# The Black Swan is episte-riffic!

April 30, 2007 12:50 PM Subscribe

*The Black Swan: The Impact of the Highly Improbable*by Nassim Taleb is out. Reviews in the Wall Street Journal, LA Times, and Financial Times. Just in time with those of us with a love of Hume's problem of induction, non-Gaussian distributions and financial intellectualism. Read an early draft of chapter 16,

*The Bell Curve, That Great Intellectual Fraud*. Read Taleb's "philisophical and literary notebook." Then, in a feat of metanarrative rarely seen outside of Metatalk, read his comments on comments on the book. Previously on Metafilter: Languagehat has already made his thoughts on Taleb known, it wasn't pretty, and someone with "vested interests in Taleb" responded. Taleb, refreshingly, does not shy away from debates about his work.

From my background psychology, his theories and writing seems a bit out of place and counter examples keep popping into my head as I read him, but given his focus in financial markets, where improbable events can be highly profitable or costly, this type of analysis and style of thinking makes sense. Different worlds I guess.

posted by bhouston at 1:12 PM on April 30, 2007

posted by bhouston at 1:12 PM on April 30, 2007

When I read the title of chapter 16, "The Bell Curve, That Great Intellectual Fraud", given by background in psychology, I immediately through it was about the very controversial book entitled "The Bell Curve" and it confused me since it seemed a bit late for him to be spending a lot of time critiquing that book.

posted by bhouston at 1:14 PM on April 30, 2007

posted by bhouston at 1:14 PM on April 30, 2007

First, I generally like geoff's posts and comments, so please take this in the spirit in which it's intended, but calling out languagehat in the post is bad form, I would think.

Anyway, I have read Taleb quite a bit, and I generally agree with the proposition that rare or singular events have huge impacts on averages in financial markets, but I can't help but thinking that he represents the worst kind of social darwinism. He's not really successful because of his philosophy and opinions, and yet we are treated to them because he's rich.

The example excerpted by languagehat is a good example of this:

His mind is actually just as closed as hers. It was his place to ask her what interested her about "randomness" and why she felt "markets" somehow lay outside of that interest. In the story, she is the one stepping furthest outside of her intellectual milieu, not Taleb, and not Maldelbrot. And yet she's criticized for not understanding why his field should be interesting?

In the face of conflicting and inconsistent"wisdom" of various Wall Street sages, I've developed my own philosophy. Buy the market, and learn to paint.

posted by Pastabagel at 1:40 PM on April 30, 2007

Anyway, I have read Taleb quite a bit, and I generally agree with the proposition that rare or singular events have huge impacts on averages in financial markets, but I can't help but thinking that he represents the worst kind of social darwinism. He's not really successful because of his philosophy and opinions, and yet we are treated to them because he's rich.

The example excerpted by languagehat is a good example of this:

*When she learned by looking at my bio on the dust jacket that I was “in markets”, she gave me the look as if I had killed her mother. She turned her back to me as I was in mid-sentence, leaving me to the discomfort of having to speak without audience. It feels extremely humiliating to be speaking to someone’s back; it felt like the worst, most demeaning insult I ever had in my life.*His mind is actually just as closed as hers. It was his place to ask her what interested her about "randomness" and why she felt "markets" somehow lay outside of that interest. In the story, she is the one stepping furthest outside of her intellectual milieu, not Taleb, and not Maldelbrot. And yet she's criticized for not understanding why his field should be interesting?

In the face of conflicting and inconsistent"wisdom" of various Wall Street sages, I've developed my own philosophy. Buy the market, and learn to paint.

posted by Pastabagel at 1:40 PM on April 30, 2007

nicolin, Taleb is trying to make a case for the non-deterministic behavior of financial markets. His philosophy goes beyond financial markets and he makes cases that we often resort to things such as narrative fallacies that are convenient for journalists but undermine that a lot of things happen due to serendipity rather than cause-effect determinism. He's basically an empirical skeptic along the lines of Hume, Popper, et al.

With that disclaimer, he teamed up with Benoit Mandelbrot to write this book (I am not finished with it, I'm in the middle of two other books at the moment). Mandelbrot did some great research on distribution laws and cotton prices way, way back in the 1960s. He observed, correctly, that the bell curve distribution is a poor way of viewing markets and finance and noticed that fractal geometry creates far better approximations than Gaussian derived financial models. This has to do, among other things, the long standing observation that stock prices often exhibit long-memory (that is stocks going up will continue to go up, the market seems to remember the behavior of stocks). Taleb wrote a book called Fooled by Randomness which talked about this and other behaviors of the markets, that it contains properties such that being a trader is more of a skill than a science.

So what, you might ask? Well many modern theories in finance are based on Bachelier's original paper which more or less defines Brownian motion. With Brownian motion, movement is said to be normally distributed. Regardless of the final destination of the particle or stock, its next movement has an equal chance of being up or down (this is a bad technical description). I am currently going over John Hull's standard text Options, Futures, and Other Derivatives and it makes a lot of assumptions, very early on, to base decisions on a normally distributed model. For example, starting on page 88:

Now many people will argue that many capital asset pricing models or GARCH, etc. make adjustments to this, and I will agree that adjustments are made. I still think that they severely underestimate the need to avoid downside risk is higher than the need to achieve returns above a certain rate of return.

Which is where the "black swan" and the problem of induction comes in. Many models look at historical data and inductively come up with estimated variance. If you never have a black swan the model won't know its there, but black wan events (stock market crashes, 9/11) happen in reality far more than is usually prepared for. Returns in the market especially are incredibly asymmetric. One black swan can wipe you out (or make you rich). He argues that the markets are not as math based as academics would like and that it is more of a skill than a formulation.

I could go on but I would fear that I would erroneously present modern finance, his ideas and induction far worse than I already have.

posted by geoff. at 1:43 PM on April 30, 2007

With that disclaimer, he teamed up with Benoit Mandelbrot to write this book (I am not finished with it, I'm in the middle of two other books at the moment). Mandelbrot did some great research on distribution laws and cotton prices way, way back in the 1960s. He observed, correctly, that the bell curve distribution is a poor way of viewing markets and finance and noticed that fractal geometry creates far better approximations than Gaussian derived financial models. This has to do, among other things, the long standing observation that stock prices often exhibit long-memory (that is stocks going up will continue to go up, the market seems to remember the behavior of stocks). Taleb wrote a book called Fooled by Randomness which talked about this and other behaviors of the markets, that it contains properties such that being a trader is more of a skill than a science.

So what, you might ask? Well many modern theories in finance are based on Bachelier's original paper which more or less defines Brownian motion. With Brownian motion, movement is said to be normally distributed. Regardless of the final destination of the particle or stock, its next movement has an equal chance of being up or down (this is a bad technical description). I am currently going over John Hull's standard text Options, Futures, and Other Derivatives and it makes a lot of assumptions, very early on, to base decisions on a normally distributed model. For example, starting on page 88:

The optimal hedge ratio, h*, is the slope of the best fit line when S∆ is regressed against ∆F and the ratio of standard deviation of S∆ to the standard deviation of F∆.With the standard deviation formulas given for a bell-curve distribution without even caveats that the distribution is non-normal. The implicit assumption is that stock variation will fit a bell-curve distribution.

Now many people will argue that many capital asset pricing models or GARCH, etc. make adjustments to this, and I will agree that adjustments are made. I still think that they severely underestimate the need to avoid downside risk is higher than the need to achieve returns above a certain rate of return.

