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Entropic Dynamics of Jump-Diffusion Option Pricing

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Pith's one-line read Maximum Entropy Derives Jump-Diffusion Pricing From Scratch

desk verdict Solid entropic derivation of Merton jump-diffusion; Esscher-transform 'derivation' is the one overstatement read the letter →

arxiv 2607.06355 v1 pith:ES6VQ2HU submitted 2026-07-07 q-fin.PR q-fin.MFq-fin.ST

Mohammad Abedi This is my paper · ORCID
classification q-fin.PRq-fin.MFq-fin.ST
keywords MaximumEntropyEntropicDynamicsJump-DiffusionEsscherTransformOptionPricingImpliedVolatilitySmileKolmogorov-FellerEquationIncompleteMarkets
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The pith

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

The reading

The paper claims that the Merton jump-diffusion process, the Kolmogorov-Feller forward equation, the Esscher transform for risk-neutral pricing, Merton's option-pricing equation, and the implied-volatility smile are not assumptions but outputs of a single Maximum Entropy inference procedure. The argument begins from a symmetry: markets reward returns rather than price levels, which forces the logarithm of price to be the dynamical variable. Five constraints are then imposed on this log-price microstate — two on the continuous channel (continuity of motion, directionality of drift) and three on the jump channel (arrival rate, mean jump size, second moment of jump size). Because these constraints act on disjoint parts of the microstate, the joint Maximum Entropy distribution factorizes into independent channel-specific factors as a theorem, not an assumption. The resulting transition density is the Poisson-weighted mixture of Gaussians that defines the Merton jump-diffusion, with Geometric Brownian Motion recovered when the jump intensity is set to zero. The forward equation for the log-price density is the Kolmogorov-Feller equation, reducing to the Fokker-Planck equation in the no-jump limit. On the pricing side, the incomplete-market non-uniqueness of the equivalent martingale measure is reinterpreted as the non-uniqueness of the observable through which no-arbitrage is imposed. Imposing no-arbitrage through the mean log-return and maximizing relative entropy from the jump-diffusion measure yields an exponential tilt — the Esscher transform — derived rather than imported. The option premium satisfies Merton's partial integro-differential equation, and the Poisson-weighted mixture of lognormals generates positive excess kurtosis and, for downward mean jumps, negative skewness, producing the implied-volatility smile and smirk. Black-Scholes results return in the no-jump limit. The paper's methodological thesis is that what changes from one model to another is never the inference engine but the information supplied to it as constraints.

What carries the argument

Maximum relative entropy updating with modular constraints; return symmetry selecting log-price as dynamical variable; factorization of joint posterior from disjoint constraints; Esscher transform as the MaxEnt tilt under mean-log-return constraint; Kolmogorov-Feller equation as the continuum limit of the entropic-instant propagation.

What would settle it

If empirical option prices were found to be better described by the minimum-entropy martingale measure (gross-return constraint) than by the Esscher measure (log-return constraint) across a range of underlyings and maturities, the claim that the Esscher transform is the natural output of the framework would be weakened.

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Extended reading notes

Core claim

The central object is the augmented microstate (ln S', n), the pair of the next log-price and the number of jump arrivals over a short step. Five constraints on disjoint parts of this microstate — continuity and directionality for the continuous displacement, arrival rate and the first two moments for the jump — force the joint Maximum Entropy distribution to factorize into a product of a Bernoulli/Poisson count factor, a Gaussian diffusion factor, and a Gaussian jump-size factor. This factorization is a theorem of the constraint structure, not an independence assumption. The marginalized transition density is the Merton jump-diffusion, and the same inferential principle, extended to pricing

Load-bearing premise

The paper claims to 'derive' the Esscher transform rather than borrow it, but this derivation depends on choosing to impose the no-arbitrage condition through the mean log-return rather than the gross return. The paper itself acknowledges that imposing no-arbitrage through the gross return instead yields a different measure (the minimum-entropy martingale measure). The selection of the log-return as the controlled observable is a motivated modeling choice, not something the推理

