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Harmonic analysis in value at risk calculations
| Content Provider | Semantic Scholar |
|---|---|
| Author | Albanese, Claudio Seco, Luis |
| Copyright Year | 1996 |
| Abstract | Value at Risk is a measure of risk exposure of a portfolio and is de ned as the maximum possible loss in a certain time frame typically days and within a certain con dence typically 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 nd approximate analytic formulas for the Value at Risk as a function of time and of the con dence 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 re sum the expansions and to obtain formulas that smoothly interpolate between low and large con dence levels as well as between short and long time horizons Introduction The notion of Value at Risk VaR introduced in the J P Morgan RiskMetrics document JPM captures the risk exposure of a portfolio in terms of the largest possible loss within a certain con dence interval In the RiskMetrics framework one deals with portfolios subject to a number of risk factors whose evolution is a geometric Brownian motion with a given covariance matrix The full valuation method C Albanese and L Seco consists of repricing the portfolio under a number of scenarios by calling all the relevant pricing functions This procedure is computationally very intensive In typical applications with portfolios that consist of several hundred thousand instruments not more than a few thousand scenarios can be priced overnight with current technologies The small number of scenarios results in large inaccuracies in the Value at Risk measurement The use of rather unsophisticated pricing models can speed up the calculation but is also at the origin of uncontrollable errors An alternative that has been advocated in the RiskMetrics tech nical document is to use the quadratic approximation for the portfolio variation as a function of the underlying risk factors To obtain this representation the knowledge of the total delta vector and of the total gamma matrix is required This leads to the problem of evaluating an integral of the form I K Z x x x K exp x A x for certain vectors and matrix To our knowledge the problem of estimating was rst considered by Ruben Rub in the case of positive de nite and zero and then extended by a number of other authors see KJB and references therein An asymptotic expansion in the large con dence limit has been obtained by Quintanilla see Q The notion of Value at Risk owes its popularity to the fact that it captures with just one parameter of intuitive meaning the risk expo sure of a portfolio However the Value at Risk evolves with time and is subject to stochastic uctuations which re ect the evolution of the risk factors and the evolution of the composition of the portfolio itself The sensitivity of the Value at Risk with respect to the dynamics of the underlying risk factors depends on the relative importance of delta and gamma risks To capture this e ect it is useful to use dual variables which give the sensitivity to the total delta and the total gamma risk of a given portfolio In our setting duality transformations involve Fourier transforms After an initial simultaneous diagonalization of the covariance matrix A and of the total gamma of the portfolio we reduce the calculation to an integral of a Gaussian over a high dimensional quadric This integral is computed using techniques from harmonic analysis which reduce all calculations to the Fourier transform of quadrics In the case of positive de nite Gammas this reduces to explicit Bessel functions the general case is not much more di cult The Paley Wiener theorem guarantees Harmonic analysis in value at risk calculations that our formulas have adequate computational properties This re sult can be strengthened by deriving analytic formulas which give the asymptotic behaviour and determine the relevant Fourier transforms up to smooth multipliers The moments of these transforms can be com puted by means of a technique related to Feynmann diagrams This gives rise to matrix invariants such as traces and determinants which yield analytic formulas for the Value at Risk as a function of the time horizon and of the con dence level In this context the use of Fourier transforms is crucial to resum the expansions and to obtain formulas that smoothly interpolate between low and large con dence levels as well as between short and long time horizons By changing coordinates the integral in can be reduced to a convolution of integrals of the same form but with a positive In this case Fourier transforms give rise to the two following representations rst I K R R n Z Jn R pjb j G b jb j db |
| File Format | PDF HTM / HTML |
| Alternate Webpage(s) | http://www-risklab.erin.utoronto.ca/seco/var.ps.gz |
| Alternate Webpage(s) | http://www.ems-ph.org/journals/show_pdf.php?iss=2&issn=0213-2230&rank=1&vol=17 |
| Language | English |
| Access Restriction | Open |
| Subject Keyword | Approximation algorithm Arabic numeral 0 Bessel filter Brownian motion Convolution De Morgan's laws Density matrix Diagram Dual Duality (optimization) EXPTIME Ectomesenchymal Chondromyxoid Tumor Estimated Instrument - device Interpolation Juvenile Polymyositis Linear algebra Microencapsulated Potassium Chloride 20 MEQ Extended Release Oral Tablet [Klor-Con] Morgan Naruto Shippuden: Clash of Ninja Revolution 3 Normal Statistical Distribution Population Parameter REV RiskMetrics Smoothing Tracing (software) Value (ethics) Value at risk cell transformation |
| Content Type | Text |
| Resource Type | Article |
