We've tested this also, but it won't improve a CT strategy. Reason is that "more mean" can also mean "more random".

Thanks Jcl for the code. You are GREAT!!!!!

Good work, but please, some marketing please! You should have called it Market Madness Index

Can someone please tell me where this statement comes from? A hint to its math. base might be also sufficient.

Thanks, Sphin

Uncorrelated with what? Or are you referring to a random walk process?

Yes. Uncorrelated with itself, i.e. Price(today) has no correlation to Price(yesterday). Then you'll get the 75% mean reversion probability.

You can prove this with a simple geometrical consideration. Suppose you have a set of today's prices Pt, and a set of yesterday's price Py. By definition, half the prices from Pt are below the median and half are above the median; same for Py. Now combine the two sets to a 2 dimensional set of vectors with coordinates (Pt,Py). Every such vector represents a possible price change from Py yesterday to Pt today. This set of price changes can then be split by the median lines into 4 sub-sets:

1. (Pt < Median, Py < Median)
2. (Pt < Median, Py > Median)
3. (Pt > Median, Py < Median)
4. (Pt > Median, Py > Median)

The 4 subsets have exactly the same number of elements when Pt and Py are uncorrelated. The value of the median does not matter.

Now, mean reversion, or more precisely moving in direction to the median value means the following condition:

(Pt > Py and Py < Median) // yesterdays Price was below the median and rises, i.e. todays price is higher
or
(Pt < Py and Py > Median) // yesterdays Price was above the median and falls


The first condition is fulfilled in half of subset 1 (the other half had Pt < Py) and in the full subset 3 (because Pt > Py always in subset 3). So, this happens for 1/2*1/4 + 1/4 = 3/8 of all elements.

The second condition is fulfilled in half of subset 4 (the other half had Pt > Py) and in the full subset 2 (because Pt < Py always in subset 2). This is true for another 3/8 of all elements.

3/8 + 3/8 = 6/8 = 75%.

Then, would it be accurate to say
"The theory behind [MMI] is that in an uncorrelated price series, the probability of a price to revert to the median on the next sampled bar is exactly 75%"

Yes, that is correct.

Sometimes I need a little longer, sorry for that. But mean reversion can be tricky and I'm interested in understanding your derivation.


You are talking of 2 sets of prices Pt/Py like in bars of today/yesterday and there are 2 medians, one for the set Pt and one for the set Py?


Combining arbitrarily or each one of Pt with each one of Py?

Thanks, Sphin

The 2 medians should be substantially the same. The Py is calculated from all the same values as Pt, except 2, at the edges. So is safe to speak of only one median.

... and in pairs in the natural order so that the sense of each pair is (Pt,Py), over time.