STEPHEN JEWSON 1658.9 INCORPORATING HEDGING CONSTRAINTS
Obvious constraints between the payoffs of contracts within a weather
derivative portfolio are not necessarily preserved by the BMVN method,
or its extension to non-normal distributions. This is an aspect of the fact that
it is never possible to exactly capture the correct multivariate distribution of
the weather indices, even if using copulas.
As an example, consider the following three contract portfolio: contract 1
is based on the number of freezing days in Chicago in November; contract
2 is based on the number of freezing days in Chicago in December; and
contract 3 is based on the number of freezing days in Chicago in November
and December. Clearly, in any particular year the index for contract 3 is the
sum of the indices from contracts 1 and 2. However, the BMVN method
(even with extensions to copulas) is not guaranteed to preserve this con-
straint i.e. may produce simulated years in which this constraint does not
hold. This is unlikely to be a problem in most cases, since the errors will
probably be small, but if contracts are being traded in such a way that one
is relying on this constraint to get exact cancellation of risk (for instance,
with long swaps as contracts 1 and 2 and a short swap as contract 3) and the
simulations miss the exact cancellation and give a finite instead of zero risk,
then this may be a problem.
The most obvious way to ensure that hedging constraints are captured
is to use some level of atomic simulation.^4 That is, simulating at a more
detailed level than the contract index level. For instance, one could con-
sider all seasonal indices as sums of monthly indices, simulate the monthly
indices, and create simulated seasonal indices by summing the simulated
monthly indices. This would have the advantage that it would correctly
capture constraints related to sums of monthly indices. However, it would
introduce various disadvantages too: first, that monthly indices are harder
to model because they are less likely to be normally distributed, second
that modelling monthly rather than seasonal indices may involve estimat-
ing many more parameters, and thus introduce extra sources of parameter
error, and third that simulating monthly indices may increase the dimen-
sionality of the problem (increasing the size of the correlation matrix), which
will make the whole modelling process slower.
An even more atomic approach would be to simulate daily temperatures,
and there have been a number of articles written on this topic, such as
those of Dischel (1998), Cao and Wei (2000), Dornier and Queruel (2000),
Moreno (2000), Torro, Meneu and Valor (2001), Alaton and Djehiche and
Stillberger (2001), Moreno and Roustant (2002), Caballero, Jewson and Brix
(2002), Brody, Syroka and Zervos (2002) and Jewson and Caballero (2003).
This is, however, a very difficult statistical problem to solve in general and
the simulations can be very slow.