Which is where the "black swan" and the problem of induction comes in. Many models look at historical data and inductively come up with estimated variance. If you never have a black swan the model won't know its there, but black wan events (stock market crashes, 9/11) happen in reality far more than is usually prepared for. Returns in the market especially are incredibly asymmetric. One black swan can wipe you out (or make you rich). He argues that the markets are not as math based as academics would like and that it is more of a skill than a formulation.

I could go on but I would fear that I would erroneously present modern finance, his ideas and induction far worse than I already have.

posted by geoff. at 1:43 PM on April 30, 2007

pastabagel, it wasn't my intention to call languagehat out, I thought the exchange was a somewhat humorous example of how Taleb can be a bit irascible in his discourse. I love that dumb fuck languagehat, yes I do.

posted by geoff. at 1:57 PM on April 30, 2007

posted by geoff. at 1:57 PM on April 30, 2007

*calling out languagehat in the post is bad form, I would think.*

I certainly didn't feel called out, and I see geoff. didn't intend it that way. I expressed myself vigorously, and apparently geoff. felt it was worth mentioning as an example of strong reactions induced by Taleb. (And before anyone gets on his case about calling me a dumb fuck, he's just quoting my own words about myself!)

posted by languagehat at 2:01 PM on April 30, 2007

So is this the sequel (of sorts) to The Misbehavior of Markets? That was a fascinating book, if this is in the same direction I expect I'll enjoy it as well.

posted by Skorgu at 2:04 PM on April 30, 2007

posted by Skorgu at 2:04 PM on April 30, 2007

You called out languagehat in a FPP? Not good. Not good at all.

posted by caddis at 2:06 PM on April 30, 2007

posted by caddis at 2:06 PM on April 30, 2007

So, wealth and power are not among the quantities that exhibit normal distributions. And people sometimes attribute meaning to more-or-less random events. IANA economist or chaos theoretician, is there anything else I need to know about this book?

posted by Mister_A at 2:19 PM on April 30, 2007

posted by Mister_A at 2:19 PM on April 30, 2007

For those who care: I should have put the John Hull equation in context. F∆ and S∆ refer to two sets of data in the change of future prices and the change of spot prices. You then perform standard statistics equations to the data to obtain the optimal hedge ratio given the variance and finally the optimal number of contracts given the portfolio. That's fine as I see that using normal distributions are easier to work with in a textbook format, but there's absolutely no discussion of the equations being for a bell curve distribution or the implications of using the bell curve (like discussing the number of times the market has experience variance far out of line what a normal distribution would suggest).

skoorgu, if you liked

posted by geoff. at 2:24 PM on April 30, 2007

skoorgu, if you liked

*Misbehavior of the Markets*I would suggest Fractal Market Analysis by Edgar Peters, which is a much more technical look at the subject. I have a PDF if you want (I doubt you'll find a copy in Barnes and Noble).posted by geoff. at 2:24 PM on April 30, 2007

What he seems to be saying about the bell curve is that it doesn't apply in all situations.

Which is true, but not particularly profound. I don't think anyone ever claimed otherwise.

In population dynamics it's often the case that what you get is something that looks a lot like a bell curve, except that it's unbalance on one side or the other, or has a bit of a flat spot on the top.

On the other hand, electronics noise often has a gorgeous agreement with the bell curve. Things like A/D jitter balances beautifully.

If you study a case and apply the wrong statistical model to it, you won't get the right answer. That seems to be what he's saying. But that, too, is true and not particularly profound.

posted by Steven C. Den Beste at 2:30 PM on April 30, 2007

Which is true, but not particularly profound. I don't think anyone ever claimed otherwise.

In population dynamics it's often the case that what you get is something that looks a lot like a bell curve, except that it's unbalance on one side or the other, or has a bit of a flat spot on the top.

On the other hand, electronics noise often has a gorgeous agreement with the bell curve. Things like A/D jitter balances beautifully.

If you study a case and apply the wrong statistical model to it, you won't get the right answer. That seems to be what he's saying. But that, too, is true and not particularly profound.

posted by Steven C. Den Beste at 2:30 PM on April 30, 2007

hi geoff,

actually I made a mistake, my comment is about previous post...(baseball fans...)

sorry

posted by nicolin at 2:33 PM on April 30, 2007

actually I made a mistake, my comment is about previous post...(baseball fans...)

sorry

posted by nicolin at 2:33 PM on April 30, 2007

*I have a PDF if you want*

Yes please, gmail in profile. It's funny, finance is the only subject that makes math make sense to me. You could put the same content into a physics or pure math context and I'd be out to sea, but add dollar signs and it clicks. Well, it comes closer to what I imagine "clicking" is like anyway.

posted by Skorgu at 2:47 PM on April 30, 2007

My apologies, but I feel I'm missing something important here-- I've read the Guardian review of this book, and my first (and so far, only) reaction has been: life is random. Uh huh. Random events determine reactions in financial systems (and everything else). Random events are unpredictable. Anyone who thinks they can forecast the future is kidding themselves. Event and reaction are interdependent, teetering structures with no end point and no beginning. This is big news?

Could somebody be kind enough to give the statistics-challenged among us some kind of idea about why this is considered important?

posted by jokeefe at 2:49 PM on April 30, 2007

Could somebody be kind enough to give the statistics-challenged among us some kind of idea about why this is considered important?

posted by jokeefe at 2:49 PM on April 30, 2007

**Steven**, yeah, but the problem is that in finance at least, many people have at least by omission assumed that markets behave in a gaussian way. And been fairly resistant to address that assumption as well. Finance isn't anywhere near as open to refutation as the purer sciences are, at least in my limited experience.

posted by Skorgu at 2:50 PM on April 30, 2007

Hmm, the gist I'm getting is the same one

The normal distribution doesn't always apply, especially to financial markets. No shit? I'd like to read more of this, but I'm actually too busy studying probability models right now, go figure (and busily checking my assumptions about the normality, non-normality of the data I model!).

Does it get any deeper than that?

Economics models in general seem to make a lot of fallacious assumptions that make the models possible, but that also make them poor models of reality: that financial market models do that too would not be in the least bit surprising to me.

posted by teece at 2:54 PM on April 30, 2007

*Steven C. Den Beste*is getting, I think.The normal distribution doesn't always apply, especially to financial markets. No shit? I'd like to read more of this, but I'm actually too busy studying probability models right now, go figure (and busily checking my assumptions about the normality, non-normality of the data I model!).

Does it get any deeper than that?

Economics models in general seem to make a lot of fallacious assumptions that make the models possible, but that also make them poor models of reality: that financial market models do that too would not be in the least bit surprising to me.

posted by teece at 2:54 PM on April 30, 2007

**jokeefe**I'll give it a whirl.

I really must learn to preview one of these years. people who actually know something about stats should please correct me, I'm an enthusiastic amateur.

The problem is that the word 'random' has a very different meaning in math than it does in common language. Things we humans think of as 'random' are usually barely connected with a strict definition of the word. To compound it, there are different kinds of randomness.

Think of raindrops, you'd be hard-pressed to argue that they're not randomly distributed, yet you can predict pretty accurately how many drops, or how much liquid will fall in a given area given some knowledge about how hard it's raining. The fancy name for this class of randomness is a gaussian distribution. Lots of natural events fall into this pattern, so much so that most of statistics was written with it in mind. Because it's so common, it's natural for scientists to unintentionally assume that a given source of random data fits into this distribution.

