Drawdowns, Drawups, and Occupation Times under General Markov Models

TL;DR

This paper develops efficient algorithms for computing drawdown-related risk measures in general Markov models, enabling practical risk assessment in finance with theoretical guarantees and validated numerical performance.

Contribution

It introduces a unified framework and algorithms for drawdown quantities under general Markov models, with proven convergence and computational efficiency.

Findings

Algorithms achieve cubic complexity for general Markov models

Algorithms are linear for diffusion models

Numerical experiments confirm accuracy and efficiency

Abstract

Drawdown risk, an important metric in financial risk management, poses significant computational challenges due to its highly path-dependent nature. This paper proposes a unified framework for computing five important drawdown quantities introduced in Landriault et al. (2015) and Zhang (2015) under general Markov models. We first establish linear systems and develop efficient algorithms for such problems under continuous-time Markov chains (CTMCs), and then establish their theoretical convergence to target quantities under general Markov models. Notably, the proposed algorithms for most quantities achieve the same complexity order as those for path-independent problems: cubic in the number of CTMC states for general Markov models and linear when applied to diffusion models. Rigorous convergence analysis is conducted under weak regularity conditions, and extensive numerical experiments validate the accuracy and efficiency of the proposed algorithms.