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I have a question about the intersection of technical math as it relates to capital and commodities markets with technical law. It asks, "At what point, if any, would a crime be committed?"

First, I will provide a bit of technical background to the problem. In 1952, Harry Markowitz proposed a way to think about the relationship between risk and return. It revolutionized finance. He won a Nobel for it. The Nobel is part of the problem. It provides a safe-harbor for financial institutions to use them and avoid liability.

In 1963, Benoit Mandelbrot published a paper that said, roughly, "if Markowitz's model is true, then this data cannot be real data, and it is real data." By 1973, Fama and MacBeth completed a population study, and the Markowitz model was falsified. In 1972, Black and Scholes built an options pricing model, the math behind which is the Markowitz model. You can derive it directly from those models. None of these models have passed validation tests.

At the doctoral level, every financial economist can regurgitate this information. All of these models are built on top of the field of "Frequentist" statistics. They are Frequentist models. Nearly all financial models are Frequentist. It is mostly due to the training. It is this grounding in Frequentism that is the source of the mathematical problem in this question. It is not the only problem with those models, however.

If you wonder what "Frequentist" statistics are, they are the statistics someone made you learn in college with null hypotheses, t-tests, z-tests, and ordinary least-squares regression. Virtually all the money in the world is run on them.

In the field of probability, there is a concept called coherence. It is a property of some probability structures but not others.

A simple definition of coherence would be: A market maker's prices are coherent if a client cannot place a bet or a combination of bets such that no matter what the outcome occurs, the market maker will lose money.

In other words, things such as options or futures prices are coherent if a participant in the market cannot find a combination of contracts where the participant wins one hundred percent of the time.

Between 1930 and 1955, economists, statisticians, and probabilists worked on these problems and determined that Frequentist methods led to incoherent prices. Financial econometrics wouldn't exist until the mid-1970s as a real field. There was no mentoring overlap. Also, coherence is considered a minor feature in statistics because nobody makes a market for Mercury's location relative to the Sun for tomorrow at noon UCT. The sciences don't need coherence. Indeed, if economic models are never applied, then they never need to be coherent either.

So, focusing only on coherence, it is possible, if you know what you are doing, to construct financial contracts where the market making institution will take losses no matter what happens when they use prices built on Frequentist models. I don't think anybody else has noticed this before, or if they have, they haven't opened their mouth.

I have illustrated this in an exercise for economists. It is designed so that a college sophomore with one semester of statistics should be able to provide the standard answer as a simplified model. In it, the economists price a simple problem the way they would a complicated model like Black-Scholes using Frequentist methods. The economists then determine the market prices for the gambles. I then recalculate everything using coherent methods. Forty-eight percent of the time, I have no risk of loss. Fifty-two percent of the time, I have a seventy-five percent chance of winning when I am given even odds.

According to Frequentist methods, I should expect to break even over time. According to coherent strategies, I should expect a forty-eight percent rate of return per contract. Over thirty rounds, I have produced results ranging from turning $100 into $69,000 and up to $7,500,00,000. In the sure cases, I bet everything. In the ones with risk, I bet 48% of my money each round. Frequentist probabilities create a form of mathematical color blindness to the real probabilities.

Now for the law question. Let us assume that there is no attempt to manipulate prices. Indeed, this strategy works because someone wins no matter what price is obtained. The money captured is unrelated to the "risk-free" rate. Frequentist methods allow people to earn the risk-free rate when taking no risk, but these gains are from errors and so are independent of the interest rate on riskless investments. Indeed, if borrowing is allowed, it is possible to invest no money at all.

Let us assume that incoherent prices are present in the market. Let us also assume incoherently priced contracts exist and market makers are unaware. So a clever actor could build a strategy to only enter into a set of contracts where loss is impossible from contracts that are individually highly risky. In normal conditions, the market maker or a broker may extend credit so that the participant may need little of their own money.

