motivated by a message from Bob S.
Remember when we talked about Market Crashes?
>How can I forget? As I recall, you predicted a crash on ...
Uh ... yes. Let's forget that, okay?
Sornette attempted to identify the stock price behaviour leading up to a crash.
Although we mentioned that in an earlier tutorial, a more recent paper updates the methodology, like so:
 We look at (for example) the S&P500 crash in October, 1987:
 We consider the logarithm of that index:
 We introduce a function that mimics the general upward trend (before the crash):
Note that it rises more rapidly as the time of the crash approaches.
This function has the form: E(t) =  B (tc  t)^{α} / SQRT[1 + { (tc  t)/Δt }^{2α}]
where tc is the time of the crash and B, α and Δt are constants.
 We also introduce an oscillatory component like so:
Note that it oscillates more rapidly as the time of the crash approaches.
This function has the form: Osc(t) = 1 + C COS[ w Log(tct) + (Δw/2α) Log(1+{ (tc  t)/Δt }^{2α}) ]
where tc is the time of the crash and C, w and Δw are constants.
 Then we combine the two to generate a function:
Sornette(t) = A + E(t) Osc(t) = A  B (tc  t)^{α} / SQRT[1 + { (tc  t)/Δt }^{2α}]
[1 + C COS[ w Log(tct) + (Δw/2α) Log(1+{ (tc  t)/Δt }^{2α}) ]]

>Mamma mia! It has how many parameters? Uh ...let me count ...
There are a bunch, namely: A, B, C, w, Δt and α.
>You forgot Δw.
Oops!
Anyway, we pick the parameters so as to mimic the behaviour of the S&P ... before the crash.
>And it looks like ... what?
Our Sornette curve? Like this
It oscillates just like the S&P did, before the crash. Do you see that?
>But what parameters did you choose?
Aah, you'll have to read Sornette's paper.
 
If you'd like to play with the parameters yourself, there's a spreadsheet: (Click on the picture to download.)
Note that there are sliders for each of the 7 parameters. Slide 'em left or right to see if you can get a good match.
Note, too, that the Index and the "match" go back to Jan, 1980.
>It ain't perfect ...
Perfect? You want perfect? Try this.
If you'd like to play with other stocks or indexes, for a different time period, there's this spreadsheet:
>And it's guaranteed to work?
Sometimes ...
Check it out:
>Okay, but where does Sornette get that sexy function. It looks pretty complicated and ...
You can take a peek here
