Bernoulli

  • Bernoulli
  • Volume 20, Number 3 (2014), 1126-1164.

Small-time asymptotics of stopped Lévy bridges and simulation schemes with controlled bias

José E. Figueroa-López and Peter Tankov

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Abstract

We characterize the small-time asymptotic behavior of the exit probability of a Lévy 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 Lévy 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 Lévy 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.

Article information

Source
Bernoulli, Volume 20, Number 3 (2014), 1126-1164.

Dates
First available in Project Euclid: 11 June 2014

Permanent link to this document
https://projecteuclid.org/euclid.bj/1402488936

Digital Object Identifier
doi:10.3150/13-BEJ517

Mathematical Reviews number (MathSciNet)
MR3217440

Zentralblatt MATH identifier
1312.60056

Keywords
adaptive discretization barrier options bridge Monte Carlo methods exit probability killed Lévy process Lévy bridge small-time asymptotics

Citation

Figueroa-López, José E.; Tankov, Peter. Small-time asymptotics of stopped Lévy bridges and simulation schemes with controlled bias. Bernoulli 20 (2014), no. 3, 1126--1164. doi:10.3150/13-BEJ517. https://projecteuclid.org/euclid.bj/1402488936


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