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REVIEW 4 major objections 6 minor 36 references

Entropic Dynamics of Stocks and European Options

T0 review · 4 major / 6 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read This paper derives geometric Brownian motion and Black–Scholes-Merton pricing from entropic inference alone.

desk verdict A clean MaxEnt reformulation of GBM and Black-Scholes that is honest about its assumptions; the derivation is a repackaging rather than a first-principles result, but it is a useful pedagogical piece for the entropic inference community. read the letter →

arxiv 1908.06355 v1 pith:ACTQOX5B submitted 2019-08-18 q-fin.PR

Mohammad Abedi , Daniel Bartolomeo This is my paper · ORCID
classification q-fin.PR MSC 91G2091G80
keywords maximumentropymethodentropicdynamicsgeometricBrownianmotionEuropeanoptionsBlack-ScholesmodelBlack-Scholes-Mertonequationput-callparityrisk-neutralvaluation
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

This paper claims that the two standard workhorses of quantitative finance—the geometric Brownian motion model of stock prices and the Black–Scholes–Merton option-pricing equation—do not have to be assumed as hypotheses about the market. They can be derived from entropic inference, a maximum-entropy procedure for updating probabilities when information is incomplete. The derivation chooses the logarithm of price as the dynamical variable because of scale invariance, then extracts a Gaussian transition density from constraints of continuity and directionality. Imposing a risk-neutral drift reproduces the Black–Scholes call price and the Black–Scholes–Merton differential equation. If the derivation is correct, these models are consequences of a single inferential principle, and modifications of the constraints offer a systematic route to new pricing models.

What carries the argument

The machinery is maximum-relative-entropy inference applied in three steps. Scale invariance forces the choice $x=\ln S$ as the dynamical variable, since only the log turns a multiplicative rescaling of price into an additive shift. Continuity is encoded as a constraint $\langle(\Delta\ln S)^2\rangle=k$ that makes the prior a sharp Gaussian with precision $\alpha=1/(\sigma^2\Delta t)$; this parameter defines the entropic clock that converts inference steps into time steps. A directionality constraint $\langle\ln(S'/S)\rangle=k'$ introduces the drift, and a second-order Taylor expansion relates it to $\mu\Delta t-\tfrac{1}{2}\sigma^2\Delta t$. The same constraints with $\mu=r_f$ produce the risk-neutral measure, and the Fokker–Planck equation for the density follows from the entropic definition of an instant.

What would settle it

Estimate the conditional distribution of high-frequency log-returns for individual stocks as a function of price level. If the mean or variance of $\Delta\ln S$ depends on $S$, or if the distribution is systematically non-Gaussian, then the uniformity assumption that produces the lognormal at maturity and the Black–Scholes formula is violated in that market.

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Extended reading notes

Core claim

The central claim is that the log-price transition density is fully determined by information constraints, not by postulating a stochastic process. The paper obtains $$P(\ln S'|\ln S)=\frac{1}{Z}\exp\left[-\frac{1}{2\$sigma^{{2}}$\$\Delta$ t}\left(\ln S'-\left(\ln S+\mu\$\Delta$ t-\frac{1}{2}\$sigma^{{2}}$\$\Delta$ t\right)\right)^{2}\right],$$ so the log-price performs a Wiener process with drift $\mu\Delta t-\frac{1}{2}\sigma^{2}\Delta t$ and variance $\sigma^{2}\Delta t$, equivalent to geometric Brownian motion for the price. Setting the drift equal to the risk-free rate $r_f$ gives the risk-neutral measure, and integrating the discounted payoff yields the Black–Scholes call price $C=S_0 N(d_1)-e^{-r_f T}K N(d_2)$; differentiating the expected payoff with respect to time yields the Black–Scholes–Merton partial differential equation. The paper presents this as a derivation, not a new model: the formulas are the familiar ones, but they now follow from maximum entropy.

Load-bearing premise

The load-bearing premise is that the drift and volatility are externally supplied constants and that investors are indifferent to absolute price levels; if either fails, the derived lognormal maturity distribution and the Black–Scholes formulas no longer follow.

Editorial extensions

If this is right

  • The familiar GBM transition density is recovered: log returns are Gaussian with variance $\sigma^2\Delta t$, and prices at finite maturity are lognormal.
  • If drift or volatility are not constant in time or price, the finite-time distribution solves a Fokker–Planck equation and is no longer lognormal, so the Black–Scholes formula should be replaced by the corresponding solution.
  • The risk-neutral valuation principle is obtained by one constraint, $\mu=r_f$, not added as a separate assumption.
  • The call and put formulas satisfy put-call parity, which the paper interprets as evidence that the expected-payoff pricing is arbitrage-free.
  • The Black–Scholes–Merton PDE follows by differentiating the expected payoff with respect to time, with the backward Kolmogorov equation supplying the evolution of the transition density.

