### Abstract

We consider the problem of finding optimal exercise policies for American options, both under constant and stochastic volatility settings. Rather than work with the usual equations that characterize the price exclusively, we derive and use boundary evolution equations that characterize the evolution of the optimal exercise boundary. Using these boundary evolution equations we show how one can construct very efficient computational methods for pricing American options that avoid common sources of error. First, we detail a methodology for standard static grids and then describe an improvement that defines a grid that evolves dynamically while solving the problem. When integral representations are available, as in the Black-Scholes setting, we also describe a modified integral method that leverages on the representation to solve the boundary evolution equations. Finally we compare runtime and accuracy to other popular numerical methods. The ideas and methodology presented herein can easily be extended to other optimal stopping problems.

Original language | English (US) |
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Pages (from-to) | 505-532 |

Number of pages | 28 |

Journal | Mathematical Finance |

Volume | 24 |

Issue number | 3 |

DOIs | |

State | Published - Jul 2014 |

### Keywords

- American options
- Dynamic grid
- Early exercise boundary
- Free-boundary problem
- Optimal stopping
- Stochastic volatility

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## Cite this

*Mathematical Finance*,

*24*(3), 505-532. https://doi.org/10.1111/mafi.12002