### Abstract

In this paper we apply the recently established Wiener-Hopf Monte Carlo simulation technique for Lévy processes from Kuznetsov et al. (2011) to path functionals; in particular, first passage times, overshoots, undershoots, and the last maximum before the passage time. Such functionals have many applications, for instance, in finance (the pricing of exotic options in a Lévy model) and insurance (ruin time, debt at ruin, and related quantities for a Lévy insurance risk process). The technique works for any Lévy process whose running infimum and supremum evaluated at an independent exponential time can be sampled from. This includes classic examples such as stable processes, subclasses of spectrally one-sided Lévy processes, and large newfamilies such as meromorphic Lévy processes. Finally, we present some examples. A particular aspect that is illustrated is that the Wiener-Hopf Monte Carlo simulation technique (provided that it applies) performs much better at approximating first passage times than a 'plain' Monte Carlo simulation technique based on sampling increments of the Lévy process.

Original language | English |
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Pages (from-to) | 129-148 |

Number of pages | 20 |

Journal | Journal of Applied Probability |

Volume | 52 |

Issue number | 1 |

Publication status | Published - Mar 2015 |

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### Keywords

- Exotic option pricing
- First passage time
- Insurance risk process
- Lévy process
- Monte Carlo simulation
- Multilevel Monte Carlo
- Overshoot
- Wiener-Hopf decomposition

### Cite this

*Journal of Applied Probability*,

*52*(1), 129-148.