召唤师技能加点>张冠李戴造句
A comparison of analytical VaR methodologies
for portfolios that include options
Stefan Pichler
Karl Selitsch
December 1999
Department of Finance
Vienna University of Technology
Favoritenstras 11, A-1040 Wien
email: spichler@pop.tuwien.ac.at
Abstract
It is the main objective of this paper to compare different approaches to analytically calculate value-at-ri
sk (VaR) for portfolios that include options. We focus on approaches that are bad on a cond order Taylor-ries approximation of the nonlinear option pricing relationship. The main difficulty common to all the methods is the estimation of the required quantile of the profit and loss distribution, since there exists no analytical reprentation of this distribution. In our analysis we examine different moment matching approaches and methods to directly approximate the required quantile. For this purpo, we perform a backtesting procedure bad on randomly generated risk factor returns which are multivariate normal. The VaR-numbers calculated by a specific methodology are then compared to the simulated actual loss. We conclude that the accuracy of methodologies that rely only on the first four moments of the profit and loss distribution is rather poor. The inclusion of higher moments, e.g. through a Cornish-Fisher expansion ems to be appropriate. In addition, we find that an approximation of the profit and loss distribution by a normal distribution might be appropriate in some cas even for correlated risk factors.
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1Motivation
During the last decade value-at-risk (VaR) has become one of the most important risk measurement
tools in financial institutions as well as in other corporations facing considerable market risk. One of the major features that make VaR attractive to risk managers across different institutions is its analytical tractability. Although numerical methods to calculate VaR have been developed leading to more accurate results depending on less restrictive assumptions, many institutions em to still rely on analytical methodologies. The most important advantage of analytical methods over their numerical counterparts - the saving of computing time that makes real-time calculations possible - ems to outweigh their disadvantages for many practical applications. The motivation of this paper is thus bad on the need to improve the accuracy of analytical VaR methodologies and simultaneously make the underlying assumptions less restrictive.
The VaR of a portfolio is defined as the maximum loss that will occur over a given period of time at a given probability level. The calculation of VaR numbers requires some assumptions about the distributional properties of the returns of the portfolio components. The common delta-normal approach originally promoted by JP Morgan´s RiskMetrics software is bad on the assumptions of normally distributed returns of prespecified risk factors. In the ca of a strictly linear relationship between the returns of the risk factors and the market value of the portfolio under consideration there exists a simple analytic solution for the VaR of the portfolio. This simple analytic solution does
not hold for portfolios that include financial instruments with non-linear payoffs like options. Since the relationship between the normally distributed returns of the risk factors (underlyings, interest rates, etc) and the value of the options is nonlinear, the distribution of the portfolio value is no longer normal. It can be shown that for portfolios with a high degree of nonlinearity this distribution shows extremely high skewness and excess kurtosis. This makes a reasonable VaR-calculation using the delta-normal approach impossible.
A first step to solve this problem is to include the quadratic term of a Taylor-ries expansion of the option pricing relation, i.e. the gamma matrix, in the VaR calculation framework. The inclusion of quadratic terms implies a distribution of portfolio values that may be described as
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a linear combination of non-central χ2-distributed random variables. Fortunately, this distribution was shown to be equivalent to the distribution of a random form in normally distributed random variables for which at least the moment-generating function exists (e Mathai and Provost (1992)). There are veral attempts prented in the literature to incorporate higher moments of this distribution in approximation procedures to calculate the required quantile of the distribution. This paper will focus
寒露的诗句on this class of methods that are widely ud by practicioners to incorporate nonlinearity in the VaR calculation1.
In a first attempt Zangari (1996a) suggested to u the Cornish-Fisher approximation to directly calculate the quantile of a distribution with known skewness and kurtosis. Other approaches try to find a moment matching distribution for which the quantiles can be calculated. This class of approaches contains Zangari (1996b) who suggested to u the Johnson family of distributions to match the first four moments, Britten-Jones and Schaefer (1997) who suggested to u a central χ2-distribution to match the first three moments, and a simplifying approach that us the normal distribution to match the first two moments (for a discussion of this approach, e El-Jahel, Perraudin, and Sellin (1999)).
