From fluid dynamics to finance, creatures like the Weierstrass function have challenged our ideas about the relationship between mathematics and the natural world. Mathematicians around the time of Weierstrass used to believe that the most useful mathematics was inspired by nature, and that Weierstrass’ work did not fit into that definition. But stochastic calculus and Mandelbrot’s fractals have proven them wrong. It turns out that in the real world—the messy, complex real world—monsters are everywhere. “Nature has played a joke on the mathematicians,” as Mandelbrot put it. Even Weierstrass himself fell victim to the trick. He created his function to argue that mathematics should not be based only on physical observations. His followers believed that Newton had been constrained by real-life intuition and that, once free of these limitations, there were vast, elegant new theories to be discovered. They thought that mathematics would no longer need nature. Yet Weierstrass’ monster has revealed the opposite to be true. The relationship between nature and mathematics runs deeper than anyone ever imagined.
Illustration : Alessandro Gottardo
Of course, honest traders change their minds all the time and cancel orders as economic conditions change. That’s not illegal. To demonstrate spoofing, prosecutors or regulators must show the trader entered orders he never intended to execute. That’s a high burden of proof in any market. One helpful fact is if most of a trader’s (canceled) orders were on one side (say to buy) when he was mostly actually trading on the other (selling). For instance Sarao allegedly put in huge orders to sell, so that he could buy a few contracts: All his trading was on one side, but most of his orders were on the other. Then he’d switch a little while later. That seems like a bad sign.
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Instead of trying to identify profitable trading algos on in-sample data that validate out-of-sample and remain profitable forward, one could instead try to identify unprofitable algos in some data sample that turn profitable in a forward sample. This often works because markets have become more mean-reverting in recent years.
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To illustrate the phenomenon, consider the S&P’s daily percentage returns in terms of quantiles, which divides the performance record into equal-sized portions. The graph below plots the sample return of the S&P (black circles) against the theoretical quantiles (red line), defined here by a random distribution. If the S&P’s daily returns were perfectly random, the black circles would match the red line.
Normal distributions are still useful for analyzing markets and designing portfolios. Indeed, even in the daily return plot above it’s clear that the distribution looks quite normal for a fair amount of the sample. We can’t rely on normality alone for modeling markets. Factoring in fat-tails risk is essential. But letting a fat-tail worldview dominate your analysis is every bit as flawed as assuming that normal distributions will prevail. Asset pricing doesn’t neatly fit into one theoretical box, which means that our analytical tool kit shouldn’t be in a conceptual straightjacket either.
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The nemesis of Wall Street’s high-frequency traders operates out of an apartment-sized office above the Bliss Salon — manicure/pedicure $45 — on Elm Street in the Chicago suburb of Winnetka.