Editorial extensions

If this is right

  • Other Lévy-family processes (variance gamma, normal inverse Gaussian, CGMY, Kou's double-exponential) should be reachable by changing only the jump-size constraints while leaving the inference machinery untouched, making the framework a unified derivation scheme for an entire class of asset-price models.
  • Stochastic volatility, rough volatility, and path-dependent volatility can be incorporated by conditioning the volatility constraint on the history of the price path, turning non-Markovian dynamics into a constraint-design problem rather than a new modeling paradigm.
  • The reinterpretation of market incompleteness as the non-uniqueness of the no-arbitrage constraint provides a principled taxonomy of pricing measures: each choice of controlled observable generates one member of the equivalent martingale measure family, with the Esscher and minimum-entropy measures as two instances.
  • The framework's modularity suggests that jump clustering (via self-exciting intensities) can be added as a constraint coupling the arrival rate to the past count, extending the Bernoulli short-step construction to a Hawkes-type process without changing the inferential foundation.

Reading between the lines

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

  • If the factorization theorem holds generally for disjoint constraints, then any multi-channel price model — for instance one combining stochastic volatility with jumps — could be derived by specifying constraints for each channel and reading off the factorized posterior, potentially unifying the Bates stochastic-volatility-jump model within the same framework.
  • The claim that diffusion is a limiting case of the jump channel (frequent, infinitesimal jumps converging to a Wiener process) suggests that the continuous and jump channels are not fundamentally distinct mechanisms but different regimes of a single inferential structure, which could simplify the taxonomy of asset-price models.
  • If the entropic-clock construction can carry memory (as the paper hints for rough volatility), then the Hurst parameter of rough volatility might be derivable as a property of the clock's temporal correlations rather than postulated, offering an inferential origin for the rough-volatility exponent.
  • The framework's treatment of the Esscher transform as contingent on the choice of log-return as the controlled observable raises the question of whether empirical option prices could be used to infer which constraint the market is effectively imposing, turning the pricing-measure selection problem into a testable identification problem.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

1 major / 5 minor

Summary. The manuscript develops an entropic-inference framework that derives the Merton jump-diffusion process, the Kolmogorov-Feller forward equation, the Esscher risk-neutral measure, Merton's PIDE, and the implied-volatility smile as outputs of a single Maximum Entropy procedure applied to the log-price microstate with five constraints (continuity, directionality, arrival rate, first two jump moments). The factorization of the joint posterior into independent continuous and jump channels is presented as a theorem following from the disjoint constraint structure. The Poisson distribution is derived from Bernoulli short-step accumulation (Appendix A), and the Kolmogorov-Feller equation is derived step-by-step from the entropic-instant identity (Appendix B). The Esscher transform is obtained as the MaxEnt update of the jump-diffusion measure subject to a mean-log-return constraint through which no-arbitrage is imposed.

Significance. The paper's central methodological contribution is the unified derivation of both dynamics and pricing from a single inferential principle, which is a meaningful reframing of results usually obtained by postulate. The factorization theorem (Section 2.3.2) is a genuine structural result: independence of diffusion and jump channels follows from the constraint structure, not from assumption. The derivation of the Poisson distribution from Bernoulli short-step accumulation (Appendix A) is self-contained and correct. The Kolmogorov-Feller derivation (Appendix B) is standard and properly executed. The cumulant analysis (Section 3.3, Eqs. 76-82) provides explicit, falsifiable predictions linking the smile geometry to jump parameters. The framework's constraint-modularity is clearly articulated, and the limiting cases (GBM when lambda=0, pure-jump when sigma=0) are verified throughout.

