Sehr geehrte Damen und Herren,
Im Rahmen des Seminars "Finanzmathematik" spricht Herr Prof. S. Pichler (TU Wien) ueber
"A comparison of analytical VaR methodologies for portfolios that include options".
Zeit: Mittwoch, 21.04.1999, 19 Uhr
Ort: Hoersaal 1 Institut fuer Mathematik Strudlhofgasse 4 1010 Wien
Der Vortrag findet auf Deutsch statt; eine englische Kurzfassung finden Sie im Anhang.
Im Anschluss an seinen Vortrag wird Prof. Pichler das
CCEFM Doctoral Program in Finance (http://info.tuwien.ac.at/ccefm)
vorstellen, eine gemeinsame Initiative der Universitaet Wien, WU Wien, TU Wien und der Wiener Boerse AG.
Mit besten Gruessen,
Markus Fulmek
P.S.: In unserer gestrigen Einladung zum Vortrag am Institut fuer Statistik wurde aus Frau Prof. Silvia Vogel (TU Ilmenau) irrtuemlich ein "Herr Prof. S. Vogel": Wir bitten, dieses Versehen zu entschuldigen.
Anhang: Kurzfassung zum Vortrag von Prof. Pichler
Stefan Pichler Karl Selitsch
April 1999
Department of Finance Vienna University of Technology Floragasse 7/4, 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 VaR for portfolios that include options. For this purpose, we perform a backtesting procedure based on randomly generated risk factor returns which are multivariate normal. The VaR- numbers calculated by a specific methodology is then compared to the simulated actual losses.
The Value-at-Risk (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 Morgans RiskMetrics software is based on the assumptions of normally distributed returns of prespecified risk factors. In the case 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. In contrast to alternative numerical methods this approach is based on rather strong assumptions. However, the analytic tractability of the delta-normal approach seems to be a desirable property at least for practical applications. For that reason a large number of methodologies to improve and/or extend the delta-normal approach have been developed.
A very important extension of the delta-normal approach is the attempt to include financial instruments with non-linear payoffs like options in the VaR calculation. Since the relationship between the normally distributed returns of the risk factors (underlyings, interest rates, etc) and the value of the options is non-linear, 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-series expansion of the option pricing relations, 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 a linear combination of non-central 2-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 (see Mathai and Provost (1992)). There are several attempts presented in the literature to incorporate higher moments or cumulants of this distribution in approximation procedures to calculate the required quantile of the distribution.
In a first attempt Zangari (1996a) suggested to use 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 use the Johnson family of distributions to match the first four moments, Britten-Jones and Schaefer (1997) who suggested to use a central 2-distribution to match the first three moments, and a simplifying approach that uses the normal distribution to match the first two moments (see El-Jahel, Perraudin, and Sellin (1999)).
The latter approach might be justified by applying the central limit theorem for portfolios with a gamma matrix of very large dimension. However, based on a simplified setting Finger (1997) argues that this application will only hold for uncorrelated risk factors. We provide additional analytic results for more general cases where the distribution of the portfolio value does not converge to a normal distribution even for weakly correlated risk factors. Since it is hard to generalize this analytic results this approach is included in our numerical analysis.
It is the main objective of this paper to compare the approaches cited above to calculate VaR for portfolios that include options. We perform a backtesting procedure based on randomly generated risk factor returns which are multivariate normal. These returns are used to calculate a simulated time-series of profits and losses given the portfolio composition determined by an N-dimensional vector of deltas and an NN dimensional matrix of gammas. The VaR-number calculated by a specific methodology is then compared to the simulated actual losses. The perfomance of the different methodologies is measured by the amount of deviation of the percentage of periods where the simulated actual loss exceeds the VaR from the required probability. Additionally, we provide likelihood ratio statistics to test for significance of our results.
------------------------------------------------- Wissenschaftlicher Verein Modernes Risk Management
WWW: http://keen.esi.ac.at/~amrm/
Institut fuer Mathematik Universitaet Wien Strudlhofgasse 4 A-1090 Wien
Kontakt: Dr.Markus Fulmek amrm@keen.esi.ac.at =========================================================================