Trapezoid and trapezoidal prism for the maximum relative drawdown: Probability, crash options pricing, and risk

Abstract

Maximum drawdown (MDD) and maximum relative drawdown (MrDD) are well-known in portfolio management and performance evaluations. They can also form the basis of a stop-loss strategy. But there is no closed-form formula for the probability that the MrDD (MDD) ever reaches some positive threshold over a period of time. This paper focuses on MrDD and employs a random walk to approximate the underlying geometric Brownian motion (GBM) for the price, taking care to match the threshold for faster convergence. Let $n$ be the number of time steps. This paper proposes an $O\left(n^{1.5}\right)$-sized trapezoid to calculate the above-mentioned probability. The trapezoid can price the crash option with a digital payoff accurately. This paper further proposes an $O\left(n^{2.5}\right)$-sized trapezoidal prism to price the crash option with a resetting payoff accurately and calculate the expected return rate and common risk measures of an MrDD-based stop-loss strategy.