### Abstract

The connection between optimal stopping of random systems and the theory of the Snell envelop is well understood, and its application to the pricing of American contingent claims is well known. Motivated by the pricing of swing contracts (whose recall components can be viewed as contingent claims with multiple exercises of American type) we investigate the mathematical generalization of these results to the case of possible multiple stopping. We prove existence of the multiple exercise policies in a fairly general set-up. We then concentrate on the Black-Scholes model for which we give a constructive solution in the perpetual case, and an approximation procedure in the finite horizon case. The last two sections of the paper are devoted to numerical results. We illustrate the theoretical results of the perpetual case, and in the finite horizon case, we introduce numerical approximation algorithms based on ideas of the Malliavin calculus.

Original language | English (US) |
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Pages (from-to) | 239-268 |

Number of pages | 30 |

Journal | Mathematical Finance |

Volume | 18 |

Issue number | 2 |

DOIs | |

State | Published - Apr 1 2008 |

### All Science Journal Classification (ASJC) codes

- Accounting
- Finance
- Social Sciences (miscellaneous)
- Economics and Econometrics
- Applied Mathematics

### Keywords

- Multiple American exercise
- Optimal multiple stopping
- Swing options

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## Cite this

*Mathematical Finance*,

*18*(2), 239-268. https://doi.org/10.1111/j.1467-9965.2007.00331.x