# Harmonic analysis in value at risk calculations.

• Volume: 17, Issue: 2, page 195-219
• ISSN: 0213-2230

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## Abstract

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Value at Risk is a measure of risk exposure of a portfolio and is defined as the maximum possible loss in a certain time frame, typically 1-20 days, and within a certain confidence, typically 95%. Full valuation of a portfolio under a large number of scenarios is a lengthy process. To speed it up, one can make use of the total delta vector and the total gamma matrix of a portfolio and compute a Gaussian integral over a region bounded by a quadric. We use methods from harmonic analysis to find approximate analytic formulas for the Value at Risk as a function of time and of the confidence level. In this framework, the calculation is reduced to the problem of evaluating linear algebra invariants such as traces of products of matrices, which arise from a Feynmann expansion. The use of Fourier transforms is crucial to resum the expansions and to obtain formulas that smoothly interpolate between low and large confidence levels, as well as between short and long time horizons.

## How to cite

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Albanese, Claudio, and Seco, Luis. "Harmonic analysis in value at risk calculations.." Revista Matemática Iberoamericana 17.2 (2001): 195-219. <http://eudml.org/doc/39677>.

@article{Albanese2001,
abstract = {Value at Risk is a measure of risk exposure of a portfolio and is defined as the maximum possible loss in a certain time frame, typically 1-20 days, and within a certain confidence, typically 95%. Full valuation of a portfolio under a large number of scenarios is a lengthy process. To speed it up, one can make use of the total delta vector and the total gamma matrix of a portfolio and compute a Gaussian integral over a region bounded by a quadric. We use methods from harmonic analysis to find approximate analytic formulas for the Value at Risk as a function of time and of the confidence level. In this framework, the calculation is reduced to the problem of evaluating linear algebra invariants such as traces of products of matrices, which arise from a Feynmann expansion. The use of Fourier transforms is crucial to resum the expansions and to obtain formulas that smoothly interpolate between low and large confidence levels, as well as between short and long time horizons.},
author = {Albanese, Claudio, Seco, Luis},
journal = {Revista Matemática Iberoamericana},
keywords = {Distribución de Gauss; Formas cuadráticas; Transformada de Fourier; Cuádricas; Desarrollo en serie de funciones; Análisis armónico; Matemática financiera},
language = {eng},
number = {2},
pages = {195-219},
title = {Harmonic analysis in value at risk calculations.},
url = {http://eudml.org/doc/39677},
volume = {17},
year = {2001},
}

TY - JOUR
AU - Albanese, Claudio
AU - Seco, Luis
TI - Harmonic analysis in value at risk calculations.
JO - Revista Matemática Iberoamericana
PY - 2001
VL - 17
IS - 2
SP - 195
EP - 219
AB - Value at Risk is a measure of risk exposure of a portfolio and is defined as the maximum possible loss in a certain time frame, typically 1-20 days, and within a certain confidence, typically 95%. Full valuation of a portfolio under a large number of scenarios is a lengthy process. To speed it up, one can make use of the total delta vector and the total gamma matrix of a portfolio and compute a Gaussian integral over a region bounded by a quadric. We use methods from harmonic analysis to find approximate analytic formulas for the Value at Risk as a function of time and of the confidence level. In this framework, the calculation is reduced to the problem of evaluating linear algebra invariants such as traces of products of matrices, which arise from a Feynmann expansion. The use of Fourier transforms is crucial to resum the expansions and to obtain formulas that smoothly interpolate between low and large confidence levels, as well as between short and long time horizons.
LA - eng
KW - Distribución de Gauss; Formas cuadráticas; Transformada de Fourier; Cuádricas; Desarrollo en serie de funciones; Análisis armónico; Matemática financiera
UR - http://eudml.org/doc/39677
ER -

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