As I wrote two weeks ago and last week, by reading Ghislaine Idabouk's briliant dissertation on the mathematics of options pricing (Black, Scholes, Merton, etc) I have been inspired to do a weekly blog on philosophy of economics. This week I am inspired by one of her central points about the development of the famous optionspricing formula. She shows nicely how in the Fundamental Theorem of Asset Pricing, the assumption of No Arbitrage (about which more in the future), or risk-neutral investment, a martingale measure could be linked to the existence of a pricing function. A martingale is a stochistic process that defines (for example, a lot of) fair games. (For example, Brownian motion is a semimartingale.) Historically and empirically, the link results from the idea that stock markets are basically random walks in which no arbitrage is possible and that there is at least one asset that is risk-free and produces a predictable cash-flow (imagina a default free US Treasury). I am not a very good mathematician, so what follows is really the result of inspired conversations with Idabouk, my best former student, Gwyneth Harrison-Shermoen (who is getting a PhD in math at Berkeley), David M. Levy (an iconoclastic economist), and Ali M. Khan (an economist, who is a bit skeptical about what follows).

Anyway, the martingale helps define a trajectory of a stochastic process. And so we can ask an innocent seeming question: do the set/class of values that end up in the trajectory have special properties? As David Levy first reminded me, we know the answer is "yes" because it turns out some distributions (median distribution) come naturally, while others (e.g. mean distributions) not. (Even my arch-villain, Eugene Fama, comes close to seeing the significance of this.) So, this means that the modal property of the values that end up in the trajectory (i.e. really possible) is different from some of the values that could have ended up in the trajectory (merely possible), and those that never could end up in the trajectory (i.e., impossible). If we define prices in markets in this way (and recall that the financial apparatus offers a normative theory), the * modal* assumptions of our mathematical theories deserve more scrutinity. Maybe philosophy departments can grow rich with courses on possible world semantics for financial markets? (Brian Weatherson at Rutgers could probably teach a course like this in his sleep!)

So, when we conceptualize the (evolution of) prices of financial instruments, our mathematical tools categorically rule out some possibilities. These are not the so-called the tails or the cause of systemic risk (both engineering categories), but the theoretically impossible, yet real world possible--that is, uncertainty.

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