Reading between the lines

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

  • One testable extension: if realized log-returns are non-Gaussian or their variance depends on price level, the derived distribution should fail; the framework would then predict option prices from the Fokker–Planck solution rather than from Black–Scholes.
  • The same scale-invariance-plus-constraints recipe could in principle be applied to any positively priced asset, suggesting a unified entropic derivation of other market models such as foreign-exchange option pricing.
  • Because the derivation makes every input an explicit constraint, it offers a diagnostic tool: estimating the constraints from option prices could show exactly which assumed symmetries are violated in real markets.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

4 major / 6 minor

Summary. The manuscript develops an entropic-inference framework for stock-price dynamics and European option pricing. It argues that scale invariance of the formalism selects the logarithm of price as the natural dynamical variable, and then, using a continuity prior and a drift constraint, maximum entropy yields a Gaussian transition density for log returns with mean (mu - sigma^2/2) Delta t and variance sigma^2 Delta t, i.e., the transition law of geometric Brownian motion. The corresponding Fokker-Planck equation is derived. Imposing mu = r_f as the risk-neutral information, and assuming r_f and sigma are constant and price-independent, the paper obtains the lognormal terminal distribution, the Black-Scholes call and put formulas, put-call parity, and the Black-Scholes-Merton partial differential equation.

Significance. The paper is a clear and largely self-contained attempt to reconstruct standard finance models from maximum-entropy updating. Its main value is conceptual: if the derivation were fully rigorous, it would show that GBM and Black-Scholes are the least-biased models consistent with scale invariance, continuity, and externally supplied mean and variance of returns, rather than ad hoc assumptions. The authors are commendably explicit about the conditional nature of the results: they state that the drift is an input to be supplied by a separate model, and that lognormality and Black-Scholes hold only under uniformity of mu, sigma, and r_f. These disclaimers define the scope of the claim. The paper does not provide empirical tests or new pricing formulas, and it does not derive the parameters from deeper information; it is best read as a reformulation of GBM and Black-Scholes within entropic inference. Within that scope, the formulas are checkable, the Fokker-Planck equation is obtained from the transition density, and the option pricing section reproduces the standard results.

major comments (4)
  1. [Section 2.1, Eqs. (2)-(3)] The requirement that probability densities be scalars under the scale transformation is imposed rather than derived. For a genuine density, the correct transformation involves a Jacobian, so Eq. (3) does not follow from Eq. (2) unless one additionally assumes that the reference measure df is scale invariant. This is a substantive assumption that selects the logarithm as the dynamical variable and hence the lognormal family; it should be presented as an axiom of the model, not as a consequence of investor preferences.
  2. [Section 2.2.3, Eqs. (17)-(26)] The derivation of the transition density injects the GBM parameters through Eq. (19), which supplies the drift mu, and Eq. (24), which supplies the volatility sigma. Eq. (24) equates <(ln S'/S)^2> with <(Delta S/S)^2> after 'squaring the Taylor expansion,' but the squared expansion contains third- and fourth-order terms, and the equality is only valid to leading order in Delta t. More fundamentally, the constraint values in Eq. (25) are exactly the mean and variance of log-returns under GBM, so Eq. (26) follows by construction. The paper's own caveats after Eq. (31) and in Section 3.1 confirm this reading: relaxing uniformity yields a Fokker-Planck solution that is not lognormal and a Black-Scholes formula that no longer follows. The central claim in the abstract that GBM is 'derived' should therefore be qualified as a conditional MaxEnt reconstruction from externally supplied mu and sigma.
  3. [Section 3.1, Eq. (34)] The paper states that the risk-neutral measure is derived by imposing the constraint mu = r_f. This is not a derivation from no-arbitrage; no-arbitrage alone does not determine the drift of a pricing measure, and a change of measure normally requires an equivalent-martingale-measure argument. The identification mu = r_f is an additional modeling assumption and should be labeled as such, since all subsequent option-pricing results depend on it.
  4. [Section 3.2, Eq. (48)] The quantity V defined by the unbounded integral of (S_T - K) is not the price of a European call or put; for a call the payoff is (S_T - K)^+ and for a put it is (K - S_T)^+. As written, Eq. (48) is the undiscounted forward price minus the strike. The backward-Kolmogorov argument should be applied to the actual payoff function g(S_T) with appropriate boundary conditions; otherwise Eq. (52) is derived for an affine function, not for an option payoff, and the BSM equation for options is not established by the argument given.
minor comments (6)
  1. [Section 3.1, Eq. (45)] The integrand for the put payoff should be (K - S), not (S - K); as printed, V_p is negative.
  2. [Section 3.1, Eq. (40)] There is a stray comma in the integral expression 'integral dS~, P(...)'; the notation should be cleaned up.
  3. [Section 2.2.3] The text says 'We attain the Weiner process'; this should be 'Wiener process.'
  4. [Section 3.2, Eq. (53)] Equation (53) contains a misplaced comma in '1/2 sigma^2 S^2, partial^2 E / partial S^2'; this is a typographical error.
  5. [References] Reference [12] has corrupted text ('Ann. Sci. c. Norm. Supr. 1990' should be 'Ann. Sci. Ecole Norm. Sup. 1900'), and several references are missing volume or page information.
  6. [Section 3.2, Eq. (50)] The backward-Kolmogorov equation in Eq. (50) is stated without derivation; it would strengthen the presentation to note that it follows as the adjoint of the Fokker-Planck equation (33) for the homogeneous case.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the paper's derivation is conditional on explicitly acknowledged external inputs (drift, volatility, risk-free rate), and no fitted quantity is relabeled as a prediction.

