130 Valuation and Risk Management of Weather Derivatives: The Application of CME Rainfall Index Binary Contracts probability density f(x;θ) with MPR parameter θ. ( ; ) = ( ( ) ) . (3.7) The Esscher transform can also be extended to stochastic processes. We make Xt, 0 ≤ t ≤ T a Lévy process with the Lévy-Khintchine triplet (σ2, υ, γ) and assume the Lévy measure υ is such that ( ) < | | 1 . According to Cont and Tankov (2004), the Radon-Nikodym derivative corresponding to the Esscher transform is | | = [ ] = ( ) , (3.8) where κ(θ) = ln E[eθX] is the logarithm of the moment generating function of X. A big advantage of Esscher transform is the great applicability to any distribution’s probability density function f (x). In this paper, we choose the normal-inverse Gaussian distribution, whose probability density function NIG(μ, α, β, σ) remains in the original form after the Esscher transform, as NIG(μ, α, β + θ, σ). 3.3 Pricing Formulas The subject for discussion in this paper is CME rainfall index binary contracts. And we define the underlying rainfall index (e.g., monthly rainfall index) I(τ1, τ2) as ( 1, 2) = 2 = 1 , [ 1, 2] , (3.9) which is the sum of the daily rainfall index Rt, and [τ1, τ2] is the particular accumulation period. The payoff of a rainfall index binary call with strike price K is 10,000, I(τ1, τ2) > K. C(τ2, τ1, τ2) = ⎧ ⎨ ⎩ (3.10) 0, otherwise. According to Table A.1 in Appendix A, the price of CME rainfall index binary
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