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

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  • Syu, Jia-Hao
  • Lyuu, Yuh-Dauh

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(n1.5)-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(n2.5)-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.

Suggested Citation

  • Syu, Jia-Hao & Lyuu, Yuh-Dauh, 2026. "Trapezoid and trapezoidal prism for the maximum relative drawdown: Probability, crash options pricing, and risk," Applied Mathematics and Computation, Elsevier, vol. 510(C).
    • Handle: RePEc:eee:apmaco:v:510:y:2026:i:c:s0096300325004345
    DOI: 10.1016/j.amc.2025.129708
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    References listed on IDEAS

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