## 8 Tail estimation

Given a set of price changes (or log-retums) X_{1}, … X_{n} for some asset, it is important to estimate the tail behavior. If the price changes X_{t} are identically distributed^{6} with X and P(X>r)~C_{r}^{−α}, then the dispersion C and the tail index α determine the central limit behavior, as well as the extreme value behavior, of the price change distribution. Mandelbrot (1963) pioneered a graphical estimation method for C and α. If y=P(X>r) ≈ r^{−α} then $\mathrm{log}y\approx \mathrm{log}C-{\rm P}\mathrm{log}r$. Ordering the data so that ${X}_{\left(1\right)}\ge {X}_{\left(2\right)}\ge \dots \ge {X}_{\left(n\right)}$ we should have approximately that ...

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