Effectiveness of Measures of Performance During Speculative Bubbles

Statistical analysis of financial data most focused on testing the validity of Brownian motion (Bm). Analysis performed on several time series have shown deviation from the Bm hypothesis, that is at the base of the evaluation of many financial derivatives. We inquiry in the behavior of measures of performance based on maximum drawdown movements (MDD), testing their stability when the underlying process deviates from the Bm hypothesis. In particular we consider the fractional Brownian motion (fBm), and fluctuations estimated empirically on raw market data. The case study of the rising part of speculative bubbles is reported

On the drawdown of completely asymmetric Levy processes

The {\em drawdown} process $Y$ of a completely asymmetric L\'{e}vy process $X$ is equal to $X$ reflected at its running supremum $\bar{X}$: $Y = \bar{X} - X$. In this paper we explicitly express in terms of the scale function and the L\'{e}vy measure of $X$ the law of the sextuple of the first-passage time of $Y$ over the level $a>0$, the time $\bar{G}_{\tau_a}$ of the last supremum of $X$ prior to $\tau_a$, the infimum \unl X_{\tau_a} and supremum \ovl X_{\tau_a} of $X$ at $\tau_a$ and the undershoot $a - Y_{\tau_a-}$ and overshoot $Y_{\tau_a}-a$ of $Y$ at $\tau_a$. As application we obtain explicit expressions for the laws of a number of functionals of drawdowns and rallies in a completely asymmetric exponential L\'{e}vy model.Comment: applications added, 26 pages, 3 figures, to appear in SP