120 Valuation and Risk Management of Weather Derivatives: The Application of CME Rainfall Index Binary Contracts and Diko (2005), the Ornstein-Uhlenbeck process driven by the Hougaard Lévy process in Noven, Veraart, and Gandy (2015), and the pure jump model embedded in an enlarged filtration framework in Hess (2016), the model we use in this paper is straightforward and comprehensive, and also captures the significant seasonal features in precipitation. Since the weather market is incomplete, we cannot simply derive the pricing formula like that in the Black-Scholes model (1973). Several methodologies have appeared in prior literature. As mentioned in Dorfleitner and Wimmer (2010), Odening, Musshoff, and Xu (2007) and Shah (2017), burn analysis is a simple but rough approach, which directly prices the weather derivatives with the historical average of the underlying indices. Cao et al. (2004) and Shah (2017) simulate rainfall paths by the Monte Carlo simulation and price the rainfall derivatives with the average of simulated payoffs. Brockett, Wang, Yang, and Zou (2006), Xu, Odening, and Musshoff (2008), and Carmona and Diko (2005) all propose the applying of the utility indifference approach, which is widely used in pricing derivatives with non-tradable underlying assets. However, we choose to follow the approach in Cabrera et al. (2013) and Noven et al. (2015), that is, pricing the precipitation derivatives with the Esscher transform, which was first introduced by Esscher (1932) and been widely used as a risk premium principle in actuarial science. Compared with other approaches, the Esscher transform has great advantages as shown by several past studies. Bühlmann (1980) proves that the Esscher transform could be derived as the Pareto-optimal solution in a market situation where all participants are characterized by an exponential utility function, and all risks are stochastically independent. Kremer (1982) discusses a characterization of the Esscher transform and proves it yields the closest distance and discrimination (usually called the KullbackLeibler divergence) between distributions Q and P when calculating net premium. Frittelli (2000) shows that the application of the Esscher transform in option pricing minimizes the relative entropy between the physical measure ℙ and an equivalent martingale measure ℚ, and also equivalently maximizes the exponential expected utility. There are also past studies consider other factors in pricing precipitation derivatives. Hess (2016) prices precipitation swaps and futures with customized approximation procedures and extends a multi-location model. Härdle and Osipenko (2017) develops a dynamic programming approach in pricing baskets of weather derivatives under default risk on the issuer side in the OTC market. Cramer, Kampouridis, Freitas, and Alexandridis (2017) compares seven machine learning methods in pricing rainfall derivatives. Building on the basis of past studies, this paper makes several contributions to the
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