STEPHEN JEWSON 167pricing of stand-alone contracts (instead of closed-form expressions, which
are faster and more accurate), and (b) that it may be slightly slower than the
BMVN method since it involves two stages of simulation.
8.11 ESTIMATING SAMPLING ERROR
As discussed in section 8.5, even if the assumptions used to model a set
of weather indices (in terms of the shapes of trends and distributions) are
correct, then the actual values of the fitted parameters are always estimated,
and this induces errors into the final results that we call sampling error
(sampling error is to be contrasted with model error, as discussed in section
8.8 above, which is caused by the models being wrong).
It may be useful to estimate the role of sampling error when pricing a
weather derivative or valuing a weather derivative portfolio. This is because
sampling error is a major contributor to overall error, and knowledge of the
level of error in pricing can lead to better decisions about whether to trade,
and how to set risk loading levels.
Unlike model error, sampling error can be estimated rather straight-
forwardly using linear theory. The case for a single weather derivative is
described in Jewson (2003a). This can be generalised to the portfolio case,
at least for the multivariate normal, although the derivation is somewhat
involved, and, to the author’s knowledge, has not been published.
8.12 ESTIMATING VaR
As discussed in section 8.3, risk in weather derivatives portfolios is princi-
pally measured by considering the distribution of outcomes of contracts at
expiry. However, it is also of interest to estimate the likely fluctuations in
the value of a weather derivative portfolio over much shorter time horizons.
This is particularly interesting for commonly traded contracts for which liq-
uidation or hedging may be an option. There are two common cases: either
one wishes to derive the possible changes in the expected expiry value,
or one wishes to derive the possible changes in the market value. With
respect to the first of these two cases, a full calculation is extremely complex,
depending, as it does, on modelling changes in weather forecasts, changes
in weather, and the correlations between the two. But a linearized estimate,
likely to be accurate for short time periods, is much simpler. Calculating such
an estimate involves linearising the non-linearities in the payoff functions
and modelling the short term changes in expected weather indices using
Brownian motion. The resulting expressions, which can be derived for both
single contracts and portfolios, are very simple (see Jewson, 2003b). With
respect to the second of these cases, that of estimating possible changes in the