Why is it Fat-tailed?

Summary

When we talk about the price fluctuation, whether it is in an academy or in the actual financial market, following three factors are taken for granted.

First, the distribution of a price fluctuation obeys logarithmic normal distribution.

Second, the random variables of prices are independent.

Third, the distribution is steady and constant at all times.

We tend to neglect the fact that a normal distribution requires some important premises: both expectation and variance must be constant. Can we separate both mean value and variance as an independent parameter in the world of price fluctuation statistics? After all our experience in the financial market, we are rather skeptical about this. Real market is a very discrete and uniformless world.

We make mistakes easily on recognizing the real market phenomenon, leading to obtain result such as “We can’t see forest for trees.”

So we would like to explain the structure and the characteristic of a price in a financial market, mainly focusing on the Dollar Yen market with some concepts: quasi-Fractal dimension in ξ t-P metric space, µ ² — σ ² test and that the structure leads a Lévy Additive process.