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

Abstract

We define the drawdown stopping time of a Brownian motion as the first time its drawdown reaches a duration of length 1. In this paper, we propose an efficient algorithm to efficiently simulate the drawdown stopping time and the associated maximum at this time. The method is straightforward and fast to implement, and avoids simulating sample paths thus eliminating discretisation bias. We show how the simulation algorithm is useful for pricing more complicated derivatives such as multiple drawdown options.

Azéma martingales for Bessel and CIR processes and the pricing of Parisian zero‐coupon bonds

“…From this, we develop exact simulation algorithms to sample from the stopping time distribution. This is a generalization of the result for Brownian motion in Dassios and Lim (2017). Through the Laplace transform, we also observe that the Parisian stopping time of the Bessel process with index α is distributed according to a truncated stable process with index α taken at an exponential time, and the Parisian stopping time of the CIR process is distributed according to a truncated Lamperti stable process taken at exponential time.…”

“…The comparison of the simulated and recursive density for the Parisian stopping time of the Squared Bessel/Bessel process under the parameter settings is illustrated in Figure 2. Since for , results were obtained for the Brownian motion in Dassios and Lim (2015), we also establish a comparison of the simulation results for Algorithm 1 with k close to 0, and Algorithm 2 against the simulation algorithm for the drawdown stopping time of Brownian motion described in Dassios and Lim (2017). The density plot is given in Figure 3.…”

Instantaneous Volatility Seasonality of High-Frequency Markets in Directional-Change Intrinsic Time

“…Drawdowns D t and drawups U t , also called rallies in Hadjiliadis and Večeř (2006), registered by the moment of time t, depend on the running price maxima S t and the running price minima S t (Dassios and Lim 2018;Landriault et al 2015;Mijatović and Pistorius 2012;Zhang and Hadjiliadis 2012). These reference points hinge on the set of historical prices S s and are mathematically defined in the following way:…”

The arctangent law for a certain random time related to one-dimensional diffusions

“…The knowledge of the distribution of S is relevant in various diffusion models used in applied sciences, such as Mathematical Finance, Biology, Physics, Hydraulics, etc., whenever the time evolution of the phenomenon under study is described by a diffusion X(t); in fact, one is often interested to find the first instant, after a given time r, at which X(t) exceeds the maximum value attained in the time interval [0, r], namely in times prior to r. For instance, in the Economy framework, if we let r vary in (0, +∞), the process S(r), so obtained, is related to the drawdown process, which measures the fall in value of X(t) from its running maxima, and is frequently used as performance indicator in the fund management industry (see e.g. (Dassios and Lim, 2017) and references therein). Indeed, S(r) can be expressed in terms of the time elapsed since the last time the maximum is achieved, that was studied in (Dassios and Lim, 2017).…”

“…(Dassios and Lim, 2017) and references therein). Indeed, S(r) can be expressed in terms of the time elapsed since the last time the maximum is achieved, that was studied in (Dassios and Lim, 2017).…”