136 Valuation and Risk Management of Weather Derivatives: The Application of CME Rainfall Index Binary Contracts 4.3 Simulation and Estimation of Index Distribution Since the rainfall of December 31, 2004 and December 31, 2005 is both 0 inches, the path of the rainfall index is simulated for 10,000 times with the same initial value (0 inches), which implies that the distribution of these two years are the same. Simulated monthly rainfall indexes from March to October are calculated and fitted with normal, exponential, and NIG distribution by MLE. The histograms and fitted PDF are presented in Figure C.1 of Appendix C, which shows that the NIG distribution fits best. We draw the QQ plot with the NIG distribution for each month in Figure C.2 of Appendix C. We also use the one-sample Kolmogorov-Smirnov test (K-S test) to check the equality between the simulated sample and the fitted NIG distribution. The result is shown in Table 5. Table 5 Result of One-Sample Kolmogorov-Smirnov Test Month Mar. Apr. May Jun. Jul. Aug. Sept. Oct. D 0.0122 0.0118 0.0129 0.0143 0.0128 0.0166 0.0190 0.0184 P-value 0.3328 0.3748 0.2713 0.1731 0.2776 0.0738 0.0263 0.0345 According to the definition, there is no significant difference between the sample and the fitted distribution when the p-value of the K-S test is greater than 0.05. In addition, it has a better performance from March to August, but does not fit so well in September and October. The fitted parameters of the NIG distribution are shown in Table 6. Table 6 Fitted Parameters of NIG Distribution (Mar. to Oct.) Param Mar. Apr. May Jun. Jul. Aug. Sept. Oct. α 79.5702 49.3685 37.6038 33.8541 16.5078 43.8524 41.1671 42.1280 β 78.2652 48.4672 37.0062 33.2425 15.6259 43.0413 40.6162 41.5929 μ -1.2892 -1.7029 -2.0678 -2.2737 -2.1244 -2.2339 -1.8373 -1.4369 δ 0.6200 0.9637 1.1913 1.3258 2.0743 1.1489 0.8826 0.7375 4.4 Theoretical Prices and Risk Management 4.4.1 Market Price of Risk Since we fit the simulated monthly rainfall index with the NIG distribution, by taking the rainfall index binary call option as an example, the market price of risk (MPR) θ under the Esscher transform can be derived as (4.2), which is proved in Appendix C.
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