Entropic Dynamics of Exchange Rates and Options

Daniel Bartolomeo; Mohammad Abedi

REVIEW 4 major objections 4 minor 41 references

Entropic Dynamics of Exchange Rates and Options

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

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Pith's one-line read This paper derives the standard model of foreign-exchange option pricing—geometric Brownian motion for the exchange rate and the Garman–Kohlhagen formula—from maximizing relative entropy under scale, continuity, and drift constraints.

desk verdict A clean MaxEnt route to Garman-Kohlhagen, but the risk-neutral measure is imposed rather than derived, making the central derivation claim overstated. read the letter →

arxiv 1908.06358 v1 pith:NGOTIC7S submitted 2019-08-18 q-fin.PR

Mohammad Abedi , Daniel Bartolomeo This is my paper · ORCID

classification q-fin.PR MSC 91G2062B1060J65

keywords EntropicinferenceMaximumentropydynamicsGeometricBrownianmotionFokker–PlanckequationGarman–KohlhagenmodelBlack–Scholes–MertonScaleinvariance

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 argues that the standard model of foreign-exchange dynamics—geometric Brownian motion for the exchange rate and the Garman–Kohlhagen formula for European options on it—can be derived, not assumed, from a single inference principle: maximize relative entropy subject to the little information one actually has. Scale invariance of returns forces the logarithm of the exchange rate to be the modeled variable; continuity and a drift constraint fix the transition distribution as lognormal. The same procedure, with the no-arbitrage drift imposed, reproduces the Garman–Kohlhagen option prices and the Black–Scholes–Merton partial differential equation. If correct, the paper offers a derivation of a standard pricing model from information-theoretic first principles rather than from an assumed stochastic process.

What carries the argument

The load-bearing object is the maximum-entropy transition density $P(\ln u'|\ln u)$: relative entropy is maximized against a Gaussian prior fixed by the continuity constraint $\langle(\Delta\ln u)^2\rangle=k$, with entropic time $\alpha=1/(\sigma^2\Delta t)$ serving as the clock. A drift constraint $\langle\ln(u'/u)\rangle\approx(\mu_d-\mu_f)\Delta t-\frac12\sigma^2\Delta t$ fixes the Gaussian's mean, yielding the lognormal transition density; replacing the real drift with the interest-rate differential $(r_d-r_f)\Delta t$ turns the same density into the risk-neutral measure. The scale-invariance requirement enters by selecting $f(u)=\ln u$ as the variable whose probability density is a scalar under $u\mapsto lu$.

What would settle it

Look at the implied volatility surface of liquid FX options: the model uses a single volatility $\sigma$ for every strike, so a pronounced smile or skew—systematically different implied volatilities across strikes—would falsify the lognormal transition density at the core of the derivation. Alternatively, high-frequency log-return data with significant excess kurtosis or jumps would violate the continuity constraint.

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Extended reading notes

Core claim

The paper claims that the standard lognormal model of foreign exchange—geometric Brownian motion for the exchange rate and the Garman–Kohlhagen formulas for European calls and puts—is not an independent postulate but follows from maximizing relative entropy under three pieces of information: scale invariance selects the log exchange rate as the variable; continuity constrains the mean-square displacement to be small; and a drift constraint sets the mean log return. With the risk-neutral drift imposed, the same transition density prices European FX options exactly as Garman–Kohlhagen, and the option premium satisfies the Black–Scholes–Merton partial differential equation. The claim is that all of this is derived from entropic inference rather than assumed from stochastic calculus.

Load-bearing premise

The argument assumes, rather than derives, that risk-neutral pricing uses the difference between domestic and foreign interest rates as the drift; if that market rule fails, the option formula does not follow from entropy alone.

Editorial extensions

If this is right

The exchange-rate process is geometric Brownian motion: log returns are Gaussian with mean $(\mu_d-\mu_f-\frac12\sigma^2)\Delta t$ and variance $\sigma^2\Delta t$, and the transition density is invariant under rescaling the exchange rate.
European call and put prices are exactly the Garman–Kohlhagen formulas, and call–put parity $C-P=e^{-r_f T}u_0-e^{-r_d T}K$ holds as a consequence of the construction.
Option premia satisfy the Black–Scholes–Merton partial differential equation $\partial_t E+(r_d-r_f)u\,\partial_u E+\frac12\sigma^2 u^2\partial_u^2 E-r_d E=0$.
Any extension of the model that includes jumps must preserve scale invariance; otherwise the paper argues an arbitrage opportunity appears.

