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Comment by thanatropism | Hacker News Reader

I'm ignorant of "heavy-tailed statistics" as a field, but the normal distribution in quantitative finance is justified by the Levy characterization of Brownian motion (basically every continuous stochastic process is a drift-diffusion driven by a Brownian) and the Levy representation of a discontinuous process as a sum of a diffusion and a pure jump process.

Mind, you might find different distributions when solving diffusion equations (the Cox-Ingersoll-Ross process involves a Bessel distribution if I'm not mistaken). But the diffusion-jump paradigm is much better justified there than distributional (or even finite moment) assumptions in discrete land.

Put it this way: if finance is abusing distributional assumptions, there's money being left on the table.

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The main issue with this being that markets don't exhibit behaviour analogous to Brownian motion. It's a nice little assumption that makes the math work, but frequency analysis of real world markets shows that they behave like pink noise instead.

And unsurprisingly there are quant funds and prop trading firms that use this very fact to make lots of money. Academic economist's love for the normal distribution is derided by pretty much any fellow real world trader I know.

You are throwing in a lot of technical words around and the specific fact you state arent wrong, but that cannot make a strong case for the choice of the Normal distribution. All you need is that the variance of bounded (in time) increments be unbounded for the Gaussian behavior to fall apart (for example non-rare jumps of sufficient magnitude).

I did say "diffusion-jump process".

Look, even if we disregard jumps as a possible term in equations, merely considering volatility to be stochastic and driven by a Brownian already gets you variances that may grow arbitrarily fast. This is without considering stranger nonlinearities on the Brownian term itself.

People are overly impressed by forceful arguments of the Taleb variety and log-log plots of empirical distributions and start parroting talking points about heavy/fat tails.

If I wasn't computerless and on my phone I would link to a paper that does the analog of the Anscombe quartet for log-log distribution plots "proving that data is Pareto/power law/etc." It's good vaccine for fat tail hipsterism, look it up.

I dont think we are disagreeing (as long as you do not claim that stock prices are merely a Gaussian process.

I thought of a simple example on my way to my commute. (It's a short walk.)

The Tanaka equation is an example (admittedly: not a diffusion, not a case o plug-and-chug the Ito lemma) of a process driven by a Brownian with a discontinuous probability distribution. How the hell? From memory,

dTNK = dB if dB>0 else -dB

Now imagine a model with three equations.

X1 is a bog-standard geometric diffusion (we could have picked something that's chi-squared distributed driven by a Brownian from standard interest rate models) as in the 1970s Black-Scholes models. But instead of having an exogenous volatility, it has its Brownian term dB1 multiplied by a second equation, X2.

dX2 could be a standard mean-reversion equation, but again its dB2 is multiplied by X3.

dX3 is something like abs(dB3 - dB4* X3).

Voilà, an equation (X1) with a sudden break driven by the level of a mean-reverting equation (X2, which tells us volatility should come down in finite time even if it grows by a lot at times) that's set to blow up at a X2-dependent but stochastic level.

Don't get me wrong, Poisson-like jumps are very common (they're precisely the limiting process for sudden jumps) but people overstate (perhaps because they didn't really read the conditions for Ito isometry) how much a Brownian motion forces a system into normality or smoothness.

But hey, people get away with being hipsters about programming languages, why shouldn't they do that for stochastic calculus too, you know?

Hey. Why do people upvote a post that says "Taleb is a jerk" and makes a reputation argument in a forceful manner (reminiscent of Taleb's own style) -- and then downvote a post with technical detail?

Probably the fact that you flatly ignored the remark about stuffing your post with opaque technical terms--I know a lot of maths, but I can't make heads or tails of what you're saying. Who are you even talking to? I assume it means something to you, but it doesn't even look like you're trying to communicate your point in a clear manner. Just that you enjoy using words. When trying to read around those technical terms, what is left has a rather nasty and arrogant tone. Therefore, doesn't add much to the discussion = downvote.

Can't recommend the Mandelbrot book highly enough - he was a student of Paul Levy, and he wrote extensively about why Gaussian is not a good choice to model financial time series.

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