Speed and duration of drawdown under general Markov models

Abstract

We propose an efficient computational method based on continuous-time Markov chain (CTMC) approximation to compute the distributions of the speed and duration of drawdown for general one-dimensional (1D) time-homogeneous Markov processes. We derive linear systems for the Laplace transforms of drawdown quantities and show how to solve them efficiently by recursion. In addition, we prove the convergence of our method and obtain a sharp estimate of the convergence rate under some assumptions. As applications, we consider pricing two financial options that provide protection against drawdown risk. Finally, we demonstrate the accuracy and efficiency of our method through extensive numerical experiments.

Keywords:

• Speed of drawdown
• Duration of drawdown
• General Markov models
• CTMC
• Option pricing

MSC (2020) Classification:

• 60J28
• 60J60
• 60J76
• 91G20
• 91G30
• 91G60

Disclosure statement

No potential conflict of interest was reported by the author(s).

Notes

1 This can happen if the process is observed before time 0.

Additional information

Funding

Gongqiu Zhang was supported by National Natural Science Foundation of China Grant 12171408, and Shenzhen Fundamental Research Program Project JCYJ20190813165407555. Lingfei Li was supported by Hong Kong Research Grant Council General Research Fund Grant 14207019. Pingping Zeng would like to acknowledge the support from the National Natural Science Foundation of China (Grant Nos. 11701266 and 12171228).