Reading between the lines

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

Because the derivation never uses the level of the exchange rate, only its log, the same argument would carry over to any positive asset whose returns are scale invariant—for example, other currencies or commodity prices—without changing the pricing formula's structure.
The entropy-maximization route suggests a natural diagnostic: if empirical FX returns show excess kurtosis, one should add jump constraints to the entropic model rather than abandon the framework; the scale-invariance requirement then predicts how jump terms must be parameterized.
A complete derivation of the risk-neutral measure from entropy alone would still be missing; the paper imports the standard no-arbitrage drift, so the framework's novel content currently lies in deriving the dynamics, not in deriving the pricing measure.

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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

The best thing in this paper is the scale-invariance argument for why the logarithm of the exchange rate is the right variable. That argument is clear, and the max-entropy machinery is applied correctly: given the continuity and drift constraints, the transition density is Gaussian in log-u, the Fokker-Planck equation follows, and the option prices come out as Garman-Kohlhagen with the expected call-put parity check. As a pedagogical re-derivation of a known result, it is coherent and mostly sound. The skipped calculations in Eqs. (20) and (43) are annoying but fillable; the citation pattern is fine, including the reference to the authors' own stock paper.

The load-bearing problem is the risk-neutral measure. The abstract and introduction say it is derived. In Section 3.1, the paper actually imposes the risk-neutral drift in Eq. (27), replacing the real drift with rd-rf with no Girsanov/Radon-Nikodym computation, no martingale verification, and no argument that this is the minimal-entropy update consistent with no-arbitrage. Without that, Eqs. (37) and (39) are just the standard Garman-Kohlhagen result obtained by assuming lognormal FX and risk-neutral valuation. This is a real gap between claim and content, not a matter of taste. Tonally, the paper should say "we impose the standard risk-neutral drift" rather than "we derive a risk-neutral measure."

A second, lesser issue is the uniqueness of the log transformation. The functional equation f(lu)=f(u)+C(l) with C(l)C(l')=C(ll') is satisfied by C(l)=ln l, and that gives f(u)=ln u, but you need a regularity assumption to rule out other solutions. Minor, because any monotone f would do for the rest of the argument.

Also worth saying plainly: the empirical content is entirely in the constraints. The continuity and drift constraints already encode the GBM drift, so the "derived" stochastic process is basically the information put in. That is not a fatal flaw for a foundational exercise, but it limits the significance.

Who should read this? People curious whether entropic inference can reproduce standard finance models, and whoever wants a self-contained example of the max-entropy route to lognormal dynamics. Practitioners will find nothing new. It does deserve a serious referee: the math is mostly right, the goal is meaningful, and the fix is to either supply a genuine derivation of the risk-neutral measure or soften the claims. As it stands, I would reject as is but invite a revision.

Referee Report

4 major / 4 minor

Summary. The paper proposes an entropic-dynamics model of exchange rates. It maximizes relative entropy subject to continuity and drift constraints to obtain a Gaussian transition density for the log exchange rate, which it identifies with geometric Brownian motion. It then introduces a risk-neutral drift and derives the Garman–Kohlhagen European option pricing formula and the Black–Scholes–Merton PDE. The paper claims that this constitutes a derivation of GBM and of risk-neutral option pricing from entropic inference and scale invariance.

Significance. If the claims were fully established, the paper would offer an information-theoretic foundation for a standard result in foreign-exchange option pricing, which is conceptually interesting. The maximum-entropy calculation is explicit, and the structure clearly shows which constraints produce the Gaussian transition density. However, the risk-neutral measure is imposed rather than derived, the uniqueness proof for the logarithmic variable is mathematically incorrect, and a key second-moment relation is assumed without derivation, so the central claims of the abstract are not supported as stated. The final formula, once corrected for a sign error, is the standard Garman–Kohlhagen price, which limits the novelty to a reformulation rather than a new pricing result.

major comments (4)