But there are some things that simply don't fall into this model. Taleb, and Mandelbrot who I'm more familiar with, argue that financial systems are in the second category. To paraphrase Misbehavior, the Black Monday stock market crash was so severe, that if the market had been gaussian (or, paradoxically, "Normal"), it should have happened once every ten thousand years or so. Yet it was smaller than the crash of 29, and nearly as big as the 2000 dotcom crash. You can't have one-in-ten-thousand events happening every fifty years without noticing that something is up.

On preview (hah!), teece has it. Nobody understands the markets, so the good-enough models lasted substantially longer than they probably should have.

posted by Skorgu at 3:03 PM on April 30, 2007 [1 favorite]

It goes beyond just a refutation of non-normal distributions, but also explores the many fallacies surrounding explaining properties that emerge in market behavior. Economists and their models are especially bad at this, but watch market announcements to understand how pervasive this is ("Market went down on Iranian news" Oh really? The news and the markets declining didn't happen appear together as coincidence?) ... and the multitude of people who believe they successfully predict stock prices. It is not about the results, the results can be right or wrong on a lot of factors, but the way these results are produced. If a monkey is picking good stocks it would be erroneous to believe he has skill in producing good stocks without looking at the generating device itself.

The problem isn't that it is bad science, but that finance will never be a science. You can never know all the inputs and you won't be able to to track stocks like you can track planets.

posted by geoff. at 3:07 PM on April 30, 2007

The problem isn't that it is bad science, but that finance will never be a science. You can never know all the inputs and you won't be able to to track stocks like you can track planets.

posted by geoff. at 3:07 PM on April 30, 2007

Languagehat acquitted himself too well in that thread ever to be very upset at having it pointed to, I think, regardless of the intent of the pointer.

I thought I felt the presence of two issues left unstated in Taleb's narration of his collision with Sontag:

1. She was a Jew and a Zionist; Taleb is a Muslim name (are his religious beliefs and ethnic background a matter of public record, geoff.?) and I think there are aspects of his thought about mathematics and markets as well as cause and effect which are trailing clouds of implication from their birth in an Islamic intellectual milieu. I would be surprised if they weren't both a little tense from all that.

2. Sontag was a beautiful woman, by all accounts a rather imperious beautiful woman. Not many male egos of any ethnic persuasion can sustain the cut direct from such a one without curling in upon themselves and whining a bit.

posted by jamjam at 3:11 PM on April 30, 2007

I thought I felt the presence of two issues left unstated in Taleb's narration of his collision with Sontag:

1. She was a Jew and a Zionist; Taleb is a Muslim name (are his religious beliefs and ethnic background a matter of public record, geoff.?) and I think there are aspects of his thought about mathematics and markets as well as cause and effect which are trailing clouds of implication from their birth in an Islamic intellectual milieu. I would be surprised if they weren't both a little tense from all that.

2. Sontag was a beautiful woman, by all accounts a rather imperious beautiful woman. Not many male egos of any ethnic persuasion can sustain the cut direct from such a one without curling in upon themselves and whining a bit.

posted by jamjam at 3:11 PM on April 30, 2007

Unfortuntately, teece, it doesn't get any deeper (at least certainly in "Fooled by Randomness"). Yes, it's true that the probability distributions are non-Gaussian. So in other words,

markets can fall by large amounts in a short amount of time.

Yup, they certainly do.

posted by storybored at 3:12 PM on April 30, 2007

markets can fall by large amounts in a short amount of time.

Yup, they certainly do.

posted by storybored at 3:12 PM on April 30, 2007

The only direct quote about religion I recall:

"[Note 1: My Stand Againt Atheism. This, and many other things explain why I just cannot understand atheism. I just cannot. If I were to take “rationality” to its limit, I would then have to treat the dead no differently from the unborn, those who came and left us in the same manner as those who do not exist yet. Otherwise I would be making the mistake of sunk costs [endowment effect]. I cannot & I just do not want to. Homo sum! I want to stay rational in the profane, not the sacred.]"

Given his biographical background, I would assume Eastern Orthodox and not Islamic. If I remember correctly he came from an aristocratic Lebanese family and they would most likely be some form of Eastern Orthodox rather than Islamic if my history of the region is correct.

posted by geoff. at 3:18 PM on April 30, 2007

"[Note 1: My Stand Againt Atheism. This, and many other things explain why I just cannot understand atheism. I just cannot. If I were to take “rationality” to its limit, I would then have to treat the dead no differently from the unborn, those who came and left us in the same manner as those who do not exist yet. Otherwise I would be making the mistake of sunk costs [endowment effect]. I cannot & I just do not want to. Homo sum! I want to stay rational in the profane, not the sacred.]"

Given his biographical background, I would assume Eastern Orthodox and not Islamic. If I remember correctly he came from an aristocratic Lebanese family and they would most likely be some form of Eastern Orthodox rather than Islamic if my history of the region is correct.

posted by geoff. at 3:18 PM on April 30, 2007

The New York Times review of Black Swan dissuaded me from wanting to read it.

posted by stbalbach at 3:43 PM on April 30, 2007

posted by stbalbach at 3:43 PM on April 30, 2007

Skorgu, thank you, that helped. (It also brought to mind the Poisson distribution in Gravity's Rainbow [/lit nerd].)

posted by jokeefe at 3:45 PM on April 30, 2007

posted by jokeefe at 3:45 PM on April 30, 2007

You are correct, geoff..

His home page (your first link, to my embarrassment) says he is Antiochan, a Syrian Christian variety of Eastern Orthodox.

Those interested in the unpublished, because unknowable in principle, according to Taleb, ETA for the next incoming flight of Black Swan Lines Ltd. might be amused by this very recent AskMe.

Even though I despise vatic nonsense of the kind in which I am about to indulge, I am too tempted not to rejoin to Taleb's assertion that we cannot predict the really big things, that it is a blessing, then, that they will predict themselves as often as they do.

posted by jamjam at 4:25 PM on April 30, 2007

His home page (your first link, to my embarrassment) says he is Antiochan, a Syrian Christian variety of Eastern Orthodox.

Those interested in the unpublished, because unknowable in principle, according to Taleb, ETA for the next incoming flight of Black Swan Lines Ltd. might be amused by this very recent AskMe.

Even though I despise vatic nonsense of the kind in which I am about to indulge, I am too tempted not to rejoin to Taleb's assertion that we cannot predict the really big things, that it is a blessing, then, that they will predict themselves as often as they do.

posted by jamjam at 4:25 PM on April 30, 2007

*Steven, yeah, but the problem is that in finance at least, many people have at least by omission assumed that markets behave in a gaussian way. And been fairly resistant to address that assumption as well. Finance isn't anywhere near as open to refutation as the purer sciences are, at least in my limited experience.*

Nicely put. I've seen too many risk management strategies blown away by the assumption of normality. Even after several consecutive days of multiple-sigma moves the folks running one book didn't see what was wrong with their model. That's okay, the deeper the hole they dug for themselves the more we got paid to fix it.

People stick to what's easy and it's sad, really. A lot of financial modeling's shortcomings come from its genesis predating cheap data processing -- models tended to be analytical rather than computational because otherwise the math just couldn't be done.

