Small-time asymptotics of stopped L\'evy bridges and simulation schemes with controlled bias

TL;DR

This paper derives small-time asymptotics for the exit probabilities and overshoot distributions of Lévy processes, and introduces an adaptive Monte Carlo scheme with controlled bias for computing related functionals.

Contribution

It provides new asymptotic formulas for Lévy process exit probabilities and overshoot laws, and develops an adaptive discretization algorithm for Monte Carlo simulations with bias control.

Findings

Asymptotic expansions with computable error bounds for exit probabilities.

A novel adaptive Monte Carlo scheme with controlled bias.

Applicability to finance and natural sciences functionals.

Abstract

We characterize the small-time asymptotic behavior of the exit probability of a L\'evy process out of a two-sided interval and of the law of its overshoot, conditionally on the terminal value of the process. The asymptotic expansions are given in the form of a first-order term and a precise computable error bound. As an important application of these formulas, we develop a novel adaptive discretization scheme for the Monte Carlo computation of functionals of killed L\'evy processes with controlled bias. The considered functionals appear in several domains of mathematical finance (e.g., structural credit risk models, pricing of barrier options, and contingent convertible bonds) as well as in natural sciences. The proposed algorithm works by adding discretization points sampled from the L\'evy bridge density to the skeleton of the process until the overall error for a given trajectory becomes smaller than the maximum tolerance given by the user.