In:
Statistics & Risk Modeling, Walter de Gruyter GmbH, Vol. 32, No. 3-4 ( 2015-12-1), p. 177-195
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 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 ${\mathbf {U}}$ 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.
Type of Medium:
Online Resource
ISSN:
2193-1402
,
2196-7040
DOI:
10.1515/strm-2015-0009
Language:
English
Publisher:
Walter de Gruyter GmbH
Publication Date:
2015
detail.hit.zdb_id:
2630783-2
detail.hit.zdb_id:
2630803-4
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