I'll try to oblige, but I doubt I'll have the time for it before Christmass break. The articles from the report rely on quite a few other publications which I feel need to be followed to get a better picture. Until then and with a caveat I've done only a very cursory reading of the topic, here we go: Their model is, in essence, the first term of a Fourier series of a more general log-periodic function progressing towards a critical point (in this case: crash) describing idealised variance of index changes. From the analysis of said variance, they show how previous crashes happened as a result of a local/short-term bubble burst happening on top of long-term bubble growing. In spirit of "picture is worth a thousand words", the self-similarity/fractal part is best seen on their graphs: however, it's just a quirk/property of the function they used. Intuitively, I think that everything log-periodic will show those kinds of patterns. As to how valid/grounded it is for their model, that will have to wait. Maths part I can get (it's even somewhat similar to what I used for analysis of phase transitions in glass during the last year internship), but economics isn't my forte. Here's the summary of mentioned "Physica A" publication, which is far more conservative than reports:So I'm curious as to what, exactly, they're modeling and where the dependent and independent variables lie.
The analysis presented above provides not only further arguments in favour of the existence of the log-periodic component in financial dynamics, self-similarly on various time scales, but also indicates that the corresponding central parameter—the preferred scaling factor—may very well be a constant close to 2. In this way it is possible to obtain a consistent relation between the patterns and it allows more reliable extrapolations into the future. It also allows the log-periodicity to pretend to the status of a law. Of course, on short time scales it is a fragile one, as the real financial market is exposed to many “external” factors, such as unexpected wars or other political events, which may distort its internal hierarchical structure on the organizational as well as on the dynamical level. In this connection it is worth remembering that the functional form of the log-periodic modulation so far is not supplied by theoretical arguments and this opens room for some mathematically unrigorous assignments of patterns, as is often needed in order to properly interpret them. Identifying a hierarchy of time scales and a universal preferred scaling ratio is crucial in this connection and very helpful for real predictions. Strict fitting of the lowest order term in the Fourier expansion of the periodic function in Eq. (3) is typically not an optimal procedure. Here it serves basically as a convenient representation to guide the eye.
EDIT because I apparently suck at pasting text: They fitted an approximation of a scaling function with the log-periodic form being chosen as being both the best to describe observed rapid oscillations and being a solution of a functional equation problem. Then they made it into a discrete function of time around the critical point and, though they could easily do a higher-order approximation, decided to use the first-order solution for demonstration purposes. I'm not savvy enough to know if it's a new, inventive, approach to this type of problems. There's very likely a lot more to it (especially seeing how I almost exclusively talked about this paper), but while you hum loudly on formal maths, my eyes glazed over when I opened a page on Lehman Brothers bankruptcy. While I'd rather avoid it at all cost, I'm going to contact my BSc adviser and ask her for an explanation. She's basically a stockbroker/quant hiding in dynamical systems institute and, unfortunately, might be the best person I know to ask about it. Hopefully, updates inbound.
Jargon in finance is obscurant in nature. For example: CAPE ratio = Cyclically-Adjusted Price/Earnings Ratio. Take how much the stock is, divide it by the amount of money it makes, throw a coefficient on it, get a Nobel Prize in Economics. Fundamentally there's no there there. "Derivatives" in finance are anything where you're not directly trading something. A contract to buy the stock later at a certain price is a "futures option" and also a "derivative." When they talk about "the velocity of money" that's not a derivative, that's a greek symbol used in deep macroeconomic analysis. Lehman Brothers, Long Term Capital Management, The Great Depression, it all boils down to "someone was expecting to get paid and didn't." For purposes of finance, "Joe owes me $10" is the same as "I have $10." That was the basic peril of WWI: Germany figured they could make France pay for the war once they conquered them and France was dependent on Germany paying once they lost. Meanwhile the French owed the British and the French and British owed the Americans (Lloyds of London informed Parliament in 1913 that, regretfully, should the Royal Navy sink any members of the German merchant fleet, Lloyds would have to cut checks). Prior to the actualization of assorted parties' inability to pay, all that war debt was assets. And as soon as you decide you're never going to get paid, that's a "write down." We still can't call it a loss because apparently then we have to feel bad. Worse, "being in debt" is called "leverage" in finance. A "highly-leveraged" hedge fund is a daring bunch of swashbucklers out to make lots of money without having a phatty war chest. They're also a bunch of debtors buying stocks with other people's money on the assumption they're smart and will pay the money back in spades. That's the other dumb thing - the higher your leverage, the more interest people will charge you which means the more money they make from lending you money. The riskier you act, the wealthier you get. Until you write down your margin debt and bread lines erupt on Wall Street. "Borrowing money to bet on things" is fundamentally how markets work. All the rest of it is hand-waving. And so long as people get paid back around the amounts they're expecting, it works. As soon as someone starts losing money, everyone starts losing money because it's a highly-correlated, interconnected mess. This is why I'm mostly interested in what they're modeling, not how they modeled it. It sounds like they're basically taking a number series - the S&P over time - and curve-fitting. Which is basically a selective application of order to chaos that allows you to go "look I predicted the future!" when in fact you predicted the past.