The reason market makers are at risk is that they often cover residual amounts. Like a bookie where $1,000,000 is riding on horse A and $1,100,000 is riding on horse B, there is often net exposure if the bookie would allow it. Contracts do not perfectly match up in real markets.

So the first version of this question is, "assume a clever actor has found these contracts and enters into trades by accepting the bid or ask prices available at the market prices. Has a crime been committed?"

The second version requires a little bit more involvement. Assume a clever actor has found these contracts but not in the volume they want, so they make offers that are better than what the market maker is seeking to purchase a larger volume of trades. They are placing limit orders but are not planning on subsequent transactions to reverse these positions as the only goal is to let the contracts expire. Has a crime been committed?

The third version implies evil intent. Imagine that the clever actor is a vengeful sociopath and has enough information to target a particular market maker. They only make trades with this one counter-party on the hook so as to transfer its capital accounts into the nefarious actor's pockets. The behavior is the same, but the goal is to attack a particular institution.

The fourth version would be casus belli, but my interest is legal instead. What if a hostile nation realized that it could effectively rob another nation's food or energy distribution system or capital markets through a set of incoherent contracts, would it have committed a crime under US law when there is no direct target?

Finally, are the publicly traded financial institutions unknowingly violating Sarbanes-Oxley in this example?

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    I presume you know what arbitrage is, then, and it resolves much of the theoretical issue - yes, some people can finds ways to bet in a way that means they'll never lose. You also assume some unrealistic things (much as you point out is a criticism of Markowitz) - the fact that there exists a non-losing bet does not at all imply that there exists the specified or desirable non-losing bet. – Nij Jan 27 at 23:46
  • There are also many risks in arbitrage. Positions can take a long time to close (price disparities have been known in multi-market shares for years) and can also move the wrong way. Because all arbitrages are leveraged, this can send the arbitrager broke before they can collect. – Dale M Jan 28 at 4:11
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So the first version of this question is, "assume a clever actor has found these contracts and enters into trades by accepting the bid or ask prices available at the market prices. Has a crime been committed?"

Only if someone acted illegally in sharing this information, in violation of non-disclosures agreements, duties of confidentiality, some form of theft of trade secrets, or other forms of insider trading. The relatively fine details of how the information got out are pertinent to this inquiry and the law of insider trading is one of the less settled areas of securities fraud law. This said, I can't imagine as a practical matter, any way that someone could obtain all of the information that they need to implement the hypothetical trading strategy without obtaining information illegally on a massive scale from thousands of separate market participants. Pretty much everyone participating in the market is subject to some sort of non-disclosure agreement or confidentiality requirement that is designed to prevent other market participants from outwitting them.

The other possibility is a market manipulation securities fraud claim, but generally, a legitimate decision to make trades based upon knowledge or guesses about what positions people hold doesn't amount of market manipulation. Certainly a plan to hedge risk or arbitrage is not a crime.

Generally speaking, the gist of a market manipulation securities fraud claim involves a calculated and implicit false statement of fact communicated through bid and ask prices and volumes, rather than through a more ordinary channel of communication.

For example, in thinly traded stock, making a lot of wildly overpriced bid to suggest that something really good is happening to stock, at a time when there is a lot of "chatter" that something big is going on in the company, when you know at the time of making the overpriced bids that there is actually nothing going on or that the news that is about to be shared with the public is actually negative.

Your theoretical belief that it is possible to do what you suppose to market makers, collectively, is also unduly optimistic. Market makers are very canny and employ strategies that do an extremely good job of protecting themselves. The mere fact that it is theoretically possible to make a market maker lose money (and indeed, market makers do lose money now and then when their particular market is in particular turmoil) fails to account for the fact that as a practical matter, it is virtually impossible and that market makers can change their trading policies and strategies in real time to prevent that from happening, for example, by increasing buy and sell spreads, if necessary to prevent them from being taken advantage of. Market makers are not just robots, even if they feel like they are acting like robots in ordinary times.

In general, your concept that coherence is a lynch pin of how actual brokers and investors act in the market and that it is something that they rely upon is empirically not the case, nor is it the case that securities law, in general, makes such an assumption.