[Section 3.1, Eq. (27)] The paper states in the abstract and in Section 4 that it 'derives a risk-neutral measure,' but Eq. (27) simply replaces the real drift with the domestic-foreign interest-rate difference as a constraint. No derivation is given of why this is the minimal-entropy update from the real-measure density, and no Girsanov or martingale verification is supplied. Consequently, the option price in Eq. (37) follows from the standard assumption of a lognormal FX rate under a risk-neutral measure, not from entropic inference alone. This is the load-bearing step for the paper's central claim and needs either a genuine derivation or an explicit admission that the risk-neutral drift is an input.
[Section 3.1, Eqs. (33)-(39)] The definition of d1 in Eq. (34) has the wrong sign on the σ^2T/2 term. For a lognormal variable with mean ln u0+(rd-rf-σ^2/2)T, the expected sale value is u0 e^{(rd-rf)T} N(d1) with d1 = [ln(u0/K)+(rd-rf+σ^2/2)T]/(σ√T), not the expression given, which is actually the standard d2. With d2 defined by d2 = d1 - σ√T, the call price in Eq. (37) and the put price in Eq. (39) do not equal the Garman–Kohlhagen prices. This sign error must be corrected for the central result to be valid.
[Section 2.4, Eq. (20)] The relation ⟨(ln u'/u)^2⟩_P = ⟨(Δu/u)^2⟩_P = σ^2Δt is stated after a skipped calculation, but it is not a consequence of the maximum-entropy posterior. From Eq. (17), ⟨(ln u'/u)^2⟩_P = (βσ^2Δt)^2 + σ^2Δt, which reduces to σ^2Δt only to leading order; the equality with ⟨(Δu/u)^2⟩ is an additional assumption. This assumption is what fixes the -σ^2/2 correction in the drift and hence the GBM form, so the derivation of GBM is circular: the GBM drift is effectively inserted through Eqs. (15) and (20) and then recovered in Eq. (24).
[Section 2, Eqs. (3)-(5)] The claim that the scale-invariance condition uniquely yields f(u)=ln u is incorrect. The functional equation C(l)+C(l')=C(ll') has the general solution C(l)=a ln l for any real a, leading to f(u)=a ln u+b. In addition, Eq. (2) is not the correct transformation law for probability densities under a change of variables; densities transform with a Jacobian. The logarithmic variable can be motivated by scale invariance, but the uniqueness proof as stated is false and should be replaced or removed.

minor comments (4)

[Section 3.1, line before Eq. (27)] The phrase 'To drive the risk-neutral measure' should be 'To derive the risk-neutral measure,' and the introduction to Section 3 contains a typo 'German-Kohlhagen' that should be 'Garman-Kohlhagen.'
[Section 2.4, Eq. (12)] The notation k'(u) is introduced as a function that tends to zero, but later k' is used as a constant; the notation should be clarified and made consistent.
[Section 2.4, Eqs. (16)-(19)] The Lagrange multiplier β is written as β(u) in Eq. (16) but treated as a constant thereafter; the paper should state the assumption that β is independent of u or justify why it may be treated as constant.
[Section 3.2, Eq. (43)] The backward Kolmogorov equation is stated with the remark 'the derivation is skipped'; for a paper whose goal is to derive standard results from entropic principles, this is a nontrivial step and should be shown or at least referenced.

Circularity Check

1 steps flagged · score 6.0 of 10

Risk-neutral measure is imposed (Eq. 27), not derived; Garman-Kohlhagen follows by construction from the assumed lognormal pricing measure.

self definitional [Abstract; Section 3.1, Eqs. (27)-(28), (37)]
"To this end, we derive a risk-neutral measure to value European Options on FX. ... To drive the risk-neutral measure, we impose the first risk-neutral constraint on Equation (15), ⟨Δu/u⟩_P = (rd − rf)Δt."

The abstract claims the risk-neutral measure is derived, but Section 3.1 obtains it by imposing Eq. (27), an external finance input. No no-arbitrage/Girsanov/martingale argument connects the real-measure density (23) to the pricing measure; Eq. (28) is Eq. (23) with μ_d−μ_f replaced by r_d−r_f. The Garman-Kohlhagen price (37) is then the discounted lognormal expectation under (28). The central option-pricing result is thus equivalent to the assumed risk-neutral lognormal measure; the entropic MaxEnt step supplies only the Gaussian form already determined by the input moments, and the pricing measure itself is an input, not a consequence.

full rationale

The GBM derivation (Section 2) is a self-contained MaxEnt exercise: the continuity and directionality constraints (8), (12), (15), (20) determine the Gaussian transition density (23), and the scale-invariance argument for the log variable is internally derived. No load-bearing self-citation is involved; the citations to the authors' other work ([40], [41], [10]) are not used to justify the derivation. The circularity is concentrated in the risk-neutral step: the paper's abstract says a risk-neutral measure is derived, but Eq. (27) imposes the risk-neutral drift as an assumption, and Eq. (28) is just Eq. (23) with the drift replaced. Consequently the Garman-Kohlhagen formula (37) follows from the assumed lognormal pricing measure by standard integration; the entropic framework does not independently produce the pricing measure. The skipped derivations for Eqs. (20) and (43) are completeness gaps, not circularities. Overall partial circularity: the option-pricing result reduces to the imposed risk-neutral constraint by construction.