Even though computational methods currently rule the roost, the underlying volatility parameters still often reflect the assumption of normality, which means when you get into fat tail city your hedging blows up.

posted by Opposite George at 4:55 PM on April 30, 2007

Random behavior, in the sense of being unpredictable, encompasses many more distributions than Gaussian (even the

So saying financial markets are not Gaussian doesn't tell you much. In particular, it tells you nothing particularly useful. Markets have lots of feedback, so a small change that generates feedback can have a surprisingly large effect. And that's about all there is to the black swan.

If you read even a little Taleb (I did it all on his website) you find he has build a tower of babble on the thinnest of foundations. If you spend much time on it and take it all seriously, you will be older and dumber than when you started.

posted by hexatron at 5:27 PM on April 30, 2007

*sum*of two Gaussian distributions is generally not Gaussian).So saying financial markets are not Gaussian doesn't tell you much. In particular, it tells you nothing particularly useful. Markets have lots of feedback, so a small change that generates feedback can have a surprisingly large effect. And that's about all there is to the black swan.

If you read even a little Taleb (I did it all on his website) you find he has build a tower of babble on the thinnest of foundations. If you spend much time on it and take it all seriously, you will be older and dumber than when you started.

posted by hexatron at 5:27 PM on April 30, 2007

There's a word I like: "stochastic". It means something which isn't deterministic, but whose behavior can be closely predicted statistically.

A long series of rolls of 3 6-sided dice added together is an example of stochastic behavior. Any given roll cannot be predicted, but a histogram of a thousand such rolls will almost always very closely conform to the classic bell curve.

Atomic breakdown in the wild is another example of that. It's impossible to predict exactly when a given U-238 atom is going to pump out an alpha particle, but if you've got a thousand kilos of pure U-238 you can predict very closely how much U-234 you'll have after ten years (the alpha being followed by two beta emissions in close succession). Answer: not very damned much, but it can be calculated very precisely, because nuclear breakdown is stochastic.

Rainfall as described above is stochastic, over the short term (minutes). Markets are not stochastic at all. But this is hardly a profound observation.

posted by Steven C. Den Beste at 5:29 PM on April 30, 2007

A long series of rolls of 3 6-sided dice added together is an example of stochastic behavior. Any given roll cannot be predicted, but a histogram of a thousand such rolls will almost always very closely conform to the classic bell curve.

Atomic breakdown in the wild is another example of that. It's impossible to predict exactly when a given U-238 atom is going to pump out an alpha particle, but if you've got a thousand kilos of pure U-238 you can predict very closely how much U-234 you'll have after ten years (the alpha being followed by two beta emissions in close succession). Answer: not very damned much, but it can be calculated very precisely, because nuclear breakdown is stochastic.

Rainfall as described above is stochastic, over the short term (minutes). Markets are not stochastic at all. But this is hardly a profound observation.

posted by Steven C. Den Beste at 5:29 PM on April 30, 2007

*...many people have at least by omission assumed that markets behave in a gaussian way. And been fairly resistant to address that assumption as well.*

Well, in that case they'll be punished by the market and either learn their lesson or be bounced from the game.

posted by Steven C. Den Beste at 5:32 PM on April 30, 2007

Ypu know, I read his first book, I'm sure I'll read this book - but I still don't know what great revelation he has revealed to the world? I mean you can approach the whole randomness element from a bunch of directions - whether you prefer you acadmic work math driven or word driven, this has all been said before.

I mean I admire the guy, and actually recommend his first book to friends all the time, but this is not revolutionary thought here. Many thoughtful investors have been on to this notion of uncertainty for a long time - some intentionally, most unintentionally.

posted by JPD at 5:40 PM on April 30, 2007

I mean I admire the guy, and actually recommend his first book to friends all the time, but this is not revolutionary thought here. Many thoughtful investors have been on to this notion of uncertainty for a long time - some intentionally, most unintentionally.

posted by JPD at 5:40 PM on April 30, 2007

Oh, and

There's a whole financial discipline called risk management that's devoted to dealing with unpredictable market movements. By "unpredictable" here I'm talking about magnitude -- obviously market participants expect movement all the time, and risk management basically accepts the idea that you can't predict market movements, but you can do a better job predicting how related assets might move relative to each other.

Risk management is a Big Deal

One very important input to the problem is how likely common market movements of different sizes might be. That's more or less what a probability distribution is -- it's a measure of how likely it is for something to happen.

The thing is, you usually can't build a perfect hedge. It's expensive to even try and often you only want to insure against catastrophic moves. And your hedge movements almost never negate the movements in the main asset's value exactly. A good sense of how probable certain kinds of moves are will give you a good idea as to how well you're protected.

The thing that's always unpredictable is what the market is actually going to do. What is more predictable is how good my hedge will be when the market moves one way or the other, but I need to assume the correct probability distribution to make that prediction. The real ugly thing about the Gaussian probability distribution is that it tends to underestimate the likelihood of really major moves -- the exact kind of moves that can wipe you out.

If you build a hedge based on a Gaussian distribution you might think you're protected 999,999 times out a million when in reality it might only insure you 999 times out of a thousand. If you knew that, you'd probably adjust your hedge to protect yourself better.

So even though it still can't tell you what the market is going to do tomorrow, using a better distribution helps you stay afloat.

posted by Opposite George at 6:01 PM on April 30, 2007

**jokeefe**, let me just elaborate a bit on**skorgu**'s excellent comment.There's a whole financial discipline called risk management that's devoted to dealing with unpredictable market movements. By "unpredictable" here I'm talking about magnitude -- obviously market participants expect movement all the time, and risk management basically accepts the idea that you can't predict market movements, but you can do a better job predicting how related assets might move relative to each other.

Risk management is a Big Deal

^{TM}because it's insurance against getting wiped out. The basic idea is to build the insurance policy or "hedge" by maintaining a separate financial position whose value moves in an opposite direction to the value of the asset you're trying to protect.One very important input to the problem is how likely common market movements of different sizes might be. That's more or less what a probability distribution is -- it's a measure of how likely it is for something to happen.

The thing is, you usually can't build a perfect hedge. It's expensive to even try and often you only want to insure against catastrophic moves. And your hedge movements almost never negate the movements in the main asset's value exactly. A good sense of how probable certain kinds of moves are will give you a good idea as to how well you're protected.

The thing that's always unpredictable is what the market is actually going to do. What is more predictable is how good my hedge will be when the market moves one way or the other, but I need to assume the correct probability distribution to make that prediction. The real ugly thing about the Gaussian probability distribution is that it tends to underestimate the likelihood of really major moves -- the exact kind of moves that can wipe you out.

If you build a hedge based on a Gaussian distribution you might think you're protected 999,999 times out a million when in reality it might only insure you 999 times out of a thousand. If you knew that, you'd probably adjust your hedge to protect yourself better.

So even though it still can't tell you what the market is going to do tomorrow, using a better distribution helps you stay afloat.

posted by Opposite George at 6:01 PM on April 30, 2007

All you guys (and Aaron Brown) are invited over to my place for a poker game. (I'll lose, but I'll learn)

posted by bashos_frog at 6:26 PM on April 30, 2007

posted by bashos_frog at 6:26 PM on April 30, 2007

Wasn't Black Monday a result of poorly-engineered, computer-mediated trading mechanisms gone into a downward feedback loop? If so, it seems that probability distributions of random motion (Brownian motion) should have little or no role in modeling the behavior of this financial market to begin with.

posted by Blazecock Pileon at 6:51 PM on April 30, 2007

posted by Blazecock Pileon at 6:51 PM on April 30, 2007

**Blazecock Pileon**,

I'm not a stats expert but there are a number of places where the stock market price process deviates from the assumptions underlying Brownian motion (technically, the classic model of the market is a simple Brownian motion model called a Wiener Process.) The feedback you're talking about is one of those deviations.

