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.