
Stock Market Prices Do Not Follow Random Walks - QuantMash
http://www.turingfinance.com/stock-market-prices-do-not-follow-random-walks/
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nickff
This post is quite interesting, and I will have to re-read and ponder it some
more, but there is one obvious flaw in the analysis. By analyzing the past
returns of current S&P500 companies, the author is allowing for survivorship
bias; companies which have done consistently well (in terms of market
capitalization) over the analysis period are likely to be over-represented in
current indices. To correct for this, the author should re-run the analysis
using the S&P500 companies from the beginning of the period instead of the
end.

This is the same problem that a study ran into some time ago when it
demonstrated that the portfolio managers with the worst returns were the best
investments; it analysed the returns of a number of managers over a period and
found that the ones with the worst returns at the beginning had the best
returns at the end. The problem is that all the consistently mediocre or bad
managers were discarded, as they did not survive until the end of the period.

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Flarbengaber
I didn't quite understand your last paragraph, but it sounds interesting. Do
you have a link?

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gjm11
Toy model: A fund can do well or badly at the start of the period, and it can
do well or badly at the end. Both happen completely at random. A fund that
does badly at both ends is closed and never heard from again.

If you analyse funds in this situation, you will find that _every_ fund that
does badly at the start of the period does well at the end. (Because the ones
that do badly at the end too are all gone.) You might be tempted to think up
clever explanations about how fund managers with bad initial results make
extra effort, or how stocks that do badly tend to rebound later as investors
recognize their true value, or something -- but that would be a mistake,
because in this situation the _only_ thing leading to the relationship between
early and late performance is the fact that the "bad at both ends" funds
aren't represented in the analysis.

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phreeza
It is quite a leap to go from 'not gaussian' to 'not random' as done here. All
that has been falsified, as far as I can tell, is a very simple model of a
random walk with normally distributed disturbances.

It would be interesting how much better it becomes if higher moments, in
particular kurtosis ('fat tails') are included.

~~~
QuantMash
The article clearly states that the test extends to many forms of randomness
beyond Gaussian:

"Nevertheless, the desired effect of stochastic volatility namely, fatter
tailed distributions ..."

"... we want a test for the random walk hypothesis which passes (it concludes
the market is random) even if the returns demonstrate heteroskedastic
increments and large drifts. Why? Because both of these properties are widely
observed in most historical asset price data (just ask Nassim Taleb) and
neither invalidate the fundamental principle underpinning the random walk
hypothesis, namely the Markov property (unforecastibility of future asset
prices given past asset prices)"

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occamrazor
His model is not heteroskedastic. The log-error terms are i.i.d, so they have
actually all the same variance. The distribution they are sampled from is a
normal variance mixture.

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QuantMash
Interesting. I'm no statistician but the blog post states that the authors of
the original paper (Lo and Mckinlay) claim the test is heteroskedasticity-
consistent. So what are you saying? Are you saying the original paper is
wrong? That his example was wrong? (This seems more likely) And if the example
is wrong, is it wrong to say it is heteroskedastic AND wrong to say it is a
stochastic volatility model? Or just that it is heteroskedastic? As far as I
can tell stochastic volatility simply means the variance itself is randomly
distributed which looks consistent with his example? Just trying to clarift.
Thanks.

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occamrazor
I haven't read the original paper, I can't comment on that.

The model in the webpage however is not heteroskedastic (literally "unequal
variance") because all the log-increments are iid. It could be legitimately
considered a geometric Brownian motion with stochastic volatility, because the
log-error is indeed normally distributed with variance picked from some
stochastic distribution, in this case a normal distribution. This term however
is normally used for models in which the volatility has more structure (e.g.
the ARCH or GARCH models which are mentioned in the page).

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curuinor
Mandelbrot and NN Taleb's claim requires non-Markovianness of the walks.
Fractional brownian motion, for example, exhibits long-range dependence.

Also "randomness" is measured using Kolmogorov complexity, a quantity which is
incomputable.

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worik
" weak form of the efficient market hypothesis which states that: future
prices cannot be predicted by analyzing prices from the past ..."

No. That is the _consequence_ of the efficient market hypothesis being true.
The hypothesis itself is subtly different. Though most people miss it:

All information from past prices is in current prices.

That is a better formulation.

I spent five years studying it and I think it is robust if you remember that
it takes time for information to be consumed. That is why HFT works, because
it acts before the information can be processed.

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worik
"All information from in past prices is in current prices."

All information from past prices is in current prices.

Duh!

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tiatia
I did not read the link but Benoit Mandelbrot basically showed this 30 years
ago. So what's the news?

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tiatia
Not sure why I get downvoted

[http://www.amazon.com/The-Misbehavior-Markets-Financial-
Turb...](http://www.amazon.com/The-Misbehavior-Markets-Financial-
Turbulence/dp/0465043577)

~~~
dang
No doubt because you reacted dismissively to the title alone instead of
engaging with the specifics of the article.