The thing is, the WP is a very good first approximation of market behavior, at least compared to its predecessors. And the simple underlying math made it very attractive for research back when computation was expensive. One major shortcoming is underpredicting the frequency of really big price moves, but like

**teece**and

**Skorgu**said, it can take a long time for stuff like that to show up (though Mandlebrot caught onto that problem in the 1960s.)

No, it's not the best thing to use but it's an excellent introduction to modeling theory and a lot of people never get any further than that. And it works well enough for long enough that you can be really surprised if you aren't aware of its shortcomings.

posted by Opposite George at 7:15 PM on April 30, 2007

Brownian motion works well when the assumption is that minute changes in price are independent of larger price trends. While this may appear true for most of the time, when stocks go down they continue to go down and vice versa. This is not exhibiting random Brownian motion but long-term memory.

I highly recommend Bachlier's Theory of Speculation as a good introduction to modern financial theory through Black-Scholes.

posted by geoff. at 7:20 PM on April 30, 2007

I highly recommend Bachlier's Theory of Speculation as a good introduction to modern financial theory through Black-Scholes.

posted by geoff. at 7:20 PM on April 30, 2007

I've always been curious as to what all those quants, the math PhDs and physicists, actually *do* on Wall Street. There is no magic bullet that's going to reveal the market's true distribution. So what value do they actually add? I can see perhaps there's an edge in figuring out and acting on obscure arbitrage situations. But what else?

posted by storybored at 8:28 PM on April 30, 2007

posted by storybored at 8:28 PM on April 30, 2007

*I'm sure I'll read this book - but I still don't know what great revelation he has revealed to the world?*

That's the thing JPD. Taleb hasn't unleashed any revelations to the world that aren't obvious from a study of probability. Probability is however counter-intuituve in many aspects. What Taleb explained is that so many people who think they are smart are, in fact, simply lucky.

He may also one of the first people to realize that Black-Scholes model (that option prices follow a random walk simlar to Brownian motion) no longer applied once investors started to use it.

posted by three blind mice at 8:29 PM on April 30, 2007 [1 favorite]

storybored, "When Genius Failed: The Rise and Fall of LTCM" is a good intro on this. They took large positions in bond prices, recognizing quirks in the way bond prices moved in relation to each other. We're talking very, very small changes in per bond prices. This meant they had to take very, very large positions.

There's all sorts of market eccentricities like that, where if you put enough quants in a room you'll find things like the volatility between Ford and GE is always "x" and if it moves out of "x" you should buy Ford and sell GE as historically that means prices converge. Which comes to the problem of not being able to see that a lot of time these complex trades and swaps have large, gaping holes where a one off event that "isn't suppose to happen" can kill everything. The idea is to hedge your position correctly so this doesn't happen, but dynamic hedging is a hard thing to do.

To get an idea of why this is probably not making much sense to those out of finance I like to quote John Merriweather when he made the statement (from the aforementioned book), and I am paraphrasing, that if everyday I drive past a tree and see leaves fallen around it in some sort of pattern or distribution and one day those leaves are arranged in neat little piles, it doesn't mean the distribution pattern is wrong. Of course the distribution pattern is wrong, it does not matter if someone raked the leaves (or caused the government to default on its bonds, or caused the computers to enter in a feedback loop) but that such events do occur, and failure to predict for them regardless of the cause is a problem nonetheless.

posted by geoff. at 8:50 PM on April 30, 2007

There's all sorts of market eccentricities like that, where if you put enough quants in a room you'll find things like the volatility between Ford and GE is always "x" and if it moves out of "x" you should buy Ford and sell GE as historically that means prices converge. Which comes to the problem of not being able to see that a lot of time these complex trades and swaps have large, gaping holes where a one off event that "isn't suppose to happen" can kill everything. The idea is to hedge your position correctly so this doesn't happen, but dynamic hedging is a hard thing to do.

To get an idea of why this is probably not making much sense to those out of finance I like to quote John Merriweather when he made the statement (from the aforementioned book), and I am paraphrasing, that if everyday I drive past a tree and see leaves fallen around it in some sort of pattern or distribution and one day those leaves are arranged in neat little piles, it doesn't mean the distribution pattern is wrong. Of course the distribution pattern is wrong, it does not matter if someone raked the leaves (or caused the government to default on its bonds, or caused the computers to enter in a feedback loop) but that such events do occur, and failure to predict for them regardless of the cause is a problem nonetheless.

posted by geoff. at 8:50 PM on April 30, 2007

*I've always been curious as to what all those quants, the math PhDs and physicists, actually *do* on Wall Street.*

After reading

*FbR*, I'm far more interested in what the

*managers*at big IBs actually do. If nearly everyone is clueless and lucky, then proprietary trading should always be a money loser compared with fees-based services. A manager's job would then be to watch their direct reports and fire them as soon as they begin to lose money.

posted by b1tr0t at 9:01 PM on April 30, 2007

Most bell curves have thick tails, is a great little piece by Bart Kosko about how most distributions are not normal and the consequences of that for science in general.

posted by afu at 11:33 PM on April 30, 2007 [1 favorite]

posted by afu at 11:33 PM on April 30, 2007 [1 favorite]

I'm not a big fan of that Kosko piece, as I think it's misleading.

The Central Limit Theorem makes many of the assumptions he implies are wrong into something that is proven to be fine. Most statistical metrics (with large n) have a normal distribution, regardless of the underlying population distribution, so using the normal to make statistical inferences is 100% A-OK, unless you get into the philosophical realm of mathematical proofs not being worth anything (which has nothing to do with normal probability distributions).

The problem is in people making mathematical models that assume the underlying distribution is normal, but not being astute enough to test that axiom.

The folks that do this:

Are NOT making that mistake. They are using the normal in a fine and dandy way. Others might not be. But in any event, I think the guy grossly overstates his point -- it's not an accident that many data, especially experimental measurement uncertainties, end up being normally distributed. There's a wonderful and beautiful proof in probability theory called the Central Limit theorem that explains exactly why this is (it also applies to the great majority of statistical metrics, as well, as I mentioned above).

There is no "challenging" the normal distribution. It models your data, or it does not. The

Math without empiricism is fine. Mathematical models of reality without empiricism is voodoo magic, not science, and that's the issue.

Which is his point, but I don't like the way he states it. I had to get out a copy of a proof of the central limit theorem, looking for this:

The only assumption in the proof is that the moment generating function M(t) be finite. The proof works for all values of sigma. So I don't really get his critique. Me doubts it's nowhere near as strong is he implies, but I'd need to do some more research to confirm that.

He also says this:

posted by teece at 12:29 AM on May 1, 2007

The Central Limit Theorem makes many of the assumptions he implies are wrong into something that is proven to be fine. Most statistical metrics (with large n) have a normal distribution, regardless of the underlying population distribution, so using the normal to make statistical inferences is 100% A-OK, unless you get into the philosophical realm of mathematical proofs not being worth anything (which has nothing to do with normal probability distributions).

The problem is in people making mathematical models that assume the underlying distribution is normal, but not being astute enough to test that axiom.

