Drawdowns Preceding Drawups in a Finite Time-Horizon

We begin our journey to the subject of drawdown in Chapter 2, where we determine the probability that a drawdown of a units precedes a drawup of b units in a finite time-horizon. Formally, the drawup of a stochastic process X· is defined as Ut := Xt − Xt, ∀t ≥ 0 where Xt := infs∈[0,t] Xs denotes the running minimum of X·. Thus, this probability assesses the relative strength of downside risk (drawdown) compared to upward momentum (drawup) over a finite time-horizon. To determine this probability, we first consider the simple case with equal-sized drawdown/drawup (i.e., a = b), and derive analytic formulas of this probability by drawing connections to the first exit problems under a simple random walk model and a Brownian motion with drift model. For the general case, we randomize the time-horizon with an independent exponential random variable — a technique known as Canadization, and reduce the probability of interest to the Laplace transform of the first passage time of the drawdown when it precedes a drawup. Using a classical approximation argument as in Lehoczky (1977), we derive analytical formulas for this Laplace transform under general linear diffusion models. Finally, we use Laplace inversion to evaluate the drawdown preceding drawup probability and the conditional density of the maximum relative drawup given a drawdown event, under a geometric Brownian motion (GBM) model.