138 Valuation and Risk Management of Weather Derivatives: The Application of CME Rainfall Index Binary Contracts The parentheses contain the absolute difference between each approach and the real rainfall index in 2005. The results of BA are around the expected rainfall index of θ = 0, ranging from θ = 0.1 to θ = -0.1. Compared with the BA, the benchmark approach, the model we use has a better performance in the first half of the year. 4.4.3 Sensitivity Analysis Since the quoted theoretical price can be expressed in a linear form with expected rainfall index Eℚθ[I(τ 1, τ2) | ℱt] and strike price K as shown in (3.11) and (3.12), we then discuss the relationship between the MPR (θ) and ERI from two aspects. (1) The Influence of the NIG Distribution Comparing the expectation before and after the Esscher transform, the bigger θ means a larger β, which positively measures the asymmetry degree of the NIG distribution. It also influences the expectation of the distribution, since a bigger θ means that the investors have a larger estimation of the monthly rainfall index. (2) The Influence of the Utility Function In the Esscher transform, we assume that all market participants are represented by a representative agent with exponential utility function u(x) = (1- e-θx) ⁄ θ, in which the parameter θ is the MPR. The Arrow-Pratt measure of Absolute Risk Aversion (ARA) represents the risk attitude of market participants as (4.3), ( ) = ( ( ) ) = = , (4.3) which means it is a Constant Absolute Risk Aversion (CARA) under the exponential utility function. When θ > 0, market participants are more risk averse as opposed to risk loving when θ < 0. By taking the data of March as an example, Figure 5 shows the monthly ERI is positively related to the MPR θ, which means market participants are more risk averse with greater ERI.
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