孕妇能喝饮料吗The latter approach might be justified by applying the central limit theorem for portfolios with a very large number of risk factors. However, bad on a simplified tting Finger (1997) argues that this application will only hold for uncorrelated risk factors. We provide additional analytic results for more general cas where the distribution of the portfolio value does not ‚converge‘to a normal distribution even for uncorrelated risk factors , whereas we can show that there are cas where the distribution of the portfolio value ‚converges‘ to a normal distribution even for correlated risk factors. It depends o
n the structure of the gamma-matrix rather than the structure of the covariance matrix whether a ‚convergence‘ is achieved or not. Since it is hard to generalize the analytic results, this approach is included in our numerical analysis.
It is the main objective of this paper to compare the approaches mentioned above to calculate 1There are different approaches that rely on the numerical inversion of the characteristic function rather than on the moment generating function (e.g., Imhof (1961), Rouvinez (1997), Glasrman, Heidelberger, and Shahabuddin (1999)). However, the approaches that are showing a high degree of accuracy are not covered by our analysis.
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VaR for portfolios that include options. We perform a backtesting procedure bad on randomly generated risk factor returns which are multivariate normal. The returns are ud to calculate a simulated time-ries of profits and loss given the portfolio composition determined by an N-dimensional vector of deltas and an N×N dimensional matrix of gammas. The VaR-number calculated by a specific methodology is then compared to simulated actual loss. The perfomance of the different methodologies is measured by the amount of deviation of the percentage of cas wh
ere the simulated actual loss exceeds the VaR from the required probability. Additionally, we provide likelihood ratio statistics to test for significance of our results.
In a recent paper, El-Jahel, Perraudin, and Sellin (1999) prented a methodology to calculate the moments of the portfolio‘s profit and loss distribution even for nonnormal risk factors. Under fairly general conditions, the knowledge of the moments of the distribution of the risk factors is sufficient to calculate the moments of the distribution of the portfolio. This leads to the same situation where the quantiles of this distribution have to be calculated. We have to stress the fact that the results of our paper are not limited to the ca of normally distributed risk factors but are also relevant for nonnormal risk factors as long as the calculation scheme is bad on a quadratic approximation of the nonlinear pricing relationship.
The outline of this paper is as follows: Section 2 describes problems arising for VaR methodologies when options are included and shows the distributional properties of a quadratic Taylor-ries approximation of the portfolio‘s profit and loss distribution. Section 3 gives an overview over differnet methodologies that were developed to calculate quantiles of this distribution. Section 4 describes the Monte Carlo backtesting procedure ud to evaluate the different approaches and summarizes the results of this evaluation procedure. Section 5 concludes the paper.
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5VaR σ∆W ]Φ(α),
∆W ˆN k 1δk ]∆S k S k ,δk j W k S k ,(1)
2Analytic VaR for portfolios that include options
The delta-normal approach originally promoted by JP Morgan´s RiskMetrics software is bad on two major assumptions.
Assumption 1 (linearity):The change in the value of the portfolio over a given interval of
time is linear in the returns of N < Q risk factors.股东合作协议书
Let W denote the market value of the portfolio under consideration at a fixed point of time and S k , k = 1, ...., N, the contemporaneous value of the k -th risk factor, then assumption 1 can be formalized as
曾经年少的我们where δk denotes the factor nsitivity of the portfolio with respect to factor k . Using matrix notation
庆三八we have where δ denotes the N×1-vector of factor nsitivities and R ∆W δT R ,denotes the N×1-vector of factor returns (∆S k / S k ).
Assumption 2 (normality):The returns of the risk factors follow a multivariate normal十二星座的
distribution.
We have where Σ denotes N×N covariance matrix of factor returns and µ denotes R ~N (µ,Σ),the N×1-vector of expected factor returns. Note, that in many applications the additional assumption µ = 0 is made. In order to simplify the notation and the interpretation of our results the analysis of this paper follows this assumption.
The assumptions imply that the distribution of ∆W itlf is normal and the VaR of the portfolio given µ = 0 can be written as
