On magnitude, asymptotics and duration of drawdowns for Lévy models

Abstract

This paper considers magnitude, asymptotics and duration of drawdowns for some Lévy processes. First, we revisit some existing results on the magnitude of drawdowns for spectrally negative Lévy processes using an approximation approach. For any spectrally negative Lévy process whose scale functions are well-behaved at $0+$, we then study the asymptotics of drawdown quantities when the threshold of drawdown magnitude approaches zero. We also show that such asymptotics is robust to perturbations of additional positive compound Poisson jumps. Finally, thanks to the asymptotic results and some recent works on the running maximum of Lévy processes, we derive the law of duration of drawdowns for a large class of Lévy processes (with a general spectrally negative part plus a positive compound Poisson structure). The duration of drawdowns is also known as the "Time to Recover" (TTR) the historical maximum, which is a widely used performance measure in the fund management industry. We find that the law of duration of drawdowns qualitatively depends on the path type of the spectrally negative component of the underlying Lévy process.

arXiv commentPublished at http://dx.doi.org/10.3150/15-BEJ748 in the Bernoulli (http://isi.cbs.nl/bernoulli/) by the International Statistical Institute/Bernoulli Society (http://isi.cbs.nl/BS/bshome.htm)

References in corpus (6)

• On the optimal dividend problem for a spectrally negative Lévy process
• On optimality of the barrier strategy in de Finetti's dividend problem for spectrally negative Lévy processes
• Parisian ruin probability for spectrally negative Lévy processes
• Suprema of Lévy processes
• Stochastic Modeling and Fair Valuation of Drawdown Insurance
• Default Swap Games Driven by Spectrally Negative Levy Processes

Cited by in corpus (8)

• A Unified Approach for Drawdown (Drawup) of Time-Homogeneous Markov Processes
• The $W,Z$ scale functions kit for first passage problems of spectrally negative Levy processes, and applications to the optimization of dividends
• Beating the Omega Clock: An Optimal Stopping Problem with Random Time-horizon under Spectrally Negative Lévy Models
• Exit problems for general draw-down times of spectrally negative Lévy processes
• Simulation of the drawdown and its duration in Lévy models via stick-breaking Gaussian approximation
• Three essays on stopping
• On Future Drawdowns of Lévy processes
• Monte Carlo algorithm for the extrema of tempered stable processes