On Future Drawdowns of L\'evy processes

TL;DR

This paper analyzes the asymptotic behavior of future drawdown extremes of Lévy processes, providing exact tail probability decay rates and explicit formulas in specific cases relevant to finance and queueing theory.

Contribution

It derives the asymptotic decay of tail probabilities for future drawdowns of Lévy processes under various jump conditions, including Cramér and heavy-tailed cases, and provides explicit distribution formulas for certain jump types.

Findings

Exact asymptotics for tail probabilities of future drawdowns.

Explicit distribution formulas for Lévy processes with single sign jumps.

Application examples including the Black-Scholes model.

Abstract

For a given L\'{e}vy process $X=(X_t)_{t\in\mathbb{R}_+}$ and for fixed $s\in \mathbb{R}_{+}\cup\{\infty\}$ and $t\in\mathbb{R}_+$ we analyse the {\it future drawdown extremes} that are defined as follows: \begin{eqnarray*} \overline D^*_{t,s} = \sup_{0\leq u\leq t} \inf_{u\leq w < t+s}(X_w-X_u), \qquad\qquad \underline D^*_{t,s} = \inf_{0\leq u\leq t} \inf_{u\leq w < t+s}(X_w-X_u). \end{eqnarray*} The path-functionals $\overline D^*_{t,s}$ and $\underline D^*_{t,s}$ are of interest in various areas of application, including financial mathematics and queueing theory. In the case that $X$ has a strictly positive mean, we find the exact asymptotic decay as $x\to\infty$ of the tail probabilities $\mathbb{P}(\overline D^*_{t}<x)$ and $\mathbb{P}(\underline D^*_t<x)$ of $\overline D^*_{t}=\lim_{s\to\infty}\overline D^*_{t,s}$ and $\underline D^*_{t} = \lim_{s\to\infty}\underline D^*_{t,s}$ both when the jumps satisfy the Cram\'er assumption and in a heavy-tailed case. Furthermore, in the case that the jumps of the L\'{e}vy process $X$ are of single sign and $X$ is not subordinator, we identify the one-dimensional distributions in terms of the scale function of $X$. By way of example, we derive explicit results for the Black-Scholes-Samuelson model.