A valuation desk can change its quote for an unhedgeable digital merely by changing the accounting unit. Take two desks trading the same assets under the same physical law, both imposing no arbitrage. One keeps its books in cash; the other uses a traded ETF as numéraire. Each selects a pricing measure by minimizing relative entropy in the familiar minimal entropy martingale measure orientation, D_KL(Q‖P). Every traded and replicable payoff receives the same price. An unhedgeable digital does not. Večeř's three-state example produces prices of 0.169521 and 0.173707 times the numéraire for the same claim, a difference of roughly 2.5%.
The paper contains no empirical result. There is no dataset, sample period, estimate of the physical law or backtest. Its quantitative evidence consists entirely of a hand-built one-period market with three states, one risky asset and the numéraire. The physical probabilities are P = (1/2, 1/6, 1/3), while the gross return of asset X measured in asset Y is R = (3/2, 1, 1/2). The remaining work is measure theory and finite-dimensional convex analysis.
Because the paper names no traded market and estimates no physical law, our exercise adapts its log-optimal primal side to ETFs. Rolling historical scenarios replace P, and portfolio constraints replace the martingale-measure set. This construction tests none of the paper's pricing results.
The MEMM is the convex dual of exponential utility, whose conjugate is V(y) = (y/γ)(log(y/γ) − 1). The forward orientation is D_KL(P‖Q) = E_P[log dP/dQ]. It is called forward here because the expectation remains under the fixed physical law. Its logarithmic dual has conjugate V(y) = −log y − 1. The paper attributes the logarithmic construction to Goll and Rüschendorf's minimal f-divergence framework, then sets out five contributions of its own.
One constant does the work
Changing numéraire multiplies a measure by the terminal likelihood ratio L = (X_T/N_T)/(X_0/N_0), where E_Q[L] = 1. Denote the transformed measure by T_L Q. The central identity is
D_KL(P‖T_L Q) = D_KL(P‖Q) − E_P[log L].
The term being subtracted is independent of Q. The transform therefore commutes with the argmin when E_P|log L| < ∞ and T_L maps the martingale-measure set for N bijectively to the corresponding set for X. The paper assumes this bijection. It cites the standard change-of-numéraire conditions to Geman, El Karoui and Rochet without checking them in a specific continuous-time model.
Pricing gives the identity its force. The two numéraires generate the same linear valuation for every claim N_T G with G bounded exactly when Q_X = T_L Q_N, a condition the paper calls likelihood compatibility. Selector invariance and agreement between the pricing functionals amount to the same statement.
The primal side has a matching relation: E_P[log W^X_T] = E_P[log W^N_T] − E_P[log L]. Restating the accounting unit leaves the growth-optimal portfolio unchanged. Exponential utility expressed in X units becomes −exp(−γ W^N_T / L), so its choice changes.
Reverse entropy fails because its expectation is taken under the candidate measure. The paper goes further and calculates the counterexample. For Q(a) = ((1−a)b, a, (1−a)c), with b + c = 1, the reverse objective has second derivative 1/a + 1/(1−a) > 0. Its unique minimizer satisfies a/(1−a) = p₂ b^b c^c / (p₁^b p₃^c).
Under the Y numéraire, b = c = 1/2, yielding √6/(12+√6) ≈ 0.169521. In X units, b = 3/4 and c = 1/4, which gives 2^(3/4)/(8+2^(3/4)) ≈ 0.173707. The forward projection for the same family reaches a* = p₂ = 1/6 exactly, regardless of how the numéraire change redistributes the remaining probability mass. Both accounting units then value the digital at Y₀/6. Its payoff, (0,1,0), lies outside the span of the traded vectors 1 and R. A hedging test could never reveal the discrepancy.
Does 2.5% justify action?
My answer remains no. The paper anticipates the objection in Remark 14: "The discrepancy in Theorem 12 is structural, not numerical." Equation (15) establishes that a nontrivial likelihood transform adds a candidate-dependent term to the reverse objective. The failure belongs to the orientation, and the mechanism is airtight.
The size of the effect receives much less attention. The abstract calls the construction "a trinomial counterexample", and the paper offers no bound for the resulting price gap. Its magnitude is established only for this three-state instance. The dependence on P and R goes unexplored, even though the closed-form minimizer makes it possible to vary p and R and follow the gap. There are no comparative statics and no bound.
Three states, one period, one claim.
The harder characterization deserves more credit. Theorem 6 proves that invariance across all elementary one-period likelihood-ratio families requires x f′(x) to be affine. It follows that f(x) = α log x + βx + γ with α ≤ 0, and α < 0 for the non-affine convex case. Among smooth f-divergences, the logarithmic integrand is therefore isolated up to positive scaling and affine equivalence.
Večeř also identifies the quantifier carrying the result. A single fixed L restricts x f′(x) at only finitely many points, so one market cannot support the characterization. The necessity argument uses models in which P is itself feasible as a martingale measure. Nonlogarithmic divergences consequently lose invariance before a market price of risk enters.
