Tuesday, April 15, 2008

von Neumann...Mandelbrot?

A 'straw man' argument is to set up a weak counterproposal, and then criticize it. It is a 'straw man', easy to knock down. I thought this was a discredited technique that would generally make the author look bad, but the more I read popular books, the more I see this approach taken. Best selling critics of the free market, such as John Kenneth Gallbraith and Lester Thurow, would criticize any misfortune as an anomaly to libertarians who thought markets were perfect.

But then I was in my local Borders, and was reading Michael Mauboussin's More Than You Know, which is about investing and such, with an academic/practical slant, something I would like. The book has its moments, and its short chapters and easy reading make it a great book to read while drinking a cup of coffee, and then return to the stack. The book has lots of graphs and tables. I love books with graphs and tables, and wish more books did, as I find it odd that a 300 page book on investing would not have these thing (eg, Against the Gods, or Random Walk Down Wall Street has but a few), as these make points so much better than mere words. So bully for Mauboussin. But he had a chapter on "frequency versus magnitude in expected value", and highlighted that some really smart people like Warren Buffet assess expected value as probability times payoff, so an improbable event may actually be a good buy, because its payoff is sufficiently large. Who knew? He even quotes my favorite flâneur, Nassim Taleb, for noting that as a trader he would be short sometimes even when he thought the market was going up, because the size of a downturn would make the expected return negative [forget about the fact that trading floors hate it when their traders have directional bets]. Supposedly, most rubes merely look at the probabilities, ignoring payoffs.

An expected value is the weighted average of the payoff times the probability. It has always been that way. To think this is subtle, or rare, strikes me as daffy, after all, most options expire worthless, yet have a positive price. Contradiction? No.

Then I moved over in the stacks and found Benoit Mandelbrot's Misbehavior of Markets, and there's a blurb by Paul Samuelson:
On the scroll of great non-economists who advanced economics by quantum leaps, next to John von Neumann we read the name Benoit Mandelbrot

ORLY? von Neumann is famous for his von Neumann-Morgenstern utility function, the workhorse of most economic analysis. He had a macro model too, but I think it's safe to say that was a noble effort, but a dead end. But Mandelbrot? He is usually the first reference of fat tails for financial distributions based on a 1963 article on cotton prices, but I fail to see this as a truly earth-shattering finding because almost everything natural has fatter tails than a normal (gaussian) distribution, so who was suspecting otherwise? Mandlelbrot's book points out (and so does Mauboussin's) that the stock market has fat tails, and states that models like Black-Scholes, and the CAPM, are based on the Gaussian hegemony that forces us to think that such things can only have normal distributions, a legacy of Bachelier from his work on brownian motion in 1903. But these models use the Gaussian model as an expositional device, because it generates nice, clean closed form solutions. Markowitz's earliest book (Portfolio Selection: Efficient Diversification of Investments, 1959) looked at semi-deviation, maximum loss, and other asymmetric loss functions. Thus since the very beginnings of the MPT, researchers have been aware that distributional assumptions were important, and obvious paths for alternative hypotheses. Looking at the Journal of Portfolio Management, “skewness” is mentioned in no less than 66 articles, kurtosis in 44. The bottom line is that after 40 years, this obvious fix—adding a cube or square term, effectively—appears only episodically as a solution, invariably not standing up to subsequent scrutiny.

In general, the first order effects are not great when ignoring these complications, so the cost of messing up the formulas to include more free parameters, at little benefit, is not done. But the CAPM doesn't work regardless, and Black-Scholes is fit to a volatility smile--always has. No one I have met has ever confused the map with the territory, when it comes to asset pricing models,

In 1987, James Gleick published a best selling book Chaos, highlighting Benoit Mandelbrot's fractals, and suggested he had a new insight that was about to revolutionize all of science, including finance. But as Rubinstein states in A History of the Theory of Investments , "In the end, the stable-Paretian hypothesis [ie, fractals] proved a dead end, particularly as alternative finite-variance explanations of stock returns were developed".

Ever since I have paid attention to option prices in the 1980's, options were priced anticipating fat tails, via a volatility smile, yet Mandelbrot would have you believe that no one knows this, or at least, that The Establishment thinks Gaussian distributions are perfect replications of reality. The 1987 crash was a 22 standard deviation event, happening once every 60 bazillion years according to the Gaussian knaves. Now truly people did not anticipate this, because the previous largest move was less than 10%, but I don't think his model is any great improvement because anticipating a 22 stdev movement in any option with a significant probability will cause you to overpay severely for illiquid options with wide bid-asks. I actually was actually bought an out-of-the-money S&P500 put option on October 16 that I sold in the last 15 minutes of trading on October 19, 1987, and made $38k on a $3k investment (my life savings at that time, see here). I remember not getting my mark for a day, and finally being surprised to see my option sold with an implied vol of around 70. Great return and all, but it wouldn't pay for a lifetime of buying 3-delta options (i.e., the way out-of-the-money options). However, if you include that investment in my arithmetic average annualized return, my life-to-date Sharpe in my personal account is around a 5--highlighting the problem with arithmetic averages.

Benoit Mandelbrot is a very smart man, but I have a feeling that he truly believes his great idea in finance is that prices follow these power-law distributions with extra parameters. That is simply a not a good idea, and I can imagine his brother-in-law thinking 'why does everyone think this guy is so smart?'

Anyway, are such straw man arguments necessary to sell a book? I should write a book on the theory that some assets go up, some go down, but very few stay exactly the same price. This proves that everyone who thinks prices are 'right' is a fool, because prices change, proving the old price wrong. The Man tells you that prices are the best estimate of market value, but he is proved wrong every day. QED. (patent no. D696,243).


Anonymous said...

Do we get no insight as to what prompted you to put your life savings into an out-of-the-money put option??? Did something stick out in one of your models or did you just have a hunch?

What about your reaction - were you convinced you were a supergenius, then you threw money around recklessly seeking even greater returns, or were you grounded enough to recognize your luck and walk away?

Eric Falkenstein said...

Of course there was a theory. I had been Hyman Minsky's TA the year before, and became a convert to his Financial Instability Hypothesis, and his overall bearishness. Basically, he taught me that we were always on the brink of a financial collapse, which is always predicated by the Fed tightening. The Fed was fighting to keep the dollar strong, and I saw the stresses it was causing, so I thought that the Fed funds rate would keep climbing until the market tanked. Now, I could have bought a fed funds future, but they didn't have options. so I put my money on the stock fall instead.

Turned out I was very fortunate, because after the crash, the Fed Funds rate fell about 200 basis points that week.

Anonymous said...

Eric, I share your appreciation for Mauboussin's "More Than You Know."

I either disagree or don't understand your comment that "I have paid attention to option prices in the 1980's, options were priced anticipating fat tails, via a volatility smile, yet Mandelbrot would have you believe that no one knows this..." (etc.).

According to Wilmott contributor and Morgan Stanley derivatives manager Aaron Brown, in his book "The Poker Face of Wall Street," options prior to October 1987 were priced with a flat volatility surface, not a "smile." He says it was due to a lack of computing power and mental inertia among options traders -- the "shortcut" had worked for so long that it seemed easier than doing the quant work. Brown claims that shortly after the '87 crash, all the options were more "correctly" priced for volatility.

So Brown would probably say Mandelbrot makes an important point -- markets didn't use his models pre-'87 crash, and now that they do, it's an improvement.

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