Let me begin by noting that about three paragraphs from the end of the Black-Scholes paper is the disclosure that the authors attempted to validate it and it failed to be significantly different from chance and it has never passed a validation test.
His argument is probably correct, but also for a different reason. Ito calculus violates the converse of the Dutch Book Theorem when the assumption that all parameters are known does not hold. As nobody knows the true parameters, the converse of the Dutch Book Theorem would clearly support the idea that some hedges are not hedges and it is possible to assure a sure win over all prices.
It is likely the author has noticed one of the mathematical structures that allow you to arbitrage an Ito calculus user. I actually have five examples that I use for training that I have worked out.
Without reading the paper, my guess is that the author may have found a sixth way to attack Ito based solutions, or it is a variant on the five I have already found. Dubbins and Savage wrote an entire book on it, a book on stochastic inequalities. I think it was in the 70s.
The short answer is that for any game with Frequentist pricing or point estimation, there exist a subset of contracts greater than zero, in some cases 100% of all contracts, where it is possible to construct a pure arbitrage where someone can win in all states of nature. I have proposed an alternate calculus that drops Ito's assumption that parameters are known but I cannot get it published. It complies with both the Dutch Book Theorem and its converse. I also propose a conjecture that a Frequentist calculus exists, but it may not.
I had a measure theorist help me either to prove or disprove the conjecture and we could not. It isn't clear at this time whether a non-Bayesian stochastic calculus that does not assume the parameters are known can exist.
One aspect of Frequentist decisions is that they are mechanistic algorithms. That is the construction by design. A decision rule is intentionally an algorithm. I can see potential attacks on the paper having gone down this road for a decade now, but, having not read it, I don't know. Nonetheless, they may be arguing against axiomatic design and that would be a non-starter.
By axiomatic design, I don't mean the economic assumptions, I mean things like Kolmogorov's axioms of probability or Wald's axiomatic construction of mathematical decisions.
They are correct on the continuous time budget being an intrinsic problem. It is already known in probability theory that a sure-thing arbitrage can be set up if that is your construction methodology.
It is not a problem if the parameters do not have to be estimated but fatal if they do. As I said, Dubbins and Savage wrote a whole book on that and the first chapter is mostly dedicated as to why that cannot be used. At least if my memory holds correctly, that is the part of the book it is in.
Without reading the paper, it should be possible to show definitively as it has been known this is the case since 1955, that the continuous time budget guarantees an arbitrage opportunity at least some percentage of the time.
If I were writing the article and didn't know anything about the Dutch Book argument, then I would proceed thusly.
First, unless actors are careful, there could exist a convex combination of contracts where pure arbitrage would exist if all offered contracts were accepted.
Second, if some actors have the mathematical equivalence of color blindness and so cannot use a method of detecting the presence of arbitrage, then some of those combinations will actually fill.
My third argument would be to find one and publish it. As Frequentist methods are "color blind" to arbitrage, by theorem, they are not that hard to find.
See for example:
de Finetti, Bruno (1937), “Foresight: Its Logical Laws, Its Subjective Sources”, in Henry E. Kyburg and Howard E.K Smokler (eds.), Studies in Subjective Probability, Huntington, NY: Robert E. Kreiger Publishing Co.
––– (1972), Probability, Induction and Statistics, New York: Wiley.
Kemeny, John (1955), “Fair Bets and Inductive Probabilities ”, Journal of Symbolic Logic, 20 (3): 263–273.