major comments (1)
  1. Section 3.1, final paragraph, and Abstract: The claim that the Esscher transform is 'derived rather than borrowed' is the paper's strongest framing, but it overstates what the inference framework alone delivers. The paper itself transparently acknowledges that imposing no-arbitrage through the gross return <e^x> instead of the mean log-return <x> yields the minimum-entropy martingale measure — a different, equally arbitrage-free pricing measure that does not preserve the jump-diffusion family. The choice of log-return as the controlled observable is motivated (it is the variable on which the dynamics are already built, so 'no new dynamical observable enters'), but it is a modeling choice, not an inference-theoretic necessity. The structural-preservation argument (Eqs. 46-49: the log-return tilt factorizes across channels while the gross-return tilt would not) is a genuine convenience/adv
minor comments (5)
  1. Section 2.3.1: The short-step binary count assumption (n in {0,1}) is listed as the sole ad-hoc axiom. The paper argues this is 'the exact leading-order content of the arrival process,' but the phrase 'simultaneous arrivals are excluded by the same logic that restricts the continuous channel to infinitesimally small steps' could be stated more precisely — the continuous channel is restricted by the continuity constraint (Eq. 10), not by a logic of exclusion.
  2. Eq. (50): The normalizer Z(theta) is written with exp[lambda*delta_t*(M_J(theta)-1)], but the Poisson accumulation over a finite horizon T would give exp[lambda*T*(M_J(theta)-1)]. The appearance of delta_t rather than T should be clarified — is this the short-step normalizer or the finite-horizon one?
  3. Section 3.3, Eq. (82): The expansion of implied volatility in standardized cumulants is presented without a reference. This is a known result (e.g., Backus et al. 1997, Cont & Tankov 2004 Ch. 11); a citation would strengthen the presentation.
  4. References [9] and [10] appear to cite the same arXiv preprint (1803.07493) with slightly different formatting and titles.
  5. The abstract and introduction use the phrase 'derived rather than borrowed' (or 'derived rather than assumed') repeatedly. Toning down this rhetoric would better match the qualified sense in which the derivation holds, as acknowledged in the final paragraph of Section 3.1.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for a careful and constructive reading of the manuscript. The recommendation of minor revision is well taken, and we address the major comment below.

read point-by-point responses

  1. Referee: The claim that the Esscher transform is 'derived rather than borrowed' overstates what the inference framework alone delivers. The choice of log-return as the controlled observable is a modeling choice, not an inference-theoretic necessity. The structural-preservation argument (Eqs. 46-49) is a genuine convenience/advantage but does not make the Esscher transform the unique output of the inference.

    Authors: We accept the substance of this comment. The referee is correct that the inference framework alone does not single out the Esscher transform: the framework delivers a family of martingale measures indexed by the choice of controlled observable, and the log-return constraint is a modeling choice, not an inference-theoretic necessity. The manuscript already acknowledges this transparently in Section 3.1, where we note that imposing no-arbitrage through the gross return would yield the minimum-entropy martingale measure — a different, equally arbitrage-free pricing measure — and where we state explicitly that 'the non-uniqueness of the equivalent martingale measure is the non-uniqueness of the informational constraint through which no-arbitrage is imposed.' However, the referee is right that the phrase 'derived rather than borrowed,' as used in the abstract and in Section 3.1, can be read as claiming more than the framework delivers — namely, that the Esscher transform is the unique inference-theoretic output, rather than the output corresponding to one principled (but not forced) choice of constraint. We will revise the wording to make clear that what is derived is the Esscher transform as the MaxEnt update under the log-return constraint, and that this constraint is a modeling choice motivated by the fact that the log return is the observable on which the dynamics are already built — so that no new dynamical observable enters — rather than an inference-theoretic necessity. The structural-preservation property (Eqs. 46-49) will be presented as a genuine advantage of this choice, not as a proof of its uniqueness. We believe these revisions preserve the paper's contribution — showing that the Esscher transform emerges as the MaxEnt output of a well-motivated constraint —hmm revision: yes

Circularity Check

0 steps flagged · score 2.0 of 10

No significant circularity; derivation is self-contained with transparent modeling choices