The folks that do this:

*Even quantum and signal-processing uncertainty principles or inequalities involve the normal bell curve as the equality condition for minimum uncertainty.*Are NOT making that mistake. They are using the normal in a fine and dandy way. Others might not be. But in any event, I think the guy grossly overstates his point -- it's not an accident that many data, especially experimental measurement uncertainties, end up being normally distributed. There's a wonderful and beautiful proof in probability theory called the Central Limit theorem that explains exactly why this is (it also applies to the great majority of statistical metrics, as well, as I mentioned above).

*And*, we have a great deal of empiricism that backs up the logic of the CLT.There is no "challenging" the normal distribution. It models your data, or it does not. The

*mathematics*of that are pretty well settled. The issue is whether or not the modeler actual/y bothers to employ some*empiricism*to test his or her axioms.Math without empiricism is fine. Mathematical models of reality without empiricism is voodoo magic, not science, and that's the issue.

Which is his point, but I don't like the way he states it. I had to get out a copy of a proof of the central limit theorem, looking for this:

*The theorem assumes that the random dispersion about the mean is so comparatively slight that a particular measure of this dispersion — the variance or the standard deviation — is finite or does not blow up to infinity in a mathematical sense.*He is saying we can't rely on the Central Limit theorem, basically. Which is more than a wee bit boldThe only assumption in the proof is that the moment generating function M(t) be finite. The proof works for all values of sigma. So I don't really get his critique. Me doubts it's nowhere near as strong is he implies, but I'd need to do some more research to confirm that.

He also says this:

*So far we have closed-form solutions for only two stable bell curves*, which bugs me, because the Gaussian has mathematical difficulties of it's own. There's a reason why stat books have tables of z scores -- it's because the Gaussian is not directly integrable. Integrate it from point a to point b -- you can only get the answer in terms of a number, or in terms of Erf, which is just a definition of the Gaussian. Again, I'm not sure what the hell he is trying to say there. Other symmetric prob. dists. may be harder -- or probably just less explored, I suspect.posted by teece at 12:29 AM on May 1, 2007

Wow, that NY Times review is brutal. Here's the concluding paragraph.

posted by OmieWise at 6:58 AM on May 1, 2007

Ultimately, Taleb’s book shipwrecks on the story of Yevgenia Krasnova, “a neuroscientist with an interest in philosophy” whose unheralded first novel (which “avoided the journalistic prevarications of contemporary narrative nonfiction”) became a blockbuster, while her heavily promoted second novel flopped. Taleb describes her books as exemplary black swans, succeeding when expected to fail and failing when expected to succeed. But Krasnova’s experience hardly sounds “highly improbable.” The sequence of events is a publishing-industry cliché. More important, Krasnova does not exist. Taleb fabricated her, which he admits only in a footnote and in the index. If Taleb were on to a general theme of history, he wouldn’t have to invent his examples.Taleb doesn't sound very serious.

posted by OmieWise at 6:58 AM on May 1, 2007

*storybored, "When Genius Failed: The Rise and Fall of LTCM" is a good intro on this. They took large positions in bond prices, recognizing quirks in the way bond prices moved in relation to each other.*

Yes, they shipwrecked on Russian bonds or something right? But this is yet again an example of bad risk management. I'm wondering if the quants have something better in their armory than "dangerous betting". I guess in one sense it's pointless asking because if they do have something, they'd likely keep it secret. Heck even qualitative trading rules are important to keep to oneself :-)

posted by storybored at 7:27 AM on May 1, 2007

The

posted by Skorgu at 7:36 AM on May 1, 2007

**really**fascinating thing about finance and markets is that so many really truly brilliant people disagree on a fundamental level. Take LTCM for example, they blew so big and so hard they needed to be bailed out by the government, but Warren Buffett wanted to buy the whole fund, at the 'worst' possible time. By the way, if this stuff even marginally (heh) appeals to you, subscribe to that blog. Tyler Cowen is a minor god.posted by Skorgu at 7:36 AM on May 1, 2007

Skorgu, thanks for the blog link, looks interesting!

Re: LTCM & Buffett. Was there really a disagreement here? LTCM's problem was that it didn't have the collateral, whereas Buffett did. So fundamentally the trade was good, it's just that LTCM went in with pockets that were too shallow.

posted by storybored at 8:20 AM on May 1, 2007

Re: LTCM & Buffett. Was there really a disagreement here? LTCM's problem was that it didn't have the collateral, whereas Buffett did. So fundamentally the trade was good, it's just that LTCM went in with pockets that were too shallow.

posted by storybored at 8:20 AM on May 1, 2007

There were more issues than Buffet not wanting collateral. He wanted to know what the hell LTCM was doing. The most interesting thing about LTCM was how they not only managed to reel in large, institutional investors without telling them a damn thing -- they secured loans at incredibly low rates. They were highly leveraged, they had to be, and were doing so at a large risk to Wall Street. Buffet as a rule doesn't invest in anything he doesn't know about (see: tech stocks) and was balking at throwing money to LTCM without them explaining what was going on. He also believed the price they were asking was too high at the time. He knew they held some things in the portfolio that were below book value (he is a value investor), but he thought their prices were still too high. Before they got things settled, LTCM collapsed to the point where the Fed brought all the Wall Street banks in who remembered LTCM's arrogance and were especially vengeful.

I don't like the saying that "fundamentally the trade was good", we all know the saying that you can't stay liquid for as long as the market is rational. That is fundamentally what Taleb is saying, that such things are only fundamentally good on paper. In the real world events begin to take hold that screw with your assumptions. I remember JM complaining how all the banks were furthering the collapse at LTCM, forcing LTCM to trade on its models and lose more money. That's the problem, it appeared the market "knew" LTCM was making bad trades and they would keeping forcing it until LTCM collapsed. Towards the end I remember them complaining that their models were showing that such trading days couldn't occur for several billion years, and that this once in twice a universe deviation began to happen several days at a time. Their models were fundamentally flawed.

posted by geoff. at 8:47 AM on May 1, 2007

I don't like the saying that "fundamentally the trade was good", we all know the saying that you can't stay liquid for as long as the market is rational. That is fundamentally what Taleb is saying, that such things are only fundamentally good on paper. In the real world events begin to take hold that screw with your assumptions. I remember JM complaining how all the banks were furthering the collapse at LTCM, forcing LTCM to trade on its models and lose more money. That's the problem, it appeared the market "knew" LTCM was making bad trades and they would keeping forcing it until LTCM collapsed. Towards the end I remember them complaining that their models were showing that such trading days couldn't occur for several billion years, and that this once in twice a universe deviation began to happen several days at a time. Their models were fundamentally flawed.

posted by geoff. at 8:47 AM on May 1, 2007

*I'm wondering if the quants have something better in their armory than "dangerous betting". I guess in one sense it's pointless asking because if they do have something, they'd likely keep it secret.*.

I'm way out of my element on this (I've never even met one of these people, and know little about their work), but If I had to guess I'd say quants are just

*way*smarter than the folks that ultimately manage them, and they are able to use their intelligence to stifle any criticism of their mistakes. They also possess certain advanced mathematical skills that are the pure gold in circles where those skills are few and far between.

When the 180 IQ genius guy fucks up, and you call him on it, his answer is: "Can you do any better?" Most people will automatically assume they can't. Of course, the data says genius or no, you ain't predicting the stock market forever. So in a very real sense, yes you can do better. Or at least, not do any worse. But it's hard to convince yourself of this when the 180 IQ has so much confidence.

