### Abstract

Given discrete time observations over a fixed time interval, we study a nonparametric Bayesian approach to estimation of the volatility coefficient of a stochastic differential equation. We postulate a histogram-type prior on the volatility with piecewise constant realisations on bins forming a partition of the time interval. The values on the bins are assigned an inverse Gamma Markov chain (IGMC) prior. Posterior inference is straightforward to implement via Gibbs sampling, as the full conditional distributions are available explicitly and turn out to be inverse Gamma. We also discuss in detail the hyperparameter selection for our method. Our nonparametric Bayesian approach leads to good practical results in representative simulation examples. Finally, we apply it on a classical data set in change-point analysis: weekly closings of the Dow-Jones industrial averages.

Original language | English |
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Title of host publication | 2017 MATRIX Annals |

Editors | D. Wood, J. de Gier, C. Praeger, T. Tao |

Place of Publication | Cham |

Publisher | Springer |

Pages | 279-302 |

Number of pages | 24 |

ISBN (Electronic) | 978-3-030-04161-8 |

ISBN (Print) | 978-3-030-04160-1 |

DOIs | |

Publication status | Published - 2019 |

### Publication series

Name | MATRIX Book Series book series |
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Volume | 2 |

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## Cite this

Gugushvili, S., van der Meulen, F., Schauer, M., & Spreij, P. (2019). Nonparametric Bayesian Volatility Estimation. In D. Wood, J. de Gier, C. Praeger, & T. Tao (Eds.),

*2017 MATRIX Annals*(pp. 279-302). (MATRIX Book Series book series; Vol. 2). Springer. https://doi.org/10.1007/978-3-030-04161-8_19