full rationale

The paper's derivation chain is largely self-contained. The dynamics (Sections 2.2–2.3) genuinely derive the Merton jump-diffusion from five MaxEnt constraints on disjoint microstate components: continuity and directionality produce GBM (Eq. 16), the arrival constraint produces Bernoulli→Poisson counts (Eqs. 28–29, Appendix A), and two jump-moment constraints produce the Gaussian jump size (Eq. 32). The factorization into independent channels (Eq. 31) follows from the constraints acting on disjoint variables — a genuine theorem, not an assumption. The Kolmogorov–Feller equation (Eq. 39, Appendix B) is derived step-by-step from the entropic-instant identity (38) by Taylor expansion. The implied-volatility smile (Section 3.3) follows from the Poisson mixture of lognormals not being lognormal, with cumulants (Eqs. 76–78) computed explicitly. The Esscher transform derivation (Section 3.1) is the closest candidate for circularity: MaxEnt with a linear constraint ⟨x⟩ always produces an exponential tilt e^{θx}, which is by definition the Esscher transform. However, this is not circular — it is a genuine (if straightforward) consequence of the MaxEnt procedure applied to the chosen constraint. The paper is transparent that the constraint choice (mean log-return vs. gross return) is a modeling decision, explicitly acknowledging that the alternative yields the minimum-entropy martingale measure instead (final paragraph of Section 3.1). The self-citations [44, 45] (Abedi & Bartolomeo) provide context for the continuous-only predecessor model but are not load-bearing: the continuous channel is re-derived from scratch in Section 2.2. No step reduces to its inputs by construction in a way that would constitute circularity.

Assumptions & free parameters 6 free parameters · 4 assumptions · 0 invented entities

The paper introduces no new physical or mathematical entities. All objects (log price, jump count, jump size, Esscher parameter theta) are standard. The free parameters are the standard inputs of the Merton model, here framed as constraint values rather than fitted constants. The axioms are a mix of standard MaxEnt methodology, standard finance assumptions (no-arbitrage, return symmetry), and a short-step binary approximation.

free parameters (6)
  • sigma (volatility)
    Diffusion volatility, identified through the entropic clock k = sigma^2 * dt. Treated as an input parameter.
  • mu_c (expected rate of return)
    Continuous drift rate, input to the directionality constraint.
  • lambda (jump intensity)
    Arrival rate of jumps, input to the arrival constraint.
  • kappa (mean jump size)
    Mean of the log-price jump Y, input to the first jump-size constraint.
  • delta^2 (jump size variance)
    Variance of the log-price jump Y, derived from the first two moment constraints (nu - kappa^2).
  • r_f (risk-free rate)
    Risk-free rate, external input for the no-arbitrage condition.
assumptions (4)
  • domain assumption Return symmetry: markets reward returns, not price levels, so the transition probability is invariant under uniform rescaling of prices.
    Section 2.1. Selects log price as the dynamical variable. Standard in finance but is a foundational assumption here.
  • standard math Principle of Maximum Entropy (relative entropy maximization).
    Sections 2.2, 2.3, 3.1. The core inferential tool, following Shore-Johnson axioms [36].
  • domain assumption No-arbitrage principle (Fundamental Theorem of Asset Pricing).
    Section 3.1. Standard finance axiom used to select the risk-neutral measure.
  • ad hoc to paper Short-step binary count: at most one jump occurs per infinitesimal interval.
    Section 2.3.1. Justified by the logic of infinitesimal steps but is a modeling simplification that leads to the Bernoulli-to-Poisson accumulation.

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Pith. "Pith review of Entropic Dynamics of Jump-Diffusion Option Pricing." pith.science (2026). https://pith.science/paper/ES6VQ2HU

@misc{pith2026260706355,
  author       = {Pith},
  title        = {Pith review of: Entropic Dynamics of Jump-Diffusion Option Pricing},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/ES6VQ2HU}},
  note         = {Machine review of arXiv:2607.06355}
}
read the original abstract

Standard models of stock price dynamics and option valuation usually begin by postulating stochastic processes. This paper develops an entropic inference framework that derives these processes from information constraints. The key symmetry is that markets reward returns rather than price levels, which selects log price as the dynamical variable. Price changes are represented by two channels. The continuous channel carries constraints of continuity and directionality. The jump channel carries the arrival rate and the first two moments of jump size. Since these constraints apply to disjoint parts of the microstate, the channels factorize. The resulting dynamics is the Merton jump diffusion, with Geometric Brownian Motion as the no jump limit. The log price density satisfies the Kolmogorov Feller equation, whose no jump limit is the Fokker Planck equation. The same inferential principle, with no arbitrage imposed through the mean log return, selects the Esscher transform from the many martingale measures available in an incomplete market. The option price then satisfies Mertons partial integro differential equation, and the risk neutral mixture of lognormal distributions generates the implied volatility smile. The Black Scholes results are recovered when jumps vanish. What changes from one model to another is not the inference, but the information supplied to it.

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