full rationale

The derivation is not circular in the sense relevant here. The transition density in Eq. (26) is obtained by maximizing relative entropy subject to the prior second-moment constraint (13), the directionality constraint (19) with <ΔS/S> = μΔt, and the definition of the entropic clock α = 1/(σ²Δt) in Eq. (15). The mean μΔt − σ²Δt/2 in Eq. (25) is a Taylor/Itô consistency relation between the externally specified arithmetic drift and the log-return mean, not a re-statement of the output. The paper explicitly identifies the drift as 'a piece of information that ought to be found' and notes that relaxing uniformity means one 'will no longer end up with a lognormal distribution.' Similarly, the Black–Scholes formula (44) follows from the risk-neutral constraint μ = rf in Eq. (34) together with the uniform-parameter lognormal transition; this is a conditional derivation, not a fit disguised as a prediction. The self-citations in the paper (e.g., refs. [4,5,35,36]) point to the entropic-dynamics framework and the authors' related work, but no load-bearing uniqueness theorem or fitted parameter is imported from them. The main limitation is that GBM and Black–Scholes are recovered because μ, σ, and rf are taken as inputs and uniformity is assumed; that limitation is acknowledged in the paper and is a scope condition, not circularity.

Assumptions & free parameters 3 free parameters · 5 assumptions · 0 invented entities

The derivation does not introduce new physical entities, but it does rely on several modeling choices: the entropic clock (alpha = 1/(sigma^2 Delta t)) and the constraint <Delta S/S> = mu Delta t are the two main free inputs. The scale-invariance argument is a postulate about investor behavior, not a consequence of the framework, and the uniformity of drift and volatility is required to obtain the standard formulas.

free parameters (3)
  • drift mu = not specified, treated as an external input
    Introduced in Eq. 19 as the expected return <Delta S/S> = mu Delta t; it is not derived and the paper says a separate future model would derive it.
  • volatility sigma^2 = not specified, treated as an external input
    Introduced through the entropic clock alpha = 1/(sigma^2 Delta t) in Eq. 15; it sets the variance of the Gaussian transition density and is not itself derived.
  • risk-free rate r_f = not specified, treated as an external input
    Imposed by the risk-neutrality constraint mu = r_f in Eq. 34; it is a market input, not derived by the framework.
assumptions (5)
  • domain assumption The maximum-entropy principle with relative entropy is the correct inference framework.
    The entire paper is built on entropic inference; cited to refs 1-3 but not independently justified here.
  • ad hoc to paper Scale invariance of the formalism requires probability densities to be invariant under price scaling (Eq. 2).
    The paper asserts that requiring P(f(S_tilde)) = P(f(S)) is the right way to impose scale invariance; this is a modeling choice, not a mathematical necessity.
  • domain assumption The prior distribution has a sharp Gaussian form with <(Delta ln S)^2> = k (Eq. 13) and the motion is continuous.
    This encodes the continuity of price changes; it is an assumption about the regime of the model.
  • domain assumption The drift and volatility are constant in time and independent of price when deriving the finite-time lognormal distribution and Black-Scholes formulas.
    The paper explicitly states this uniformity is assumed (Section 3.1), and without it the lognormal and Black-Scholes formulas do not follow.
  • ad hoc to paper The Taylor expansion of ln(S'/S) can be truncated at second order and the second-order term can be identified with sigma^2 Delta t.
    Eqs. 17-24 perform this approximation; the identification is standard in stochastic calculus only with a specific interpretation of the limit, which is not fully specified here.

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Pith. "Pith review of Entropic Dynamics of Stocks and European Options." pith.science (2026). https://pith.science/paper/ACTQOX5B

@misc{pith2026190806355,
  author       = {Pith},
  title        = {Pith review of: Entropic Dynamics of Stocks and European Options},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/ACTQOX5B}},
  note         = {Machine review of arXiv:1908.06355}
}
read the original abstract

We develop an entropic framework to model the dynamics of stocks and European Options. Entropic inference is an inductive inference framework equipped with proper tools to handle situations where incomplete information is available. The objective of the paper is to lay down an alternative framework for modeling dynamics. An important information about the dynamics of a stock's price is scale invariance. By imposing the scale invariant symmetry, we arrive at choosing the logarithm of the stock's price as the proper variable to model. The dynamics of stock log price is derived using two pieces of information, the continuity of motion and the directionality constraint. The resulting model is the same as the Geometric Brownian Motion, GBM, of the stock price which is manifestly scale invariant. Furthermore, we come up with the dynamics of probability density function, which is a Fokker--Planck equation. Next, we extend the model to value the European Options on a stock. Derivative securities ought to be prices such that there is no arbitrage. To ensure the no-arbitrage pricing, we derive the risk-neutral measure by incorporating the risk-neutral information. Consequently, the Black--Scholes model and the Black--Scholes-Merton differential equation are derived.

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