On Probability Characteristics of "Downfalls" in a Standard Brownian Motion

Abstract

International audienceFor a Brownian motion $B=(B_t)_{t\le 1}$ with $B_0=0$, {\bf E}$B_t=0$, {\bf E}$B_t^2=t$ problems of probability distributions and their characteristics are considered for the variables $$ \begin{array}{c} {\mathbb D} =\displaystyle\sup_{0\le t\le t'\le 1}(B_t-B_{t'}),\qquad {\mathbb D}_1=B_\sigma-\inf_{\sigma\le t'\le 1}B_{t'}, \\ {\mathbb D}_2=\displaystyle\sup_{0\le t\le\sigma'}B_{t}-B_{\sigma'}, \end{array} $$ where $\sigma$ and $\sigma'$ are times (non-Markov) of the absolute maximum and absolute minimum of the Brownian motion on $[0,1]$ (i.e., $B_\sigma=\sup_{0\le t\le 1}B_t$, $B_{\sigma'}=\inf_{0\le t'\le 1}B_{t'}$)

An Efficient Algorithm for Simulating the Drawdown Stopping Time and the Running Maximum of a Brownian Motion

“…The magnitude of drawdowns has been studied extensively in the literature. In particular the probabilistic properties of the first time the drawdown amount for a Brownian motion exceeds a certain threshold a > 0 has been studied in Taylor (1975) and Douady et al (2000). Insurance contracts designed to insure against the risk of large drawdowns have been introduced in Carr et al (2011), who proposed a way to hedge the liability.…”

Drawdown: from practice to theory and back again

“…To our knowledge, the earliest mathematical analysis of the maximum drawdown of a Brownian motion appeared in Taylor (1975), and it was shortly afterwards generalized to time-homogenous diffusion processes by Lehoczky (1977). Douady et al (2000) and Magdon-Ismail et al (2004) derive an infinite series expansion for a standard Brownian motion and a Brownian motion with a drift, respectively. The discussion of drawdown magnitude was extended to studying the frequency rate of drawdown for a Brownian motion in Landriault et al (2015).…”

Portfolio Management with Drawdown-Based Measures

“…An analytic solution for the drawdown distribution is not feasible for realistic time series with serial correlation and fat tails. Even the relatively simple case of standard Brownian motion involves a very complex derivation of expected drawdown, as documented by Douady, Shiryaev, and Yor [2000].…”

“…They also suggested a block bootstrap procedure for the calculation of CDD and showed that the procedure is robust after approximately 100 simulations. They used the bootstrap approach because analytic solutions are not feasible, and, as shown by Douady, Shiryaev, and Yor [2000], even the calculation of expected drawdown for standard Brownian motion can be very complex. Block bootstrap is particularly attractive because it preserves the serial and cross-correlational characteristics of the original dataset, likely making it superior to the rolling historical MDD algorithm of Goldberg and Mahmoud [2014] that was used for illustrative purposes.…”