Problem with Stochastic Calculus

One issue that has haunted models such as Black-Scholes, the CAPM, the APT and Fama-French is that, none of them have survived validation tests. I believe that each of these models shares a mathematical problem that has previously gone unnoticed. The solution to this problem could lie in constructing a new stochastic calculus for these class of problems.

Let’s consider a wheel marked with numbers; a roulette wheel will do. Think of this as an inverse game of roulette with information. The gamblers cannot see where the ball falls and they place their gambles only after the croupier observes the number.

This example is relatively common in decision theory. After the ball lands and a number is chosen two fair coins are tossed. If a coin comes up heads, then the croupier will reveal to the gamblers the value one unit to the left of where the ball landed. If the obverse comes up, then the croupier will reveal the value of one unit to the right of where the ball landed. This creates a sample space with three possible outcomes, {L,L}, {R,R} and {L,R}. Our concern is the optimal action to take regarding choosing a number to place a gamble on.

Let us assume the number came up 17. The signals from the croupier will be either {16,16}, {18,18} or {16,18}. It is obvious that if it is {16,18}, then you must bet on 17. However, the Bayesian solution and the Frequentist solution do not match otherwise. Both would choose 17 in the case of {16,18}, but they differ in the case where the numbers given by the croupier are the same.

The minimum variance unbiased estimators in the case of either {16,16} and {18,18} are 16 and 18 respectively. In the case of {16,16}, the Bayesian probability model supports either element of the solution set {15,17} equally. For the {18,18} case the same support is found for {17,19}. It does not minimize the variance, but it does maximize the winning frequency. In expectation, the Bayesian gambler would win 75% of the time, while the Frequentist would win 50% of the time.

This illustration shows two crucial facts. The first is that the choice of axiom systems can determine the course of action, and not agree with the decision function of the other system. The second is an illustration of the Dutch Book Theorem.

The Dutch Book Theorem is similar to the no-arbitrage assumption but weaker in its base assumptions. However, it has an unexpected result. You can always use Bayesian methods for gambling, and you cannot use Frequentist methods for gambling. Models such as Black-Scholes and the Capital Asset Pricing Model are instructions on how to gamble in a specific type and set of lotteries. They are built on Frequentist axioms. Now imagine an economist testing the above game where nobody is behaving as he or she are supposed to act under the MVUE. Economists may reject the model or may argue that people are behaving irrationally, but really it is because of the axioms used, not the behaviour. Any large-scale test of behaviour would come out “wrong.”

A market-maker or bookie using Frequentist rules would also suffer, not just the economists. Long-run optimal behaviour would grant 1:1 odds. Market makers using Ito models should, from time to time, have people eat their lunch. It is no wonder hedge funds abound. Dangerously, the eleven trillion dollars in outstanding over-the-counter options premiums are mispriced if they are built on Ito methodologies. By theorem, anything built on an Ito methodology will be systematically mispriced, even if every assumption is valid.

The failure to consider the purpose of models such as Black-Scholes in gambling and the failure to account for a proper loss function makes the utility and the appropriate evaluation of these financial models doubtful. A good part of this may have been that these models became more important than the authors likely intended. In a sense, economics took them too seriously, especially since they lack validation.

While it has been the case from time to time that researchers have used Bayesian methods to test these models, there is a problem with doing this with Frequentist models. As with the roulette example above, when constructed in a different paradigm, the two models make two different predictions. To examine a Frequentist model with Bayesian methods may be to check the wrong predictions. After all, a Bayesian check that the bookies should offer 1:1 odds would fail for the Bayesian just as readily as for the Frequentist, but the proper Bayesian prediction isn’t for 1:1 odds. A Bayesian methodology requires a complete rebuilding of the model. That is where a calculus problem begins to happen.

How would you approach resolving this problem? Please share your thoughts in the comments section below as I learn just as much from you as you do from me.