drawdown - Expectation of maximum draw down in the Brownian motion case

Let $$X_t = \mu t + \sigma B_t$$ be a linear Brownian motion with drift. Let $$S_t = \max(X_u, u \le t)$$ denote the process of the running max, then the draw down is given by $$DD_t = S_t - X_t,$$ and the maximum draw down over a period $[0,T]$ is $$max_{u \in [0,T]} DD_u.$$ What can we say about $$E[ max_{u \in [0,T]} DD_u ] ?$$ How can we calculate the expected maximum draw down? Are there analytical formulas, approximations, available (R) packages?

Answer

as I mentioned here, this paper provides some theoretical insight (and a way to approximate the true value).

The authors end up with an approximative series for the density. It is implemented in the function maxdd of the R-package fBasics. There are convenient functions dmaxdd, pmaxdd and rmaxdd. Calculating the Expected Drawdown should be easy. (to be honest, I found the paper as a reference provided on the help page of the functions mentioned above)

The function you are asking for would be maxddStats:

require(fBasics)
maxddStats(mu,sigma,t)