##
**Pricing perpetual options for jump processes. With discussion by X. Sheldon Lin and Xiaolan Zhang and a reply by the authors.**
*(English)*
Zbl 1081.91528

Summary: We consider two models in which the logarithm of the price of an asset is a shifted compound Poisson process. Explicit results are obtained for prices and optimal exercise strategies of certain perpetual American options on the asset, in particular for the perpetual put option. In the first model in which the jumps of the asset price are upwards, the results are obtained by the martingale approach and the smooth junction condition. In the second model in which the jumps are downwards, we show that the value of the strategy corresponding to a constant option-exercise boundary satisfies a certain renewal equation. Then the optimal exercise strategy is obtained from the continuous junction condition. Furthermore, the same model can be used to price certain reset options. Finally, we show how the classical model of geometric Brownian motion can be obtained as a limit and also how it can be integrated in the two models.

### MSC:

91G20 | Derivative securities (option pricing, hedging, etc.) |

60G35 | Signal detection and filtering (aspects of stochastic processes) |

PDF
BibTeX
XML
Cite

\textit{H. U. Gerber} and \textit{E. S. W. Shiu}, N. Am. Actuar. J. 2, No. 3, 101--112 (1998; Zbl 1081.91528)

Full Text:
DOI

### References:

[1] | Feller W., An Introduction to Probability Theory and Its Applications (1966) · Zbl 0138.10207 |

[2] | Gerber H.U., Insurance: Mathematics and Economics 22 pp 263– (1998) |

[3] | Gerber H.U., ASTIN Bulletin: Journal of the International Actuarial Association 24 pp 195– (1994) |

[4] | Gerber H.U., Insurance: Mathematics and Economics 18 pp 183– (1996) |

[5] | Gerber H.U., Mathematical Finance 6 pp 303– (1996) · Zbl 0919.90009 |

[6] | Gerber H.U., Joint Day Proceedings Volume of XXVIIIth International ASTIN Colloquium/7th International AFIR Colloquium pp 157– (1997) |

[7] | DOI: 10.1080/10920277.1998.10595671 · Zbl 1081.60550 |

[8] | Lamberton D., Introduction to Stochastic Calculus Applied to Finance (1996) · Zbl 0949.60005 |

[9] | Lundberg F., Skandinavisk Aktuarietidskrift 15 pp 137– (1932) |

[10] | Merton R.C., Bell Journal of Economics and Management Science 4 pp 141– (1973) |

[11] | Michaud F., Working paper 97.01 (1997) |

[12] | Samuelson P.A., Industrial Management Review 6 (2) pp 13– (1965) |

This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.