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Total Posts: 24
Joined: Jun 2012
Posted: 2013-04-28 01:10
I meant a power law alpha close to 1 (very fat tails).


Total Posts: 1775
Joined: Jul 2004
Posted: 2013-04-28 12:12
I see it now, thanks for pointing it out.

Qui fait le malin tombe dans le ravin


Total Posts: 1075
Joined: May 2004
Posted: 2013-04-28 12:17
Are 1 and 300 the best alphas to approximate financial markets and a Gaussian distribution?

Inflatable trader


Total Posts: 24
Joined: Jun 2012
Posted: 2013-04-28 16:25
complicated. I will come back with a discussion (from the text).
SP500 has an alpha ~3, but not for the tails as single stocks have lower alpha and the aggregation gives a certain illusion


Total Posts: 71
Joined: Nov 2013
Posted: 2020-02-07 15:50
Mea culpa for thread necro here, but was searching the phorums for discussion of power-law distributions and found this one.

After reading a few assorted papers & discussions, I'm wondering if modeling returns with a power-law is far superior to 'conventional' distributions (not normal/Gaussian per se), esp after noting in Gabaix et. al "A theory of power-law distributions in Financial market Fluctuations", they state:

"The probability that a return has an absolute value larger than x is found empirically to be P(|r_t)| > x) ~ x^{-Xi_r} (eqn 1), with Xi_r ~ 3...

...Empirical studies also show that the distribution of trading volume Vt obeys a similar power law: P(V_t > x) ~ x^{-Xi_V} (eqn 2)...

...with Xi_V ~= 1.5; while the number of trades N_t obeys:
P(N_t > x) ~ x^{-Xi_N} (eqn 3)

with Xi_N ~= 3.4"

"The 'inverse cubic law' of equation (1) is rather 'universal', holding over as many as 80 standard deviations for some stock markets, with Dt ranging from one minute to one month, across different sizes of stocks, different time periods, and also for different stock market indices4,8.Moreover, the most extreme events - including the 1929 and 1987 market crashes - conform to equation (1), demonstrating that **crashes do not appear to be outliers of the distribution**. We test the universality of equations (2) and (3) by analysing the 35 million transactions of the 30 largest stocks on the Paris Bourse over the 5-yr period 1994-1999. Our analysis shows that the power laws (2) and (3) obtained for US stocks also hold for a distinctly different market, consistent with the possibility that equations (2) and (3) are as universal as equation (1).
Here, we develop a model that demonstrates how trading by large market participants explains the above power laws."

Apologies for the lack of LaTex.

The takeaway that I perceived from this and another source was that conventional modeling by returns, without volume/speed-of-actions made by the bigger participants, is more 'tea leaves' and lags the true dynamics of the market, as well as greatly skewers the utility of a VaR measure. And so, using power-law distributions model the tails far more accurately.

Anyone care to comment?

"How be caught up in a game and have no idea of the rules." - C.S.
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