## Abstract

The lookback option with fixed strike in the case of finite horizon was examined with help of the solution to the optimal stopping problem for a three-dimensional Markov process in [P. Gapeev, Discounted optimal stopping for maxima in diffusion models with finite horizon, Electron. J. Probab. 11 (2006), pp. 1031-1048]. The purpose of this paper was to illustrate another derivation of the solution in [P. Gapeev, Discounted optimal stopping for maxima in diffusion models with finite horizon, Electron. J. Probab. 11 (2006), pp. 1031-1048]. The key idea is to use the Girsanov change-of-measure theorem which allows to reduce the three-dimensional optimal stopping problem to a two-dimensional optimal stopping problem with a scaling strike. This approach simplifies the discussion and expressions for the arbitrage-free price and the rational exercise boundary. We derive a closed-form expression for the value function of the two-dimensional problem in terms of the optimal stopping boundary and show that the optimal stopping boundary itself can be characterized as the unique solution to a nonlinear integral equation. Using these results we obtain the arbitrage-free price and the rational exercise boundary of the option.

Original language | British English |
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Pages (from-to) | 510-526 |

Number of pages | 17 |

Journal | Stochastics |

Volume | 86 |

Issue number | 3 |

DOIs | |

State | Published - May 2014 |

## Keywords

- American lookback option
- finite horizon
- fixed strike
- nonlinear integral equation
- optimal stopping
- parabolic free-boundary problem