Assumptions & free parameters 3 free parameters · 7 assumptions · 0 invented entities

The derivation's central inputs are the volatility, the drift difference, and the interest rates, which are supplied by the domain rather than derived internally. The MaxEnt framework itself is taken as the foundational axiom. The scale-invariance uniqueness claim is mathematically under-specified. No invented entities are introduced.

free parameters (3)

volatility sigma
Introduced in Eq. (10) as the clock scale alpha = 1/(sigma^2 Delta t); treated as an input, not fitted, and assumed uniform in Section 3.1 for the closed-form option price.
drift difference mu_d - mu_f
Introduced in Eq. (15) as the expected arithmetic return; treated as an input set by another model, and later replaced by rd - rf in the risk-neutral measure.
domestic and foreign risk-free rates rd, rf
Introduced in Eq. (27) as the risk-neutral drift; taken as given market data.

assumptions (7)

domain assumption Maximum entropy / relative entropy updating is the correct inference framework.
The entire paper builds on entropic inference (Eq. 6, Section 2.1); this is the foundational premise of the framework, not proven here.
domain assumption Scale invariance is a symmetry of exchange rate dynamics.
Argued in Section 2 from investor preference for returns; this is an economic argument, not a mathematical proof.
standard math The unique solution to C(l)+C(l')=C(ll') is C(l)=ln l, and hence f(u)=ln u.
Used in Section 2 to select the log variable; actually C(l)=c ln l is also a solution, so uniqueness as stated is wrong, though c=1 can be chosen by normalization.
domain assumption The dynamics is continuous (no jumps).
Imposed in Section 2.2 via the small-variance constraint Eq. (8); jumps are explicitly excluded.
domain assumption The drifts and volatility are uniform (or the model is used for small time steps).
Assumed in Section 3.1 to get the lognormal distribution for finite T and the closed-form Garman-Kohlhagen formula.
domain assumption No-arbitrage implies the risk-neutral drift is rd - rf.
Imposed in Eq. (27), Section 3.1; standard finance result, but taken as input rather than derived.
standard math Backward Kolmogorov equation for transition densities.
Used in Section 3.2, Eq. (43), to derive the PDE; the derivation is skipped in the paper.

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Pith. "Pith review of Entropic Dynamics of Exchange Rates and Options." pith.science (2026). https://pith.science/paper/NGOTIC7S

@misc{pith2026190806358,
  author       = {Pith},
  title        = {Pith review of: Entropic Dynamics of Exchange Rates and Options},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/NGOTIC7S}},
  note         = {Machine review of arXiv:1908.06358}
}

read the original abstract

An Entropic Dynamics of exchange rates is laid down to model the dynamics of foreign exchange rates, FX, and European Options on FX. The main objective is to represent an alternative framework to model dynamics. Entropic inference is an inductive inference framework equipped with proper tools to handle situations where incomplete information is available. Entropic Dynamics is an application of entropic inference, which is equipped with the entropic notion of time to model dynamics. The scale invariance is a symmetry of the dynamics of exchange rates, which is manifested in our formalism. To make the formalism manifestly invariant under this symmetry, we arrive at choosing the logarithm of the exchange rate as the proper variable to model. By taking into account the relevant information about the exchange rates, we derive the Geometric Brownian Motion, GBM, of the exchange rate, which is manifestly invariant under the scale transformation. Securities should be valued such that there is no arbitrage opportunity. To this end, we derive a risk-neutral measure to value European Options on FX. The resulting model is the celebrated Garman-Kohlhagen model.

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Pith tools

Reviewed August 14, 2026 · model on record in the stance chip above.

[1] [1]

Entropic Inference

Caticha, A. Entropic Inference. AIP Conf. Proc. 2011, 1305, 20; doi:10.1063/1.3573619

doi:10.1063/1.3573619 2011

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Golan, A. Foundations of Info-Metrics ; Oxford University Press: Oxford, UK, 2018; doi:10.1093/ajae/aay085

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