## Bernoulli

• Bernoulli
• Volume 23, Number 1 (2017), 432-458.

### 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.

#### Article information

Source
Bernoulli, Volume 23, Number 1 (2017), 432-458.

Dates
Revised: March 2015
First available in Project Euclid: 27 September 2016

https://projecteuclid.org/euclid.bj/1475001360

Digital Object Identifier
doi:10.3150/15-BEJ748

Mathematical Reviews number (MathSciNet)
MR3556778

Zentralblatt MATH identifier
06673483

#### Citation

Landriault, David; Li, Bin; Zhang, Hongzhong. On magnitude, asymptotics and duration of drawdowns for Lévy models. Bernoulli 23 (2017), no. 1, 432--458. doi:10.3150/15-BEJ748. https://projecteuclid.org/euclid.bj/1475001360

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