A static option menu that grows too broad can destroy existence of the utility maximizer. In the paper's words, the resulting closedness failure makes utility maximization a delicate issue in these models. The note responds by narrowing the menu.
No empirical claim appears in the paper. It contains no data, sample period, return series or backtest. This is a discrete-time existence result on an abstract filtered space with finitely many dynamically traded assets. Every numeric result below that does not come from the paper comes from our run and is identified accordingly.
Our run uses choices the paper never makes
The paper supplies neither a signal nor a utility. We selected the universe, payoff cone, utility, cost model and schedule. Static options also cost nothing in the model, while our implementation paid quoted premiums and spreads. Our exercise is therefore a constrained implementation of our own rather than a test of the theorem. Since the paper is entirely theoretical, it supplies no figures for comparison.
We ran daily bars from 2020-01-01 to 2024-07-01. Total return was 21.38%. Volatility reached 13.53%, max drawdown was -34.10%, Calmar was 0.13 and Sortino was 0.39.
Sharpe was 0.35.
Over 4.5 years, a Sharpe of 0.35 paired with a -34.10% max drawdown is weak. The window produced a 21.38% total return, below a plain long ETF book. Those figures belong to our run, not the paper.
One static payoff with dynamic hedging afterward
Dates run from t = 0 through T. F_0 is trivial, and X is a d-dimensional discounted price process. At time 0, the investor selects one payoff h from Y, a set of zero-price static options. Trading in the d assets then proceeds dynamically. Terminal wealth combines h with the martingale-transform P&L. Write K_0 for the zero-cost dynamic outcomes; the attainable set is K_0 + Y. Because the static leg is purchased once and left untouched, the position amounts to a buy-and-hold option hedged through time.
Arbitrage theory depends on K_0 + Y - L^0_+ being closed in probability. Closedness permits a Kreps-Yan style separation to yield a pricing measure. It also gives a maximizing sequence a possible limit. Acciaio, Larsson and Schachermayer established in 2017 that closedness can fail. Their construction has d = 1, T = 2, a bounded price process and Y equal to all integrable σ(X_T)-measurable claims. The outcome set is not closed, and its intersection with L^1(P) is not closed either.
Before supplying a remedy, Rásonyi strengthens that counterexample. The original construction restricts the dynamic component to a supermartingale. Expanding it to the full K_0 leaves the reasoning intact because a martingale transform bounded below by an integrable random variable is a true martingale, by Jacod and Shiryaev. Removing the supermartingale constraint therefore leaves the failure in place.
Which part of smallness matters?
A small cone is generated by a convex, closed and bounded set H in L^0 that excludes 0. Definition 2.3 contains the full definition. Alongside Lemma 2.4, it drives the proof. Small cones are closed. More generally, whenever C is a closed cone satisfying C ∩ (-L) = {0}, the sum C + L is closed as well. Lemma 2.4 proves this through Yan's theorem, which provides a bounded-density measure, and Komlós, which supplies almost-surely convergent Cesàro means.
Remark 2.5 isolates the condition doing the real work. Consider L^∞+, generated by the unit ball of non-negative bounded variables. Its generator is convex, closed and bounded in L^0, yet it contains 0. The cone L^∞+ is famously not closed in L^0. Boundedness comes easily. Distance from the origin carries the result.
Assumption 3.3 requires Y to be a finite sum of m small cones, with no redundancy introduced at each stage: (K_0 + C_1 +... + C_n) ∩ (-C_{n+1}) = {0}. When m = 1, the requirement becomes K_0 ∩ C_1 = {0}. Thus no non-zero static option can be replicated from zero capital.
Theorem 3.5 then gives the desk-level equivalence. No arbitrage in the semi-static market holds if and only if one equivalent martingale measure Q* has bounded density and prices every admissible static payoff at or below zero. The theorem treats arbitrary Y ⊂ L^0, while Assumption 3.3 provides the implications that depend on it. Without Assumption 3.3, each option receives its own pricing measure, one Q(Y) at a time, a considerably weaker object. Given Assumption 3.3 and NA(Y), Theorem 4.1 produces an optimizer whenever u is non-decreasing, concave and bounded from above. Rásonyi states that the unbounded case "requires more involved arguments and will be done elsewhere".
Example 3.1 is the example worth retaining. There is one period, X_0 = 0, X_1 is standard Cauchy, and Y contains every zero-mean random variable. NA(Y) holds while M(Y) is empty. A Cauchy one-period return combined with the broadest possible zero-mean menu already breaks the fundamental theorem.