The conclusion avoids an unconditional ranking of the measures, since each follows naturally from its own optimization problem. Combined with exact log-growth duality, however, the analysis favors the forward entropy projection when the objective is representation-invariant completion. Section 7 then confronts the main limitation. In continuous-time incomplete models, the logarithmic dual optimizer may not exist as an equivalent martingale measure and may instead occupy an enlarged domain of supermartingale deflators. The same discussion explains why the MEMM remains especially useful: under finite-entropy assumptions, it supplies an equivalent martingale measure. The author raises the objection directly, and it is the right one. Invariance applies on domains stable under T_L. Beyond those domains, the result remains a program.
Clean finite-state theory, open implementation choices
The finite-state argument is self-contained. Begin with a strictly positive feasible martingale measure. Extending the forward objective by +∞ at the boundary makes it lower semicontinuous on a compact polytope. Strict convexity then gives a unique, strictly positive minimizer.
The Lagrange conditions yield dP/dQ = R^(w), where R^(w) is the log-optimal gross return, together with max_w E_P[log R^w] = min_Q D_KL(P‖Q). In the numerical example, the optimal allocation to X is w = 2/5. The associated return is R^(w) = (6/5, 1, 4/5) = P/Q_Y, and the residual divergence is exactly zero. Zero residual says that the selected completion places the log-optimal payoff in the span of the traded assets. The market can still be incomplete.
Implementation requires choices the paper leaves outside its scope. P enters as an input and fully determines both the projection and the growth-optimal portfolio. We did not find a discussion of estimation error in P.
Constraints matter even more. Theorem 10 permits unconstrained long-short cost weights, requiring only R^w > 0 in every state. Long-only restrictions or a position cap alter the first-order conditions. Under those restrictions, p_i/q*_i may no longer be representable as a portfolio return, and the exact primal-dual identity is no longer assured.
Our build lives inside those restrictions.
Our ETF adaptation
There is nothing in the paper to reproduce because it trades no named market. We adapted the primal allocation mechanism to liquid US ETFs. Rolling empirical return scenarios stand in for the physical law, while portfolio constraints substitute for the feasible martingale-measure set.
Pricing arbitrary nonreplicable state-contingent claims would require both a derivative payoff model and a calibrated state distribution. We have neither. The paper's pricing-consistency result is therefore untouched by the exercise below, which addresses only the allocation mechanism.
We rebalanced monthly at the final trading-day close and selected the top 30 ETFs by one-year dollar volume. Up to 504 daily adjusted returns generated overlapping 21-trading-day scenarios of joint gross returns. Months lacking enough complete scenarios were skipped.
The optimization maximized Σ p_i log(w_cash + Σ_j w_j R_ij) under long-only weights summing to one. Cash was limited to ≤ 50%, each ETF to ≤ 10%, HHI to ≤ 0.125, and the portfolio had to hold at least 8 effective positions while respecting an 80% asset-class cap. A second optimization used ETF-numéraire units on the same scenarios. We mapped those holdings back into cash values and recorded the differences solely as a diagnostic, with no signal role.
The window ran from 2020-01-01 to 2024-07-01. Every fill incurred commissions of $0.0040 per share, subject to a $1.00 minimum and a cap of 1% of trade value, before performance statistics were calculated. We assumed a market-on-close auction and charged no slippage.
From 2020-01-01 to 2024-07-01, the book earned 50.15% in total. Its Sharpe was 0.54, Sortino 0.69, Calmar 0.25 and volatility 20.75%. Maximum drawdown reached -37.61%. That loss dominates the record. The portfolio was long-only and cap-constrained, and it remained invested through February and March 2020.
None of these figures can be compared with a paper that publishes no empirical results. They speak first to our chosen constraints. The log objective cannot place more than 10% in its largest position, while an HHI limit of 0.125 requires at least 8 effective names. The resulting book resembles a diversified long-only ETF basket that may keep up to half its capital in cash. Gross exposure remains at one, leaving unmodelled short borrow and margin financing costs irrelevant to this run.
The window carries the largest caveat. Each month's scenario distribution uses at most the previous 504 trading days. The sample begins only weeks before the February 2020 crash and ends in mid-2024. Creating overlapping 21-day windows from 504 daily observations also yields a few hundred highly dependent scenarios. The equal-weighted empirical law contains much less information than the raw scenario count implies, and that weighting was our choice.
The numéraire diagnostic has a narrow purpose: checking agreement between the solver and our constraint mapping.
A bound would change my view of the paper. Demonstrate the cross-numéraire MEMM gap in a calibrated multi-period model containing a real barrier or digital book, and 2.5% in a three-state toy becomes a control worth owning on a valuation desk. Alternatively, show the gap shrinking toward numerical noise as the state space becomes finer. Until then, the invariance theorem remains correct and rather elegant algebra about the unit in which a desk keeps its books.
Our backtest stops at 2024-07-01, and everything after that date is deliberately left untouched so the same strategy can be checked out of sample later.