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arxiv: 2606.22796 · v1 · pith:TDHQUL4Znew · submitted 2026-06-22 · 💱 q-fin.MF

Enhancing the Black-Scholes Model for Option Valuation via L\'evy Processes and Malliavin Calculus

Shantanu Awasthi , Minglian Lin , Blair Faber , Michael Roberts , Hassan Butt This is my paper

Pith reviewed 2026-06-26 06:07 UTC · model grok-4.3

classification 💱 q-fin.MF
keywords Black-Scholes modelLévy processesMalliavin calculusoption pricingstochastic volatilityvolatility smileVIX indeximplied volatility
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The pith

Incorporating Lévy processes for jumps and Malliavin calculus for volatility into Black-Scholes improves capture of volatility smiles and VIX data.

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

The paper extends the Black-Scholes model by replacing its geometric Brownian motion assumption with a Lévy process that includes jumps and stochastic volatility. Using multidimensional Itô calculus, it derives a pricing formula for European call options. Malliavin calculus then provides an exact expression for at-the-money implied volatility. Analysis shows this version matches empirical volatility smiles and VIX observations better than the original model. This matters because the classic model struggles with real market features like varying volatility and sudden jumps.

Core claim

The enhanced model incorporates stochastic volatility with jumps modeled by a Lévy process. Leveraging multidimensional Itô calculus, a pricing formula for European call options is derived under the new framework. Additionally, Malliavin calculus enables the derivation of an exact expression for at-the-money implied volatility. The proposed model better captures empirical features like volatility smiles, and analysis of VIX data demonstrates its ability to match observed market volatility.

What carries the argument

The combination of a Lévy process for modeling jumps in the asset price dynamics and Malliavin calculus to obtain an exact at-the-money implied volatility formula.

Load-bearing premise

The specific Lévy process chosen and the Malliavin-derived volatility expression remain tractable and empirically superior once calibrated, without the calibration itself being the source of the reported fit.

What would settle it

A direct comparison of the model's predicted volatility smile and VIX match against market data in a period or asset class not used for calibration would falsify the claim if the fit degrades significantly.

Figures

Figures reproduced from arXiv: 2606.22796 by Blair Faber, Hassan Butt, Michael Roberts, Minglian Lin, Shantanu Awasthi.

Figure 1
Figure 1. Figure 1: ATM implied volatility proxy. Market proxy is [PITH_FULL_IMAGE:figures/full_fig_p014_1.png] view at source ↗
read the original abstract

The Black-Scholes model has been extensively used for option pricing, but exhibits limitations in its reliance on geometric Brownian motion and fixed volatility assumptions. This paper proposes an enhanced model incorporating stochastic volatility with jumps modeled by a L\'evy process. Leveraging multidimensional It\^o calculus, we derive a pricing formula for European call options under the new framework. Additionally, Malliavin calculus enables the derivation of an exact expression for at-the-money implied volatility. The proposed model is shown to better capture empirical features like volatility smiles. Analysis of VIX data demonstrates the model's ability to match observed market volatility. The integration of L\'evy processes and Malliavin calculus represents a valuable advancement in addressing deficiencies in the classic Black-Scholes model. Further empirical testing is warranted to validate the approach across varying market conditions and option types.

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

3 major / 0 minor

Summary. The paper proposes an extension of the Black-Scholes model that incorporates stochastic volatility driven by a Lévy process for jumps. It claims to derive a closed-form pricing formula for European calls via multidimensional Itô calculus and an exact expression for at-the-money implied volatility via Malliavin calculus. The manuscript further asserts that the resulting model better reproduces volatility smiles and matches observed VIX dynamics.

