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

Abstract

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 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 ${\mathbf {U}}$ 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.