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(a)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(a)keen.esi.ac.at
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