A General Method for Analysis and Valuation of Drawdown Risk under Markov Models

Abstract

Drawdown risk is a major concern in financial markets. We develop a novel algorithm to solve the first passage problem of the drawdown process of general one-dimensional time-homogeneous Markov processes. We compute its Laplace transform based on continuous time Markov chain (CTMC) approximation and numerically invert the Laplace transform to obtain the first passage probabilities and the distribution of the maximum drawdown. We prove convergence of our algorithm for general Markovian models and provide sharp estimate of the convergence rate for a general class of jump-diffusion models. We apply the algorithm to calculate the Calmar ratio for investment analysis, price maximum drawdown derivatives and hedge the risk of selling such derivatives with a highly volatile asset as the underlying. Various numerical experiments document the computational efficiency of our method. We also develop extensions to solve the drawdown problem in models with time dependence or stochastic volatility or regime switching and for portfolio drawdown analysis.