Many Mathematical Finance “Proofs” Little More Than A Shell Game


For decades the fondest wish of the finance professoriate has been to prove that money managers who believe they earn alpha are kidding themselves and their customers. The latest attempt, titled “Active Portfolio Management and Positive Alphas: Fact or Fantasy?” is the work of Cornell’s Robert A. Jarrow, a prestigious name in mathematical finance. Jarrow, based on some previous work with Philip Protter, sets out to prove that the source of all (or nearly all) alpha must be a true arbitrage. Since true arbitrage is vanishingly rare, he then argues alpha must be as well.

Jarrow’s paper follows a form that has been common since the advent of modern portfolio theory created investment theory as an academic discipline. Here’s how it goes:
Along comes a professor. (Where’d he come from? Don’t be silly; we are doing economics. We assume the professor.)

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Now, where were we? Oh right, along comes a professor. The professor proclaims that he will prove “if A then B.” Surprising no one, he succeeds. His proof is rigorous and correct, just as one would expect from a man who has any number of degrees to his name and even better gets to pick A, plus gets to assume that A is the case and therefore has nothing to prove except that “A implies B,” which he also gets to pick.

Next the professor, with becoming modesty, admits what we already know: proving “if A then B” doesn’t really get us very far. Genuine A’s are scarce on the ground. So showing that these rare beasts imply B doesn’t amount to much in the real world. But wait, the prof tells us, there’s more to come. Then follows a certain amount of discussion and hand waving, plus a great deal of impressive math. Finally the professor announces that now he has something really important to tell us.

See Full PDF here WBA-How-Big-is-Almost (take down of EMH)