I've seen this principle in action many times. A very smart bull-shitter (and the bull-shitting from the genius can often be unwitting) is rarely, if ever, called on their bull shit. Geniuses are given a degree of credulity that few other people ever receive.

posted by teece at 11:04 AM on May 1, 2007

*Their models were fundamentally flawed.*

No, no, no,

**geoff.**,

*reality*was flawed. Fix the reality, not the model (e.g., make the Fed, the feds, the economy, the corporation, whatever, behave as their model said. But don't change the model).

Unbelievably, many people will give some variation of that argument quite seriously. It seems to be a common failing of neoclassical economics in general.

posted by teece at 11:07 AM on May 1, 2007

*More important, Krasnova does not exist. Taleb fabricated her, which he admits only in a footnote and in the index. If Taleb were on to a general theme of history, he wouldn’t have to invent his examples.*

When I read that section, I thought it was pretty obvious that Krasnova didn't exist, and he was just using her as a way to get the reader to think about probability in a way they weren't used to, (i.e. the probability of literary success). The story was only an introduction to the themes of the book, so for the NY times reviewer to criticize it as some fundamental flaw is fairly disingenuous.

posted by afu at 8:43 PM on May 1, 2007 [1 favorite]

*If I had to guess I'd say quants are just way smarter than the folks that ultimately manage them, and they are able to use their intelligence to stifle any criticism of their mistakes.*

I think i agree with this. Especially if there's a hot streak going on. The managers are blinded by (temporary) abnormal returns, the traders get overconfident and when they start losing, try to "get it back", and boom it's Amaranth all over again.

posted by storybored at 10:57 AM on May 2, 2007

*it's not an accident that many data, especially experimental measurement uncertainties, end up being normally distributed. There's a wonderful and beautiful proof in probability theory called the Central Limit theorem that explains exactly why this is*

This is a fascinating topic and one i don't know enough about. When you're talking about a proof of the CLT though how does that relate to what the result would be in the real world?

The CLT says that if you randomly select multiple samples from a population that the means of each of these samples will be normally distributed. I believe it. But why does this necessarily have to be so? Isn't it weird that the universe should behave in step with the math?

posted by storybored at 11:27 AM on May 2, 2007

I think it has to do with the binomial distribution that accompanies CLT. Something either is or it isn't. You either have heads or tail, positive or negative, up or down, etc. The same reason pi occurs so much in nature, due to its relation to the sphere and the circle.

posted by geoff. at 11:42 AM on May 2, 2007

posted by geoff. at 11:42 AM on May 2, 2007

**storybored**: the "why" in the Central Limit Theorem is only so-so satisfying.

It states, in general terms, that the

*sum*of a series of random variables (of

*any*underlying distribution) will tend to be modeled by the normal distribution's probability function as

*n*goes to infinity (where

*n*is the number of r.v.s you are summing up). At infinity, any sum of random variables is a normal random variable. For any

*n*greater than about 30, the difference between the real random variable that models the sum of r.v.s, and the normal approximation to that sum, is small. The normal is accurate enough for essentially all purposes with large

*n*(unless you have a

*very*skewed and pathological underlying distribution, in which case you may need larger

*n*. One should check these thing, but they are pretty rare.)

Why is that the case? It's just the math. The proof of the CLT is not all that hard, but it's really abstract, and ultimately, for me at least, does not give a satisfactory answer as to the "why." (It involves moment generating functions and L'Hopital's rule, and a lemma about moment generating functions, so it's really too much of a pain to go into here). But it is a rigorous mathematical proof of the fact that, as n -> infinity, any sum of any R.V.s is modeled by the standard normal.

I think it's pretty amazing that that is the case. But once you know that, suddenly a lot of really strange coincidences start to make sense. When I measure the speed of a car or the length of a stick, there is always measurement error. Each measurement is a

*random variable*. I have no idea what the probability distribution of that r.v. is. But, the CLT makes it obvious that, as

*n*gets large,

*it does not matter*. This is why all measurement error is so well represented by the normal random variable. The great lion's share of statistics involves sums of random variables, so once

*n*> 30 (or 40, in some case), the normal is good enough to model everything. Indeed, good statistics draws it's data in such a way as to make sure one can make good advantage of the normal approximation to a sum of any r.v. That's where things like random samples come into play.

(As I understand the history, people knew lots of things were modeled fairly well by the normal r.v., but they didn't know why for quite some time).

Now, this is how people get in trouble: If you are modeling something, like a financial market, and you

*assume*it is modeled by the normal r.v., but you are

*not*summing a large number of random variables, you have no

*a priori*reason to be using the normal random variable. One must can't justify such a use logically and mathematically, or one would at least need empirical data to back up that assumption.

Now when a model fails to predict empirical data quite spectacularly, as has been mentioned in this thread, someone obviously had neither the math nor the empiricism to back up their willy-nilly normal assumption, and thus really just got lucky for a while.

posted by teece at 3:59 PM on May 2, 2007 [1 favorite]

This has been one of the better metafilter threads I've been apart of. I didn't realize there were so many people interested in both (1) the metaphysics of finance, which is what a lot of this really reduces to and unlike mathematics where the metaphysics can be ignored by the inherent logical nature of mathematics, it cannot in finance, (2) statistical probability and abstract mathematics.

Perhaps it is the nature of finance, that as someone above pointed out, so many people disagree about so fundamentally, that this didn't turn into a geekery pissing contest.

I also was toying with starting my dissertation, posting my research as I did it, etc. in a blog-like format to where people can comment and offer suggestions (I'm somewhat vain and do not like people reading my research/papers unless I feel as if they hit some kind of Platonicity that I know is unobtainable). I think I may do that, once I get some preliminary research done first to see about the plausibility (researching fractal patterns in energy markets, especially looking at oil shocks which I've done considerable research in).

posted by geoff. at 7:07 PM on May 2, 2007

Perhaps it is the nature of finance, that as someone above pointed out, so many people disagree about so fundamentally, that this didn't turn into a geekery pissing contest.

I also was toying with starting my dissertation, posting my research as I did it, etc. in a blog-like format to where people can comment and offer suggestions (I'm somewhat vain and do not like people reading my research/papers unless I feel as if they hit some kind of Platonicity that I know is unobtainable). I think I may do that, once I get some preliminary research done first to see about the plausibility (researching fractal patterns in energy markets, especially looking at oil shocks which I've done considerable research in).

posted by geoff. at 7:07 PM on May 2, 2007

*It states, in general terms, that the sum of a series of random variables (of any underlying distribution) will tend to be modeled by the normal distribution's probability function as n goes to infinity (where n is the number of r.v.s you are summing up).*

Thanks for the elaboration, teece! The cool part about this is that the nature of the underlying distributions are irrelevant. This makes me want to go back and crack open my old stats books. I memorized the formulas but didn't have the time to really appreciate what they meant. L'Hopital's rule sounds vaguely familiar but I'll be hanged if i can remember moment generation.

So now talking about the "why" it works in the real world, does the secret lie in the definition of a random variable? That the mathematical definition of randomness happens to match the nature of the universe?

posted by storybored at 7:13 PM on May 2, 2007

**storybored**:

My stat. book mentions the Central Limit Theorem, but provides no proof, and it does not ever mention moment generating functions (it's Devore, and meant for into. to applied stat., really).

