Market Based Control

Australia’s only Fields Medallist, Terry Tao, has a post about the [tag]Black-Scholes equation[/tag], here.

POSTSCRIPT:  Portfolio magazine carried an article critical of Black-Scholes earlier this year:

“Good theory. The glitch was discovered only after the fact: When a market is crashing and no one is willing to buy, it’s impossible to sell short. If too many investors are trying to unload stocks as a market falls, they create the very disaster they are seeking to avoid. Their desire to sell drives the market lower, triggering an even greater desire to sell and, ultimately, sending the market into a bottomless free fall. That’s what happened on October 19, 1987, when the sweet logic of Black-Scholes was shown to be irrelevant in the real world of crashes and panics. Even the biggest portfolio insurance firm, Leland O’Brien Rubinstein Associates (co-founded and run by the same finance professors who invented portfolio insurance), tried to sell as the market crashed and couldn’t.

Oddly, this failure of financial theory didn’t lead Wall Street to question Black-Scholes in general. “If you try to attack it,” says one longtime trader of abstruse financial options, “you’re making a case for your own unintelligence.” The math was too advanced, the theorists too smart; the debate, for anyone without a degree in mathematics, was bound to end badly. But after the crash of 1987, individual traders at big Wall Street firms who sold financial-disaster insurance must have smelled a rat. Across markets—in stocks, currencies, and bonds—the price of insuring yourself against financial disaster rose. This rise in prices and the break with Black-Scholes reflected two new beliefs: one, that huge price jumps were more probable and likely to be more extreme than the Black-Scholes model assumed; and two, that you can’t manufacture an option on the stock market by selling and buying the market itself, because that market will never allow it. When you most need to sell—or to buy—is exactly when everyone else is selling or buying, in effect canceling out any advantage you once might have had.

“No one believes the original assumptions anymore,” says John Seo, who co-manages Fermat Capital, a $2 billion-plus hedge fund that invests in catastrophe bonds—essentially bonds with put options that are triggered by such natural catastrophes as hurricanes and earthquakes. “It’s hard to believe that anyone—yes, including me—ever believed it. It’s like trying to replicate a fire-insurance policy by dynamically increasing or decreasing your coverage as fire conditions wax and wane. One day, bam, your house is on fire, and you call for more coverage?” “