Significance. If the Itô and Malliavin derivations are rigorous, the pricing formula is explicit, and the VIX match is demonstrated with a fully specified Lévy triplet, transparent calibration, and benchmark comparisons (e.g., against Heston or standard Lévy models) on held-out data, the work would supply a concrete analytic advance in Lévy-augmented option pricing. The absence of these elements in the current manuscript prevents any such assessment.

major comments (3)
  1. [Abstract] Abstract: the claim that 'Analysis of VIX data demonstrates the model's ability to match observed market volatility' supplies neither the Lévy triplet (or measure), the objective function and constraints used for calibration, any error metric, nor an out-of-sample or benchmark comparison. Without these, the reported agreement cannot be distinguished from a fit driven by the added degrees of freedom in the jump component.
  2. [Abstract] Abstract: the statement that the model 'better capture[s] empirical features like volatility smiles' is unsupported by any figure, table, or quantitative comparison to Black-Scholes or other benchmarks; the central empirical claim therefore rests on an unverified assertion rather than evidence.
  3. [Abstract] The pricing formula and Malliavin-derived ATM volatility expression are asserted to have been derived, yet the manuscript provides no explicit equations, no statement of the underlying SDE, and no verification that the Malliavin expression remains tractable once the Lévy measure is introduced. These derivations are load-bearing for the paper's analytic contribution but are not shown.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for the detailed and constructive comments. We agree that several claims in the abstract require stronger empirical support and that the analytic contributions should be presented more explicitly. Below we respond point-by-point and indicate the revisions we will make.

read point-by-point responses

  1. Referee: [Abstract] Abstract: the claim that 'Analysis of VIX data demonstrates the model's ability to match observed market volatility' supplies neither the Lévy triplet (or measure), the objective function and constraints used for calibration, any error metric, nor an out-of-sample or benchmark comparison. Without these, the reported agreement cannot be distinguished from a fit driven by the added degrees of freedom in the jump component.

    Authors: We agree that the abstract claim is stated too strongly without supporting details. The current VIX analysis is preliminary and lacks the requested specification. In the revised manuscript we will either remove the claim from the abstract or add a dedicated calibration subsection that reports the Lévy triplet, objective function, constraints, error metrics, and benchmark comparisons (including Heston and standard Lévy models) on held-out data. revision: yes

  2. Referee: [Abstract] Abstract: the statement that the model 'better capture[s] empirical features like volatility smiles' is unsupported by any figure, table, or quantitative comparison to Black-Scholes or other benchmarks; the central empirical claim therefore rests on an unverified assertion rather than evidence.

    Authors: The referee is correct that no quantitative evidence for the volatility-smile claim is supplied. The assertion follows from the model structure but is not demonstrated. We will add figures and tables in the revision that compare model-generated implied-volatility smiles against Black-Scholes, market data, and at least one benchmark model, together with quantitative error metrics. revision: yes

  3. Referee: [Abstract] The pricing formula and Malliavin-derived ATM volatility expression are asserted to have been derived, yet the manuscript provides no explicit equations, no statement of the underlying SDE, and no verification that the Malliavin expression remains tractable once the Lévy measure is introduced. These derivations are load-bearing for the paper's analytic contribution but are not shown.

    Authors: The referee correctly observes that the abstract itself contains no explicit equations. The underlying SDE is stated in Section 2 and the pricing formula together with the Malliavin ATM-volatility expression are derived in Sections 3 and 4 using multidimensional Itô and Malliavin calculus. To address the concern we will insert the key SDE and the final closed-form expressions into the abstract (or a new “Main Results” paragraph) and explicitly verify tractability for the chosen Lévy measure. revision: partial

Circularity Check

0 steps flagged

No circularity: derivations use standard Itô/Malliavin tools; empirical claim lacks quoted reduction to fit.

full rationale

The abstract describes deriving a European call pricing formula via multidimensional Itô calculus and an ATM implied-volatility expression via Malliavin calculus; these are presented as analytic steps from the Lévy-augmented SDE. No equations are supplied that equate a derived quantity back to a fitted parameter by construction. The VIX-matching statement is phrased as a demonstration but supplies no calibration details, objective function, or in-sample/out-of-sample distinction that would allow identification of a fitted-input-called-prediction step. Because the provided text contains neither self-citations that are load-bearing nor any explicit reduction of a claimed prediction to its own inputs, the derivation chain remains self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Only the abstract is available; no explicit free parameters, axioms, or invented entities can be extracted or audited.

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