124 Valuation and Risk Management of Weather Derivatives: The Application of CME Rainfall Index Binary Contracts 1 0 = 1 0 0 1 10 = 1 0 01, (2.10) which means p10 should be a multiple of p01, and seems to be not very reasonable. This strong condition will limit the efficiency of capturing seasonal features, since four transition probabilities will all share one truncated Fourier series. Therefore, we consider not keeping the steady states in this paper. 2.2 Magnitude Model The magnitude model describes the amount of rain on the condition that it rains on day t, which is estimated by fitting an appropriate distribution to the historical data. According to Table 2 in subsection 4.1.2, the descriptive statistics of daily precipitation magnitude show that the distribution should be asymmetric, positively skewed and contain only positive values. In previous literature, many distributions with a nonnegative domain have been proposed, such as the gamma distribution, the exponential distribution, and the mixed exponential distribution. According to Wilks (2011), the gamma distribution is often used to describe the distribution of various atmospheric variables such as precipitation magnitudes and wind speeds. However, the estimators of the gamma distribution can yield poor results from small values of the shape parameter. A special case of the gamma distribution is the exponential distribution, in which α = 1. In this paper, we prefer applying the mixed exponential distribution, which is also widely applied in hydrology and in weather derivatives pricing, and can better represent extreme events. This distribution is a weighted combination of two exponential distributions, where α is the weight of the first exponential distribution with parameter β1, and 1-α is the weight of the other exponential distribution with parameter β2. ( ) = 1 1 + 1 2 2 , 0 1, 0< 1 < 2 ( ) ( ) . And since the magnitude of precipitation varies seasonally as described in aforementioned characteristics (3) and (4), the parameters of mixed-exponential distribution α, β1, and β2 are also approximated by the truncated Fourier series like (2.6).
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