### Abstract

We present a fast and accurate method to compute exponential moments of the discretely observed maximum of a Lévy process. The method involves a sequential evaluation of Hilbert transforms of expressions involving the characteristic function of the (Esscher-transformed) Lévy process. It can be discretized with exponentially decaying errors of the form O(exp (-aM^{b})) for some a,b > 0, where M is the number of discrete points used to compute the Hilbert transform. The discrete approximation can be efficiently implemented using the Toeplitz matrix-vector multiplication algorithm based on the fast Fourier transform, with total computational cost of O(N M log (M)), where N is the number of observations of the maximum. The method is applied to the valuation of European-style discretely monitored floating strike, fixed strike, forward start and partial lookback options (both newly written and seasoned) in exponential Lévy models.

Original language | English (US) |
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Pages (from-to) | 501-529 |

Number of pages | 29 |

Journal | Finance and Stochastics |

Volume | 13 |

Issue number | 4 |

DOIs | |

State | Published - Jul 1 2009 |

### Keywords

- Discrete lookback options
- Discrete maximum
- Esscher transform
- Exponential moments
- Fourier transform
- Hilbert transform
- Lévy processes
- Sinc expansion

### ASJC Scopus subject areas

- Statistics and Probability
- Finance
- Statistics, Probability and Uncertainty