## Abstract

It is standard in quantitative risk management to model a random vector X:={X_{tk}} k=1,..,d X:=X_{tk}k=1,d of consecutive log-returns to ultimately analyze the probability law of the accumulated return X_{t1} +X_{td}X_{t1}+X_{td}. By the Markov regression representation (see [25]), any stochastic model for X can be represented as X_{tk} =f_{k} (X_{t1},..,X_{tk-1},U_{k})X_{tk}=f_{k}(X_{t1},X_{tk-1},U_{k}), k=1,..,d k=1,d, yielding a decomposition into a vector:=Ukk=1,..,d U:=U_{k}k=1,d of i.i.d. random variables accounting for the randomness in the model, and a function f:={f k } k=1,..,d f:=f_{k}k=1,d representing the economic reasoning behind. For most models, f is known explicitly and U_{k} may be interpreted as an exogenous risk factor affecting the return X_{tk} in time step k. While existing literature addresses model uncertainty by manipulating the function f, we introduce a new philosophy by distorting the source of randomness U and interpret this as an analysis of the model's robustness. We impose consistency conditions for a reasonable distortion and present a suitable probability law and a stochastic representation for U based on a Dirichlet prior. The resulting framework has one parameter c[0]c[0]tuning the severity of the imposed distortion. The universal nature of the methodology is illustrated by means of a case study comparing the effect of the distortion to different models for X. As a mathematical byproduct, the consistency conditions of the suggested distortion function reveal interesting insights into the dependence structure between samples from a Dirichlet prior.

Original language | English |
---|---|

Pages (from-to) | 177-195 |

Number of pages | 19 |

Journal | Statistics and Risk Modeling |

Volume | 32 |

Issue number | 3-4 |

DOIs | |

State | Published - 1 Dec 2015 |

## Keywords

- Dirichlet copula
- Model robustness
- Value-at-Risk models
- model uncertainty