162 THE MODELING OF WEATHER DERIVATIVE PORTFOLIO RISKWe will refer to this basic method as the BMVN (Basic MultiVariate Normal)
method.
8.5 THE INCORPORATION OF SAMPLING ERROR IN
SIMULATIONS
The BMVN method described above is not strictly correct, from a statisti-
cal point of view, even in the case in which all the assumptions hold (for
example, the weather indices really are multivariate normal and we know
the correct model for the trends). This is because it ignores the sampling,
or estimation, uncertainty on the covariance matrix, and does not propa-
gate that uncertainty into the simulations. For instance, in this method the
variance of a particular weather index is estimated using the available data,
and the simulations are driven by that estimated variance. However, the
estimated variance is only an estimate, and the variance of the simulations
should include an extra term that takes this into account (this issue is identi-
cal to the question of how to derive prediction intervals in classical statistics).
Deriving the extra terms in the expressions for the covariances can be some-
what complicated, but may make a significant difference to the final results.
In the case in which the weather indices have not been detrended the extra
terms are rather simple, and are typically rather small. In the case in which
the weather indices have been detrended with a linear trend, the extra terms
are more complex, and become much more significant. In the case in which
other trend shapes are used, deriving the extra terms is complex, but very
important. Jewson and Penzer (2004) give expressions for these extra terms
for a number of cases. It should be noted that these extra terms arein addition
to the standard corrections to expressions for covariances that account for
changes in the number of degrees of freedom due to detrending.
8.6 ACCURATE ESTIMATION OF THE CORRELATION
MATRIX
The BMVN method, step 4, involves estimating the correlation matrix
among the historical weather indices using the empirical correlation matrix;
that is, calculating the observed correlations between historical weather
indices. Curiously, however, the empirical correlation matrix is not a par-
ticularly good estimator of the real correlation matrix, particularly for large
portfolios. This is because the elements of the correlation matrix are very
poorly estimated given the little data available. For example, for a portfolio
of 1,000 contracts, we must estimate roughly 500,000 correlations, but with
perhaps only 50,000 historical data values. This situation can be improved
using a technique known as shrinkage, in which the correlation matrix is
estimated using a combination of the empirical correlation matrix with a