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 <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>𝐗</m:mi> <m:mo>:</m:mo> <m:mo>=</m:mo> <m:msub> <m:mrow> <m:mo>{</m:mo> <m:msub> <m:mi>X</m:mi> <m:msub> <m:mi>t</m:mi> <m:mi>k</m:mi> </m:msub> </m:msub> <m:mo>}</m:mo> </m:mrow> <m:mrow> <m:mi>k</m:mi> <m:mo>=</m:mo> <m:mn>1</m:mn> <m:mo>,</m:mo> <m:mo>...</m:mo> <m:mo>,</m:mo> <m:mi>d</m:mi> </m:mrow> </m:msub> </m:mrow> </m:math> ${\mathbf {X}:=\lbrace X_{t_k}\rbrace _{k=1,\ldots ,d}}$ of consecutive log-returns to ultimately analyze the probability law of the accumulated return <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:msub> <m:mi>X</m:mi> <m:msub> <m:mi>t</m:mi> <m:mn>1</m:mn> </m:msub> </m:msub> <m:mo>+</m:mo> <m:mo>⋯</m:mo> <m:mo>+</m:mo> <m:msub> <m:mi>X</m:mi> <m:msub> <m:mi>t</m:mi> <m:mi>d</m:mi> </m:msub> </m:msub> </m:mrow> </m:math> ${X_{t_1}+\cdots +X_{t_d}}$ . By the Markov regression representation (see [25]), any stochastic model for <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>𝐗</m:mi> </m:math> ${\mathbf {X}}$ can be represented as <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:msub> <m:mi>X</m:mi> <m:msub> <m:mi>t</m:mi> <m:mi>k</m:mi> </m:msub> </m:msub> <m:mo>=</m:mo> <m:msub> <m:mi>f</m:mi> <m:mi>k</m:mi> </m:msub> <m:mrow> <m:mo>(</m:mo> <m:msub> <m:mi>X</m:mi> <m:msub> <m:mi>t</m:mi> <m:mn>1</m:mn> </m:msub> </m:msub> <m:mo>,</m:mo> <m:mo>...</m:mo> <m:mo>,</m:mo> <m:msub> <m:mi>X</m:mi> <m:msub> <m:mi>t</m:mi> <m:mrow> <m:mi>k</m:mi> <m:mo>-</m:mo> <m:mn>1</m:mn> </m:mrow> </m:msub> </m:msub> <m:mo>,</m:mo> <m:msub> <m:mi>U</m:mi> <m:mi>k</m:mi> </m:msub> <m:mo>)</m:mo> </m:mrow> </m:mrow> </m:math> ${X_{t_k}=f_k(X_{t_1},\ldots ,X_{t_{k-1}},U_k)}$ , <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>k</m:mi> <m:mo>=</m:mo> <m:mn>1</m:mn> <m:mo>,</m:mo> <m:mo>...</m:mo> <m:mo>,</m:mo> <m:mi>d</m:mi> </m:mrow> </m:math> ${k=1,\ldots ,d}$ , yielding a decomposition into a vector <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>𝐔</m:mi> <m:mo>:</m:mo> <m:mo>=</m:mo> <m:msub> <m:mrow> <m:mo>{</m:mo> <m:msub> <m:mi>U</m:mi> <m:mi>k</m:mi> </m:msub> <m:mo>}</m:mo> </m:mrow> <m:mrow> <m:mi>k</m:mi> <m:mo>=</m:mo> <m:mn>1</m:mn> <m:mo>,</m:mo> <m:mo>...</m:mo> <m:mo>,</m:mo> <m:mi>d</m:mi> </m:mrow> </m:msub> </m:mrow> </m:math> ${\mathbf {U}:=\lbrace U_{k}\rbrace _{k=1,\ldots ,d}}$ of i.i.d. random variables accounting for the randomness in the model, and a function <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>f</m:mi> <m:mo>:</m:mo>