### 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) |
---|---|

Pages (from-to) | 505-532 |

Number of pages | 28 |

Journal | Mathematical Finance |

Volume | 24 |

Issue number | 3 |

DOIs | |

State | Published - 2014 |

### Fingerprint

### Keywords

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

### ASJC Scopus subject areas

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

### Cite this

*Mathematical Finance*,

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

**Boundary evolution equations for american options.** / Mitchell, Daniel; Goodman, Jonathan; Muthuraman, Kumar.

Research output: Contribution to journal › Article

*Mathematical Finance*, vol. 24, no. 3, pp. 505-532. https://doi.org/10.1111/mafi.12002

}

TY - JOUR

T1 - Boundary evolution equations for american options

AU - Mitchell, Daniel

AU - Goodman, Jonathan

AU - Muthuraman, Kumar

PY - 2014

Y1 - 2014

N2 - 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.

AB - 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.

KW - American options

KW - Dynamic grid

KW - Early exercise boundary

KW - Free-boundary problem

KW - Optimal stopping

KW - Stochastic volatility

UR - http://www.scopus.com/inward/record.url?scp=84901978664&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84901978664&partnerID=8YFLogxK

U2 - 10.1111/mafi.12002

DO - 10.1111/mafi.12002

M3 - Article

VL - 24

SP - 505

EP - 532

JO - Mathematical Finance

JF - Mathematical Finance

SN - 0960-1627

IS - 3

ER -