Risk Latte - “Random Walk” or the “Lehman Walk”?

“Random Walk” or the “Lehman Walk”?

Rahul Bhattacharya
September 14, 2008

What is this "Lehman Walk" that we mention in our previous article?

Tanaka San was able to say something about it, though we must say that his exposition of the phenomenon was not really mathematical. A math model needs to be fitted to the observations we've made so far.

Research is still in progress; but while we slog away to unravel the latest phenomenon in the field of quantitative finance our heart goes out to all those bright, young Indian Ivy league MBAs working at Lehman worldwide whose "dowry" and "future marriage value" have gone down, in tandem with Lehman's stock price, some 95% in the past year. We also feel sad for the occasional Rhodes Scholar and the Oxford graduate who while running the bank's operation in Asia Pacific never lost his focus on the long term strategy which was to make sure that his job was secure. And who can forget the great Dick; slogging, working, mesmerizing, coercing, fooling and screwing employees and clients in a gigantic effort to keep the Lehman family together. Let's light a candle for him.

Then we have that army of quants, the Ph.D.s, working 15 hours a day on the trading floor to build models that no one else could understand or replicate. So sad, so traumatic! Our heart weeps for them. Tanaka san, all this was not really bullshit. These were good - no, great - engineers, physicists and mathematicians who understood their fluid dynamics and Fourier transforms. They simply didn't understand the fundamental concept of negative skew in the financial markets and in the real world. Please don't berate them or belittle their work.

Anyway, coming back to our research our preliminary analysis shows that "Lehman Walk" is some sort of a geometric Brownian motion (GBM) with a very complex drift term, an exploding volatility and a jump factor added in. The drift is not mean reverting, but rather "initial value reverting". It reverts back to the value of the security at the start, i.e. time equal to zero. Almost, fifteen years ago, on May 11, 1994, Lehman stock first started trading at $16.51, which adjusted for stock splits and dividends was equal to around $3.74. It was at or around that level last Friday*.

Then we suspect that the volatility scales not with the square root of time but with time itself. This gives us an exploding volatility! Also, despite the GBM assumption, it is quite likely that Lehman stock can become zero - exactly zero - rather than approaching zero. This is where the Lehman walk decisively moves away from a random walk.

And the jump term is so complex that we don't have a clue as to where it comes from and how it impacts the stock price. No jump diffusion model discovered so far has such a scary looking jump term. Is this some kind of a Poisson process? No idea at all! Tanaka san would perhaps say that we call it a "Fuld process".

Anyway, please bear with us. Research is in progress.


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