Weather Forecasting for Weather Derivatives

differ noticeably across cities both in terms of amplitude and detail of pattern.

In Figure 2 we show how the seasonality in daily average temperature manifests itself in
unconditional temperature densities. The densities are either bimodal or nearly so, with peaks characterized
by cool and warm temperatures. Also, with the exception of Las Vegas, each density is negatively skewed.
The distributional results are in line with von Storch and Zwiers (1999), who note that although daily
average temperature often appears Gaussian if studied over sufficiently long times in the troposphere, daily
average surface temperatures may have different distributions, and with Neese (1994), who documents
skewness and bimodality in daily maximum temperatures.

The discussion thus far suggests that a seasonal component will be important in any time-series
model fit to daily average temperature, as average temperature displays pronounced seasonal variation,
with both the amplitude and precise seasonal patterns differing noticeably across cities. We use a low-
ordered Fourier series to model this seasonality, the benefits of which are two-fold. First, it produces a
smooth seasonal pattern, which accords with the basic intuition that the progression through different
seasons is gradual rather than discontinuous. Second, it promotes parsimony, which enhances numerical
stability in estimation. Such considerations are of relevance given the rather large size of our dataset
(roughly 15,000 daily observations for each of four cities) and the numerical optimization that we
subsequently perform.

One naturally suspects that non-seasonal factors may also be operative in the dynamics of daily
average temperature. One such factor is trend, which may be relevant but is likely minor, given the short
forty-year span of our data. We therefore simply allow for a simple low-ordered polynomial deterministic
trend. Another such factor is cycle, by which we mean any sort of persistent covariance stationary
dynamics apart from seasonality and trend. We capture cyclical dynamics using autoregressive lags.

The discussion thus far has focused on conditional mean dynamics, with contributions coming
from trend, seasonal and cyclical components. We also allow for conditional variance (volatility)
dynamics, with contributions coming from both seasonal and cyclical components. We approximate the

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