Advances in Risk Management

166 THE MODELING OF WEATHER DERIVATIVE PORTFOLIO RISK

This particular issue highlights that there is no single best solution for
modeling weather derivative portfolios. In many cases, simulating at the
contract index level may be best. In other cases, in which constraints between
monthly contracts are very important, simulating at the monthly level may
be better. Finally, for some contracts, and for some trading situations,
simulation at a daily level may be optimal.

8.10 CONSISTENCY BETWEEN THE VALUATION OF

SINGLE CONTRACTS AND PORTFOLIOS

Single weather contracts can be priced using closed-form expressions or
simulations. Whichever method is used, however, the estimate of the
expected payoff for a single contract will not be the same as the estimate
of the expected payoff of the same contract when included as part of a
portfolio which is being valued using the BMVN method. If the single
contract is valued using closed-form expressions then this difference arises
because the BMVN method uses simulations, which necessarily introduces
a small random error. If the single contract is valued using simulations then
this difference arises because different simulation engines must be used
for the two sets of simulations, since one is univariate and the other is
multivariate.
As long as many years of simulations are being used the differences are
not likely to be material relative to sampling and model error, but they can
be rather inconvenient and confusing. It would therefore be useful to be able
to make these two sets of results numerically consistent. One way to do this
is as follows:

1 When pricing stand-alone contracts, use simulations from a univariate

random number generator.

2 When modelling the portfolio, start by running simulations for the

marginal distributions using the same univariate random number gen-

erator as used for pricing individual contracts, with the same seeds on a

contract by contract basis.

3 Then, induce the desired correlation matrix between the independent

univariate simulations of the marginals by reordering the simulated val-

ues. The reordering is based on the observed rank correlation matrix and

a set of correlated multivariate normal simulations.

The result is that the simulated marginal distribution for each contract is the
same in both the univariate and multivariate cases. The only disadvantages
of this method appear to be (a) that simulations have to be used for the