129 NTU Management Review Vol. 31 No. 1 Apr. 2021 { } = ( ) =1 , >0 1 , (3.5) the uniform random variable u3(t) can be viewed as 1 - F(m) with a specific magnitude m. Therefore, we can reach the simulated magnitude Mt by (3.3). The simulated rainfall index is calculated as the production of two simulations: Rt = Xt ∙ Mt . (3.6) Since the distribution of the rainfall index is usually asymmetric, skewed and heavytailed, normal distribution is obviously not suitable. Therefore, the distribution of the simulated monthly rainfall index should be fitted with non-normal distribution, like lognormal, exponential and normal-inverse Gaussian (NIG) distribution by MLE. 3.2 Esscher Transform Besides the skewness and semi-heavy tails of the rainfall index distribution, the properties of the weather derivatives market also make it hard to price. Traditionally, derivatives are priced with the Black-Scholes model (1973), in which the risk-neutral probability measure ℚ equivalent to the physical measure ℙ is specified, such that all discounted tradable assets in the market are martingales under the ℚ measure. This unique risk-neutral measure ℚ is obtained by the Girsanov transform (also known as the Girsanov theorem), changing the drift term of the Brownian motion. But the general consensus is that weather is not a tradable asset, which means the markets for precipitation derivatives, even for weather derivatives, are inherently incomplete. It is impossible to construct a riskless hedge portfolio containing such derivatives. In this case, we cannot find a unique risk-neutral measure ℚ equivalent to the physical measure ℙ. Instead, there are many equivalent martingale measures. Hence, we choose to apply the Esscher transform to price the precipitation derivatives. According to Gerber and Shiu (1994), the Esscher transform provides a class of risk-neutral measures, which can be justified by assuming a representative investor who wants to maximize the expected utility with the market price of risk (MPR) parameterized by θ. The MPR is defined as a measure of the extra return or risk premium that investors demand to bear risk, which also reflects the risk attitude of market participants. The Esscher transform changes a probability density f(x) of a random variable X into a new
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