Sornette
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 }]
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(tc-t) + (Δw/2α) Log(1+{ (tc - t)/Δt }) ]
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(tc-t) + (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.

 another Sornette

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