# Seeking criticism of model assumptions

I have been trying to publish a new calculus and options model for seven years. I have been consistently desk rejected, so what I am trying to do is get criticism of my assumptions because they trigger very different results than Black-Scholes. Some relationships are reversed.

The first assumption that I make is that the parameters of securities are unknown. Ito and Statonovich both assume that the parameters are known with certainty. Dropping that assumption creates a range of additional statistical requirements not typically considered in models, such as sufficiency and admissibility.

The second assumption is that returns are $$r_t(p_t,p_{t+1},q_t,q_{t+1})=\frac{p_{t+1}}{p_t}\frac{q_{t+1}}{q_t}-1$$. That treats returns as a statistic and not data. It treats prices and volumes as data and the distribution of returns has to be derived from first principles as the product distribution of two ratio distributions. Due to that, the distributions cannot have a first or higher moment, so certainly no covariance is possible. Equivalently, returns could be modeled as $$p_{t+1}q_{t+1}=Rp_tq_t+\varepsilon.$$

The third assumption is that models must conform to the Dutch Book Theorem and its converse; otherwise, a clever opponent could transfer capital from a market maker to itself without breaking any laws. Indeed, with six hundred trillion dollars in outstanding OTC derivatives, an arbitrage operation could net billions, tens of billions, or hundreds of billions. Hundreds of trillions are a lot of money. Conforming to the Dutch Book theorem drops the need for the stronger no-arbitrage assumption because the market maker will capture the profits if arbitrage happens.

The fourth assumption is that structural breaks are necessary due to firms' microeconomic changes and environmental changes such as COVID or an interest rate change.

The last editor was a mathematician and kicked it back to me. He felt it was brilliant but that the readers would reject it because it was so different. One other thing I did was include an indirect utility function in the statistical process. Most estimators can be defined as $$\min_{\hat{\theta}}\int-\pi(\theta)U(\theta,\hat{\theta})\mathrm{d}\theta$$ or $$\min_{\hat{\theta}}\int{}R(\theta,\hat{\theta})f(x|\theta)\mathrm{d}x$$ Where $$U$$ is a utility function and $$R$$ is a risk function, which follows from a utility function. I add an indirect utility function since we cannot see the market participants' actual utility functions, but we can see the system behavior. Participants do not need to use that if they know their own utility function.

The math fully explains the heavy tails, the models first-order stochastically dominate existing models, the option price is a sufficient statistic, the price is an admissible statistic. It minimizes the K-L divergence between the probabilities and nature; the probabilities in the model meet the statistical concept of coherence. Probabilities are coherent if they cannot be used to force a market maker to take a loss.

I am looking for criticism of the assumptions; also, any publishing ideas would help too. Seven years is a long time to sit on a model that can identify where others create arbitrage opportunities.

EDIT The link to the stochastic calculus paper, which needs to be published first is here https://www.linkedin.com/feed/update/urn:li:activity:6795937274921127937/. It is also at SSRN.

Responding to a comment In 1953, John von Neumann wrote a short note warning that he believed there was a possible math mistake in what is now called mean-variance finance. Economists were using folk theorems and assumptions to prove things that the field of mathematics had yet to determine what was true. He died before taking the issue up again.

In 1958, another mathematician, John White, showed that models like Black-Scholes cannot be solved empirically. As he did not work in finance and economics has been streaming along on its own, nobody was aware the economists didn't know. I found out by grabbing every bibliography on problems with mean-variance finance, reading all the articles that might provide a solution, then reading their bibliographies and so forth. That proof is also the underpinnings of the Dickey-Fuller unit root test.

In 1963, Benoit Mandelbrot published On the variation of certain speculative prices where, and this is a vast understatement of the article, he pointed out that if this is theory, then this cannot be your data and this is your data. Fama then followed on for a period of time and the descendants of this work are the people that work with alpha stable distributions.

In 72 or 73 Fama and MacBeth falsified the CAPM decisively. Articles ran about the death of mean-variance finance and then it was given a Nobel.

In 72, Black and Scholes published their model, but about three paragraphs from the end, they note that they tested their model and it failed its validation test. It won a Nobel.

So, what it does better.

It creates a different axiomatic basis and avoids the non-existence proof by White. Ito's assumption that the parameters are known is a very strong assumption. If that is not true, no non-Bayesian solution can exist.

It also is coherent. Frequentist statistics are not coherent. You can force a fund using non-Bayesian and some Bayesian methods to take losses if they are using a neural network that maps to a Frequentist method, or if using Frequentist estimators for prices, volumes or allocations.

The loss of no-arbitrage only implies that if arbitrage appears, the market making financial institution will see it and keep the profits. It permits mistakes, it doesn't permit external parties to profit from it. A private investor will have no arbitrage opportunities but they can create them for the market making institution.

Right now, it is possible if you are clever and know where to look, to construct convex combinations of contracts that are profitable in all states of nature. That is because Frequentist methods violate the Dutch Book Theorem.

It creates fair minimum prices. Of course, if you can get away with a higher premium, that isn't a problem because there is no assumption of a risk-neutral market maker. Monopolist pricing is permitted.

• You might get more feedback if you post your manuscript on a preprint site, such as arXiv. Commented Nov 4, 2021 at 5:49
• Thanks for sharing. Definitely interesting. I suspect more than the technical underpinnings, the first step would be to provide some practical context: what is the purpose of the model, who would use it and for what goal? For instance, I am not surprised a sell side trading desk would have cold feet without further explanation: where is the hedging dimension (since no arb assumption seems to be relaxed) ? How can I use the model to hedge and how does the PnL look in that case? What does the model do better/worse than existing ones? Commented Nov 4, 2021 at 7:04
• This is beyond my competence, but I am curious nevertheless. Here are a couple of question. 1. I suppose $q$ denotes quantity. Why are quantities (in addition to prices) used for defining returns? How do I interpret returns based on quantities? 2. As I have mentioned to you earlier, prices on any given day are positive and bounded, and therefore their ratios have all moments. This is because stock exchanges do not allow infinitely large price changes on any given day nor do they allow prices below a certain (positive) minimum. (But perhaps there could be latent prices that are unbounded.) Commented May 21, 2022 at 10:20
• @RichardHardy I should close this question. I will present the replacement calculus at WEAI and maybe a couple of other conferences. As to your first comment, the issue is two-fold. If P=P(Q) then the answer depends on Q. However, if it does not, it still depends on the future state of Q. If Q=0 because the firm declared bankruptcy, the distribution is a point at -100% return. Likewise, because Bayesian methods are generative instead of sampling based, it is necessary to know the likelihood function. If $PQ=k$ where $k$ is an offer to merge for cash, then the distribution will differ. Commented May 22, 2022 at 22:54
• @RichardHardy it is correct that the future price, assuming you purchased a security for cash, is uncertain but there is a finite budget constraint. What is it? If the budget constraint is stochastic, then presumably it is defined on all of the reals, even if some values are trivially small in probability. Imagine a 50,000% hyperinflation hit the US. Prices are nominal. My real return might be two percent, but my measured return would be 50,0002%. I have found two ways that seem to credibly measure the budget constraint but the right hand side is unknown. Commented May 22, 2022 at 22:57