The implication that traders and market participants do not use Bayesian statistics, as well as Frequentist statistics, is also dubious and almost certainly not true.

The second version requires a little bit more involvement. Assume a clever actor has found these contracts but not in the volume they want, so they make offers that are better than what the market maker is seeking to purchase a larger volume of trades. They are placing limit orders but are not planning on subsequent transactions to reverse these positions as the only goal is to let the contracts expire. Has a crime been committed?

No, subject to the same caveats about the source of information and market manipulation.

The third version implies evil intent. Imagine that the clever actor is a vengeful sociopath and has enough information to target a particular market maker. They only make trades with this one counter-party on the hook so as to transfer its capital accounts into the nefarious actor's pockets. The behavior is the same, but the goal is to attack a particular institution.

There might be tort liability for intentional interference with contract, but subject to the same caveats as the other scenarios, probably not a crime.

Also, if a market maker gets a sense that something is amiss, a market maker can refuse to do business with you, and a pattern of conduct seemingly targeting a particular market maker is probably sufficient to get the sociopath party banned from the exchange. Singling out a particular market maker to do business with is highly suspicious activity and would raise red flags right away.

Exchanges are private and self-regulated institutions owned by brokers, and they are quick to respond if they suspect that some novel form of malicious conduct is afoot. If a handful of influential brokers get even a glimpse of insight into what is going on and how they are being played, they can adopt new exchange policies to address the concern in a matter of a few days or weeks. They don't need new laws enacted by Congress or new SEC regulations to do that.

The fourth version would be casus belli, but my interest is legal instead. What if a hostile nation realized that it could effectively rob another nation's food or energy distribution system or capital markets through a set of incoherent contracts, would it have committed a crime under US law when there is no direct target?

Primarily, this would be a political question, for diplomats and national security leaders to address on a basis not subject to strict legal analysis of the securities laws. But as a practical matter, there are requirements that agents of foreign sovereigns register their presence and loyalties when they are in the United States or deal with it, and there are disclosure requirements in the securities laws for holders of large positions that would make it as a practical matter exceeding difficult to accomplish a scheme like this without violating disclosure laws.

Finally, are the publicly traded financial institutions unknowingly violating Sarbanes-Oxley in this example?

Sarbanes-Oxley is a many faceted omnibus law that says all manner of things, and it is not obvious which of its provisions you would be alluding to in this case.

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  • Thanks. I seem to have accidentally created a new account for this one post when I changed computers. Your answer concurs with my beliefs on the matter. Just for background purposes, I am a financial and monetary economist. I am proposing a new integral and derivative. The Ito calculus options prices are based on has an assumption that all parameters are known and that nobody is performing estimation. John White in 1958 proved that if estimation must be performed, then no solution can exist to models like Black-Scholes inside the Frequentist framework. – Dave Harris Jan 28 at 5:15
  • I dropped the assumption that the parameters are known with certainty and restructured the rules of calculus for finance and macroeconomics. I have spent a decade in financial services. – Dave Harris Jan 28 at 5:16
  • I have considered all your objections, as I would have made them myself. I also assume that a handful of traders use Bayesian methods or traditional Graham and Dodd analysis. So I asked myself, how to get around them. The neat part about looking at Frequentist gambles in this manner is that really is invisible in a way similar to color blindness. So I looked at the question of LEAPS to delay execution. I agree that the guarantee is only at maturity, prices can swing wildly in the interim. Also, about spreading trades over many makers so that no one source could easily be identified. – Dave Harris Jan 28 at 5:19
  • I should have included deception in the discussion. What if a trader is executing an arbitrage pattern but spreading it out widely and long over time so that losses are not obvious until large positions have opened up? – Dave Harris Jan 28 at 5:20
  • I also understand this is not legal advice. I am actually asking these questions to cover my bases in an article as the law is its own animal. I am trying to see the range of research I need to do. – Dave Harris Jan 28 at 5:23

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