Each family gets one direction
The paper states the implication without pursuing it. No vector subspace of L^0 can be a small cone because convexity forces 0 into any H that generates a subspace. Definition 2.3 itself imposes no sign restriction, allowing a family of short centred options to generate a small cone. Take the negatives of the put family H_2. Those payoffs obey h ≥ q on {X_T ≥ K_♯}, precisely the separation from the origin required by Definition 2.3. Holding a family together with its negative falls outside the framework, since Assumption 3.3's intersection condition fails immediately when C_2 = -C_1.
Section 5 works with long call families and long put families, using d = 1 and m = 1. The note leaves a two-sided menu over the same strike band untreated. For a static book that trades both directions, this is the first condition to examine.
Calls or puts across a continuum of strikes
Section 5 supplies the paper's most usable result. Fix [K_♭, K_♯], where 0 < K_♭ < K_♯, and centre each call payoff by its price. Suppose call prices have the uniform upper bound D = sup_K price < ∞, while X_T ≥ 0 is unbounded and P-integrable. The resulting cone is small under Proposition 5.5.
The put construction uses q = inf_K price > 0 and an unbounded X_T ≥ 0. Payoffs remain in [-K_♯, K_♯], and every convex combination obeys P(h ≤ -q) ≥ P(X_T ≥ K_♯) > 0. This keeps the generator away from the origin, as Proposition 5.8 requires. The framework therefore permits a continuum of strikes. Its two quote-level checks are a finite sup_K call price and a positive inf_K put price.
No redundancy is the tougher requirement. Propositions 5.6 and 5.9 establish K_0 ∩ L = {0} only when the F_{T-1}-conditional law of X_T has support [0,∞) almost surely. Bounded terminal prices violate that hypothesis. It concerns the law, beyond what a screen displays. Both propositions assume d = 1, and the paper restricts attention to m = 1. A call cone combined with a put cone is consequently absent from the examples. Remark 5.10 speaks, accordingly, of "either call or put options".
Put-call parity explains the concern. If one measure prices a strike's call and put, the centred call minus the centred put equals X_T - X_0, which belongs to K_0. Measures that differ by strike leave a constant. Any non-zero constant in K_0 would create an arbitrage, and NA excludes it. For m = 2, the intersection condition still requires a separate check.
What we implemented
Our dynamic sleeve trades ten US ETFs: HYG, IVV, IWM, LQD, QQQ, SOXL, SPY, SQQQ, TLT, TQQQ. A position is long only when its 252-session trend is positive. Weights use the inverse of 20-session realized volatility and are capped at 25% per name. Gross exposure cannot exceed 4.0x. Signals trade at the next close.
A monthly overlay adds defined-loss spreads. Candidates comprise the top ten ETFs and the top ten stocks, with legs near 0.30 and 0.15 absolute delta. We rank them by estimated sample utility contribution using a risk-aversion coefficient of 3.0. Admission is limited to five sleeves, 25 contracts apiece, 2% of NAV in premium and 2% in defined loss.
Purchases fill at the following day's ask and sales at the bid. We charge $0.65 per contract per leg. ETF transactions cost $0.004 a share subject to a $1 minimum, and ETF slippage is modelled at zero.
We did not allocate the 13.53% volatility or -34.10% drawdown between the sleeves, so both remain book-level figures. The overlay's contribution is bounded by the 2%-of-NAV premium cap, though we have no estimate of that contribution. Two assumptions are our leading explanations for the weak outcome. Modelling zero ETF slippage favors a daily-rebalanced book running at 4.0x. Our non-replication screen is also a finite-sample residual test using at most 252 trailing observations. It substitutes for a condition on the law, with no established connection between them. The 4.5-year span includes the COVID crash and 2022. Such a window says little about a monthly overlay that accepts no more than five sleeves.
The result worth using
Rásonyi's advance over Acciaio, Larsson and Schachermayer is a practical sufficient condition with a short price test. Over a compact strike band, call prices must be bounded above or put prices bounded away from zero, while X_T ≥ 0 must be unbounded and P-integrable. The demanding piece is no redundancy. Propositions 5.6 and 5.9 require the F_{T-1}-conditional law of X_T to have support [0,∞) almost surely, a condition that fails in practice.
For a one-sided static option overlay on a compact strike band, this is the structure to examine before claiming that the expected-utility problem has a solution. Section 5 covers long call and long put families. Since the paper does not construct a short-side generator, a two-sided book needs its own Assumption 3.3 analysis before Theorem 4.1 can bear any weight.
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.