Mar

20

James Sogi wrote:

I wonder why drops such as the last few weeks materialize out of a normal expectation as they seem to exceed the normal expectation of the current low volatility regime…

 There are theorems in statistics that show that nearly any well-behaved distribution ultimately converges to the normal, given enough iterations. However, such theorems almost always require that the variance be stable and finite. Notwithstanding Professor Mandelbrot's assertion to the contrary, the variance of markets is finite. There has never been an infinite price quote nor will there ever be.But that still leaves the issue of stable variance. That is the catch in two ways. First, the variance as measured by the VIX and even the realized standard deviation has been known to swing wildly in a short period of time. It has also demonstrated varying regimes on the order of 14 years in length.
The other issue is slightly subtler and relates to the underlying process. In the traditional random walk the process is an additive one. Each day's net change is added to the previous to arrive at the new price. If the standard deviation were stable then such distributions would converge to normal.

But if the underlying distribution is multiplicative then that alone causes the standard deviation to grow with time as measured arithmetically. A multiplicative process is consistent with the long-term compounded growth found by Dimson, et al. The way to measure the standard deviation and variance is in the transformed (natural log) variable just as the usual option models do. This is reminiscent of the recent discussion on non-linearity.

From the above it is reasonable to expect large tails whenever the variance increases on a short-term basis. In effect it is like a new higher variability distribution is being superimposed on the more common low variability distribution.


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