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

Abstract It is standard in quantitative risk management to model a random vector 𝐗:={X t k } k=1,...,d ${\mathbf {X}:=\lbrace X_{t_k}\rbrace _{k=1,\ldots ,d}}$ of consecutive log-returns to ultimately analyze the probability law of the accumulated return X t 1 +⋯+X t d ${X_{t_1}+\cdots +X_{t_d}}$ . By the Markov regression representation (see [25]), any stochastic model for 𝐗${\mathbf {X}}$ can be represented as X t k =f k (X t 1 ,...,X t k-1 ,U k )${X_{t_k}=f_k(X_{t_1},\ldots ,X_{t_{k-1}},U_k)}$ , k=1,...,d${k=1,\ldots ,d}$ , yielding a decomposition into a vector 𝐔:={U k } k=1,...,d ${\mathbf {U}:=\lbrace U_{k}\rbrace _{k=1,\ldots ,d}}$ of i.i.d. random variables accounting for the randomness in the model, and a function f:={f k } k=1,...,d ${f:=\lbrace f_k\rbrace _{k=1,\ldots ,d}}$ representing the economic reasoning behind. For most models, f is known explicitly and Uk may be interpreted as an exogenous risk factor affecting the return Xtk 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${\mathbf {U}}$ 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${\mathbf {U}}$ based on a Dirichlet prior. The resulting framework has one parameter c∈[0,∞]${c\in [0,\infty ]}$ 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${\mathbf {X}}$ . 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.