152 Valuation and Risk Management of Weather Derivatives: The Application of CME Rainfall Index Binary Contracts Appendix D Proof of Formula = ( + 100 ) 2 + + 100 2 . (4.2) Proof. Assume a random variable X follows NIG distribution: X~NIG(μ, α, β, σ), and Xℚθ~NIG(μ, α, β+θ, σ) under the risk-neutral measure ℚθ of Esscher transform. The Expectation of Xℚθ is ( ) = + ( + ) 2 ( + )2. Since the prices of rainfall index binary options are quoted as the underlying index, which could be considered as the expectation of rainfall index by investors, the market price of risk θ of binary call can be calculated as follows: ( ) = + ( + ) 2 ( + )2 = + 100 . [ + ( + )]2 = [ 2 ( + )2] + 100 2 , 2+ + 100 2 ( + )2 +2 ( + ) = 2 + 100 2 2. So we can solve this quadratic equation, and θ is the solution minus β: = ( + 100 ) 2+ + 100 2 .
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