The market value of a portfolio consisting of
different options is dependent on changes of the risk free
interest rate
, the prices
of the financial underlying,
the time to maturity
and the individual implied
volatilities
. Changes in the portfolios value can be
analytically approximated using the following Taylor
approximation, where it is assumed that the options are all based
on the same underlying.
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A well known strategy in utilizing the forecasted changes in the
maturity structure of implied volatilities consists of buying and
selling so called ``Straddles'' with varying maturities. A
straddle is constructed by simultaneously buying (``Long
Straddle'') or selling (``Short
Straddle'') the same number of ATM Call and
Put options with the same time to maturity. If a trader expects a
relatively strong increase in the implied volatility in the short
maturities and a relatively weaker increase in the longer
maturities, then he will buy straddles with a short time to
maturity and sell longer maturity straddles at a suitable ratio.
The resulting option portfolio is (
) neutral and
over a short time frame
neutral, i.e., it is insensitive
with respect to losing value over time. The Taylor series given
above can thus be reduced to:
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(20.6) |
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(20.7) |
The following ``Maximum Loss'' (ML) concept
describes the probability distribution of a short-term change in
the portfolios value dependent on changes in the value of the
underlying factors. The change in value of a
neutral option portfolio is substantially determined by the
changes in the implied volatilities of the options contained in
the portfolio. To determine the ``Maximum Loss'' it is necessary
to have an adequate exact representation of the future
distribution of the changes to the volatility of the
options with varying time to maturity.
The ``Maximum Loss'' is defined as the largest possible loss of a
portfolio that can occur over a specific factor space and
over a specific holding period
. The factor space
is
determined by a closed set with
. Here
is set to 99% or 99.9%. The ML definition resembles
at first sight the ``Value-at-Risk'' Definition (see Chapter
15). There exists, however, an
important difference between the two concepts: In calculating the
``Value-at-Risk'' the distribution of the returns of the given
portfolio must be known, whereas the ML is defined directly over
the factor space and thus has an additional degree
of freedom, see Studer (1995).
In our analysis we have divided the maturity structure of the
implied volatilities into two principal components, which explain
a considerable portion of the variability of the structure curve.
Thus the first two principal components represent the risk factors
used in the ML model. The profit and loss
profile of each portfolio held is determined by the corresponding
changes in the risk factors using a suitable valuation model. In
order to obtain this, a valuation of the underlying portfolios
must theoretically occur for every point in the factor space. In
the practical application the factor space is probed over a
sufficiently small grid of discrete data points
, during which the other risk factor is in each case
held constant. Due to the orthogonality properties of the
principal components, the profit and loss function
is
additive with
.
Under the assumption of multivariate, normally distributed principal components confidence intervals can be constructed for the ``Maximum Loss'' over the total density
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(20.8) |
with
. Here the matrix
represents
the
diagonal matrix of the eigenvalues
,
= 1,2. The random variable
has a Chi-square distribution. The confidence interval for
an existing portfolio is then
,
, where
is the
quantile of a random variable with a Chi-square
distribution and 2 degrees of freedom.