## Weather derivative risk measures for extreme events.(English)Zbl 1414.91371

Summary: We consider pricing weather derivatives for use as protection against weather extremes by using max-stable processes to estimate risk measures. These derivatives are not currently traded on any open markets, but their use could help some institutions manage weather risks from extreme events. The central challenge is to model the dependence of payments, which increases the risk of holding multiple weather derivatives. The method described utilizes results from spatial statistics and extreme value theory to first model extremes in the weather as a max-stable process, and then simulate payments for a general collection of weather derivatives. As the joint likelihood function for max-stable processes is unavailable, we use two approaches: The first is based on the composite likelihood, and the second is based on approximate Bayesian computing (ABC). Both capture the spatial dependence of payments. To incorporate parameter uncertainty into the pricing model, we use bootstrapping with the composite likelihood approach, while the ABC method naturally incorporates parameter uncertainty. We show that the additional risk from the spatial dependence of payments can be quite substantial, and that the methods discussed can compute standard actuarial risk measures in both a frequentist and Bayesian setting.

### MSC:

 91G20 Derivative securities (option pricing, hedging, etc.) 91G70 Statistical methods; risk measures 62G32 Statistics of extreme values; tail inference

R; ismev; QRM
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### References:

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