Analyzing model robustness via a distortion of the stochastic root: A Dirichlet prior approach

Jan-Frederik Mai; Steffen Schenk; Matthias Scherer

doi:10.1515/strm-2015-0009

Article

Analyzing model robustness via a distortion of the stochastic root: A Dirichlet prior approach

Jan-Frederik Mai , Steffen Schenk and Matthias Scherer

Published/Copyright: May 12, 2016

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From the journal Statistics & Risk Modeling Volume 32 Issue 3-4

Abstract

It is standard in quantitative risk management to model a random vector 𝐗:={Xtk}k=1,...,d of consecutive log-returns to ultimately analyze the probability law of the accumulated return Xt1+⋯+Xtd. By the Markov regression representation (see [25]), any stochastic model for 𝐗 can be represented as Xtk=fk(Xt1,...,Xtk-1,Uk), k=1,...,d, yielding a decomposition into a vector 𝐔:={Uk}k=1,...,d of i.i.d. random variables accounting for the randomness in the model, and a function f:={fk}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_{t_k} 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,∞] 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.

Keywords: Model robustness; model uncertainty; Value-at-Risk models; Dirichlet copula

MSC: 60G57; 60G09; 60E07

We thank Franz Lorenz, Giovanni Puccetti, Steven Vanduffel, and two anonymous referees for valuable comments on earlier versions of this manuscript. Furthermore, we appreciate the feedback and discussions after the presentations at the workshops “New horizons in copula modeling” in Montreal, “Copulae: On the crossroads of Mathematics and Economics” in Oberwolfach, and “Recent developments in dependence modelling with applications in Finance and Insurance” in Brussels.

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Received: 2015-4-1

Revised: 2016-4-11

Accepted: 2016-4-14

Published Online: 2016-5-12

Published in Print: 2015-12-1

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Articles in the same Issue

https://doi.org/10.1515/strm-2015-0009

Keywords for this article

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