My prob. theory text talks about moment generating functions and gives a proof of the CLT.

A moment generating function is actually a very nifty thing. In prob. theory and stat, you study random variables. All random variables have a PMF/PDF( probability mass/distribution function) and a CMF/CDF ( cumulative mass/distribution function) which gives Prob(X=x) for the former and Prob(X < x) for the latter, where x is the random variable, and x is the particular value of interest. you can answer most questions about a random variable with those two things. however, what we are generally interested in are measures of location about the r.v.: the mean, the std. dev., the skewness, and the kurtosis (measure of center, measure of spread, measure of asymmetry, and measure of peakedness, respectively). there is a function, m(t)=E[ exp( t x) ]. that is, for a r.v.'s pmf/pdf which, when takes an expected value of exp(t x) for that r.v. (e.g., plugging exp(t x) into the expected value definition, rather than just plain old x), you get the moment generating function. when you take the nth derivative of this function m(t), and evaluate it at 0, you get the nth moment. the 1st moment is the i>mean of the random variable. The 2nd moment (manipulated a bit to make it a central moment) is the

*std. deviation*. The 3rd can give you skewness, and the 4th can give you kurtosis, and so on.

It's very cool, but very abstract. You don't really need moment generating functions to do most applied statistics, but you need them to prove the CLT. You need L'Hopital's rule to show that a function of M(t) tends to a certain value as n-> infinity. They are more a part of prob. theory than stat.

Sorry, I'm babbling on. But in any event, as it turns out, many random things humans care about turn out to be sums of random variables, so the CLT pops up it's head every where.

And I just happen to be neck-deep in these things right now, studying them! ;-)

posted by teece at 8:38 PM on May 2, 2007

*Sorry, I'm babbling on.*

It's not babble to me! Your explanations are very helpful. There's a connection here to today's FPP on innumeracy - so many math textbooks are written as volumes of cold formulas. Your explanation above of moment generating functions is an example of what's missing in many math books. More of them should say "look at this! this is cool!" and then proceed to show it. Instead they start with the minutiae and its a real slog.

Here is one Proof of the CLT. It involves Fourier transforms, yet another concept which i've forgotten. The article also claims there's a proof of CLT in six lines, though it probably draws on even higher mathematical concepts.

posted by storybored at 10:54 AM on May 3, 2007

Huh, I never would have thought to use a Fourier transform to prove the CLT. That's nifty -- I bet that would work better as a proof for EEs, as they live in the Fourier transform.

It's a lot more complicated than proving it with moment generating functions, but it doesn't involve introducing that arcane concept, either. I guess it's not all that surprising since both work: an inverse Fourier transform and a moment generating function are both integral transforms involving exp( - x ).

HOWEVER: the definition with moment generating functions works for discrete random variables where one is summing, rather than integrating. Is there a discrete Fourier transform? I don't remember. Cool to think about, thanks.

posted by teece at 11:28 AM on May 3, 2007

It's a lot more complicated than proving it with moment generating functions, but it doesn't involve introducing that arcane concept, either. I guess it's not all that surprising since both work: an inverse Fourier transform and a moment generating function are both integral transforms involving exp( - x ).

HOWEVER: the definition with moment generating functions works for discrete random variables where one is summing, rather than integrating. Is there a discrete Fourier transform? I don't remember. Cool to think about, thanks.

posted by teece at 11:28 AM on May 3, 2007

I'm going to have to spend some more time on CLT!

But back to the main thread: Here is an article from last month's Fortune about Citadel, one of the hottest hedge funds. They bought out Amaranth after the latter's big kablooie. The dude who runs it has a net worth of several billions.

An interesting quote:

Whoa. 3% of trading volume on the three biggest exchanges! That is unbelievable. Plus as you can read in the article they're generating 20-30% p.a. returns.

Am I alone in thinking there is something really wacko about this? With such high volumes, they must be taking some pretty big positions. Plus the fact that they're generating double-digit returns, what does this imply about their leverage?

posted by storybored at 8:11 PM on May 3, 2007

But back to the main thread: Here is an article from last month's Fortune about Citadel, one of the hottest hedge funds. They bought out Amaranth after the latter's big kablooie. The dude who runs it has a net worth of several billions.

An interesting quote:

*Citadel is basically a quant fund. Its trading programs furiously buy and sell on behalf of the firm's two main funds, Kensington Global and Wellington. (Citadel accounts for more than 3 percent of average daily trading volume on the New York, Tokyo and London exchanges, 15 percent of the options market, and as much as 10 percent of the Treasury bond market.)*

Whoa. 3% of trading volume on the three biggest exchanges! That is unbelievable. Plus as you can read in the article they're generating 20-30% p.a. returns.

Am I alone in thinking there is something really wacko about this? With such high volumes, they must be taking some pretty big positions. Plus the fact that they're generating double-digit returns, what does this imply about their leverage?

posted by storybored at 8:11 PM on May 3, 2007

It shows that they are paying up the ass in fees to someone. These fees usually enable them to be the first to know when analysts change the ratings for stocks and even influence stock ratings for analysts, q.v., SAC Capital. I don't mean to be incredibly cynical but I don't think they're "multiple investment strategies" are in some esoteric physics formula. I have a feeling a lot of their positions are what one would consider, if not quite a legal definition of, market manipulation. Read Business Week's article on SAC Capital to give you an idea of how these funds operate.

In any case Citadel especially, is incredibly secretive. Anything we'd be doing is speculating. They issue bonds though, assuming that the bigger they are the bigger they fall, I wonder what the cost for short position in their bonds is.

posted by geoff. at 11:14 AM on May 4, 2007

In any case Citadel especially, is incredibly secretive. Anything we'd be doing is speculating. They issue bonds though, assuming that the bigger they are the bigger they fall, I wonder what the cost for short position in their bonds is.

posted by geoff. at 11:14 AM on May 4, 2007

Good article on Citadel's finance including the use of bond's to avoid the strict repayment of bank loans should Citadel's positions collapse.

posted by geoff. at 11:22 AM on May 4, 2007

posted by geoff. at 11:22 AM on May 4, 2007

Err ... I should clarify that when a firm gets large enough to be a gorilla I begin to suspect they are as good at manipulating Wall Street as they are at prospecting in the markets ... if that makes any sense?

posted by geoff. at 11:27 AM on May 4, 2007

posted by geoff. at 11:27 AM on May 4, 2007

Nice article geoff. I dug into it some more and verified that Citadel is indeed using 7.8x leverage. How does this compare to Fortress Investment Group I wonder.

posted by storybored at 11:53 AM on May 4, 2007

posted by storybored at 11:53 AM on May 4, 2007

*I should clarify that when a firm gets large enough to be a gorilla I begin to suspect they are as good at manipulating Wall Street as they are at prospecting in the markets*

I think this is certainly true, at least to some degree. For the really huge players, it's as much about controlling the market, to the extent that they can, as it is about predicting the markets.

And, honestly, I'd be

*extremely*surprised if average folks would find such gorilla's tactics ethical, were they exposed to the harsh light of day.

posted by teece at 4:09 PM on May 4, 2007

Good, long discussion with Taleb over at EconTalk (~1hr, 30min).

posted by geoff. at 11:31 AM on May 7, 2007

posted by geoff. at 11:31 AM on May 7, 2007

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sure I'm going to read it

but could you point to some place where the baseball game is explained to complete beginners ?

posted by nicolin at 1:10 PM on April 30, 2007