Handbook of Heavy Tailed Distributions in Finance

8 Tail estimation

Given a set of price changes (or log-retums) X₁, … X_n for some asset, it is important to estimate the tail behavior. If the price changes X_t are identically distributed⁶ 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 $\log y\approx \log C-{\rm P}\log r$. 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 ...