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Quantile estimation for the generalized pareto distribution with application to finance

Generalized Pareto distributions (GPD) are widely used for modeling excesses over high thresholds (within the framework of the POT-approach to modeling extremes). The aim of the paper is to give the review of the classical techniques for estimating GPD quantiles, and to apply these methods in finance - to estimate the Valueat-Risk (VaR) parameter, and discuss certain difficulties related to this subject. | Yugoslav Journal of Operations Research 22 (2012), Number 2, 297-311 DOI:10.2298/YJOR110308013J QUANTILE ESTIMATION FOR THE GENERALIZED PARETO DISTRIBUTION WITH APPLICATION TO FINANCE Jelena JOCKOVIĆ Faculty of Pharmacy, University of Belgrade, Serbia haustor@pharmacy.bg.ac.rs Received: March 2011 / Accepted: May 2012 Abstract: Generalized Pareto distributions (GPD) are widely used for modeling excesses over high thresholds (within the framework of the POT-approach to modeling extremes). The aim of the paper is to give the review of the classical techniques for estimating GPD quantiles, and to apply these methods in finance - to estimate the Valueat-Risk (VaR) parameter, and discuss certain difficulties related to this subject. Keywords: Generalized Pareto distributions, excesses over high thresholds, quantiles of the distribution, value at risk. MSC: 62P20. 1. INTRODUCTION The two-parameter generalized Pareto distribution with the shape parameter γ and the scale parameter σ (denoted GPD (γ, σ)) is the distribution of the random variable X = σ 1 − e−γ Y γ where Y is a random variable with the standard exponential ( ) distribution. GPD (γ, σ) has the distribution function 1 ⎧ γ γ x ⎛ ⎞ ⎪1 − 1 − , γ ≠ 0, σ > 0, ⎪ ⎜ σ ⎟⎠ Fγ ,σ ( x) = ⎨ ⎝ ⎪ ⎛ x⎞ ⎪1 − exp ⎜ − σ ⎟ , γ = 0, σ > 0, ⎝ ⎠ ⎩ (1.1) J. Jocković / Quantile Estimation for the Generalized Pareto 298 where 0 ≤ x 0. γ A number of important and commonly used probability distributions belong to GPD family: 1. For γ = 0, GPD reduces to the exponential distribution with mean σ. 2. For γ = 1, GPD reduces to the uniform U [0, σ] distribution. 3. For γ u is GPD (γ, σ-γu). This property has a key role in the POT-approach to modeling extremes. POT-approach consists of fitting the GPD to the distribution of the excesses over a sufficiently high threshold, i.e. to the conditional distribution of X - u given X > u, when u tends to the right endpoint of the support of the distribution. This type of approximation is .