and state the resulting pricing and
[Proposition 5.3, condition 4] Condition 4 states the arbitrage-free drift restriction as r(t)=ξ(t)+∫(e^x−1)λ(t,x)ψ_X(t,x)dx. Proposition 5.2 and Eq. (22) correctly use ψ_Y, the compensator density of the innovation Y. The fVG specification used in the paper is Y=W=θγ+σX, whose jump compensator includes the θγ component and is not ψ_X. As stated, the sufficient condition in Proposition 5.3 is not the correct condition for the fVG model unless Y is taken to be exactly X. Please replace ψ_X by ψ_Y throughout the proposition and either derive ψ_W for the W specification or clarify that the proposition applies only to the generic pure-jump innovation Y of Section 5.1.

minor comments (4)

[Proposition 4.5, Eq. (16)] The gamma argument in the raw moment formula contains an undefined symbol: Γ(h/v + m − 2k(1−H)) should almost certainly be Γ(h/v + n − 2k(1−H)). Please correct.
[Example 3.1] The jump measure is written as Σ_{s: ΔY(s)=1} δ_{(s,1)}, but in the example the process is N. Replace ΔY with ΔN.
[Section 7, Table 1] The claim that deviations from H=0.5 are 'significant and economically meaningful' is not supported by any standard errors, confidence intervals, or specification test. Given the instability of H across p and across restricted specifications (H ranges from 0.35 to 0.48), please present the empirical section as descriptive and qualify the abstract's 'estimated H≈0.45' accordingly.
[Abstract and Section 4.1] The phrase 'retains the defining properties of fBm' is too strong: the time-changed process is not self-similar and is not Gaussian. It would be more precise to say it preserves selected distributional features such as roughness, moment-scaling behavior, and long-range dependence in the relevant parameter regime.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the semimartingale theorem rests on external results and the empirical analysis is in-sample GMM fitting, not disguised prediction.

full rationale

The central claim that X = B_H(γ) is a semimartingale is supported by Theorem 4.2's proof using Yor (2007) for bounded variation and Protter (2005) for finite-variation processes being semimartingales; neither input is the conclusion, and neither is a self-citation. Proposition 4.3 constructs A_X from the marked-point-process compensator, and the claim that X − A_X is a true martingale is an independent mathematical assertion, although its proof in Appendix A.3 contains a technical gap: it uses the identity A(t) = E[X(t) | F(t−)] for a compensator, which is not generally valid for pure-jump processes (e.g., a Poisson process has E[N(t)|F(t−)] = N(t−), not λt). That gap is a correctness concern, not a circular reduction. The moment formulas in Propositions 4.1 and 4.5 are derived from the Gaussian–Gamma mixture and then used in a GMM objective to estimate parameters from S&P 500 data; this is standard in-sample estimation. The paper does not call the estimated Hurst exponent a prediction or forecast, and it does not use the same data to both fit and validate an out-of-sample claim. The only self-citations (Jarrow 2021; Tan & Jarrow 2026; Tan 2026) are background or positioning and are not load-bearing. I therefore find no significant circularity.

Assumptions & free parameters 5 free parameters · 6 assumptions · 0 invented entities

The paper introduces no new physical or economic entities. The gamma activity time is a pre-existing construct from the VG literature (Madan et al. 1998) and is treated as an unobservable latent variable, which is the source of the main modeling difficulty. The five GMM parameters are the model's free parameters, each fitted to the data.

free parameters (5)

ξ (log-return drift) = 0.1048 to 1.0125 depending on restriction
Estimated via GMM from first moment conditions; not specified a priori.
θ (skewness parameter) = -0.9077 to -0.1851 across specifications
Estimated via GMM from third moment conditions; controls asymmetry.
σ (volatility scale) = 0.1113 to 0.1451
Estimated via GMM from second moment conditions.
v (gamma variance intensity) = 0.0037 to 0.0849
Estimated via GMM; governs kurtosis and activity time variability.
H (Hurst exponent) = 0.3491 to 0.4776 across specs; about 0.45 in preferred row
Estimated via GMM from moment scaling across lags; the paper's headline empirical finding.

assumptions (6)

domain assumption Independence of fBm B_H and gamma process γ
Section 4 states B_H and γ are assumed independent; required for the Gaussian-Gamma mixture representation and all moment formulas.
domain assumption Gamma process has unit mean rate and variance v
Section 3.2 normalizes the mean activity rate to 1, a modeling choice without independent economic justification.
domain assumption Market is frictionless with continuous trading and satisfies the usual hypotheses
Section 5.1 uses the standard continuous-time, continuous-trading no-arbitrage setup.
standard math Volterra kernel representation and Girsanov theorem for fBm (Decreusefond and Üstünel)
Section 3.3 invokes these as background results for fBm properties and measure change.
standard math Yor (2007) Proposition 2: gamma-time-changed continuous process has bounded variation on compacts
Appendix A.2 relies on this external theorem to prove semimartingale property of X.
domain assumption No-arbitrage and EMM results are formulated w.r.t. the enlarged filtration where γ is observable
Section 4.3 admits A_X is not adapted to the natural filtration of X; the paper does not provide the analogous pricing framework in the observable market filtration.

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Cite this review

Pith. "Pith review of Option Pricing with Time-Changed Fractional Brownian Motion: A Fractional Variance Gamma Model." pith.science (2026). https://pith.science/paper/X4ZP5IWO

@misc{pith2026260803925,
  author       = {Pith},
  title        = {Pith review of: Option Pricing with Time-Changed Fractional Brownian Motion: A Fractional Variance Gamma Model},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/X4ZP5IWO}},
  note         = {Machine review of arXiv:2608.03925}
}

read the original abstract

Fractional Brownian motion (fBm) exhibits attractive features for financial modeling, including long-range dependence, path roughness, and anomalous diffusion. However, its non-semimartingale nature precludes the use of conventional no-arbitrage approaches to option pricing. We address this limitation by introducing a time-changed fBm, obtained by evaluating fBm at stochastic gamma activity time, where activity time represents cumulative executed trading time. The resulting process retains the defining properties of fBm while recovering the semimartingale structure. Building on this construction, we develop the fractional Variance Gamma (fVG) model and propose a generalized method of moments (GMM) estimation procedure for option pricing. An empirical analysis of the S\&P 500 yields an estimated Hurst exponent of approximately 0.45, consistent with mildly sublinear temporal scaling of return moments.

Figures

Figures reproduced from arXiv: 2608.03925 by the authors.

**Figure 1.** Autocorrelation of the increments in X and PDF of X across various v, H effect on the shape of the PDF. This suggests that v is the primary parameter governing the kurtosis of the time-changed fBm process. Next, we derive the unconditional moments of the time-changed fBm process in Proposition 4.1. Proposition 4.1 (Unconditional raw moments of X). For any n ∈ N, E[B H(γ(t))n ] = ( v nH Γ(t/v+nH) Γ(t/v) (n − 1)!!, n … view at source ↗

**Figure 2.** Simulated sample paths of X under various v, H [PITH_FULL_IMAGE:figures/full_fig_p013_2.png] view at source ↗

**Figure 3.** Moments of the increments in W given in Corollary 4.6 across parameters {H, v, θ}. 18 [PITH_FULL_IMAGE:figures/full_fig_p018_3.png] view at source ↗

Discussion (0). Continue with ORCID to comment.

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