128 Valuation and Risk Management of Weather Derivatives: The Application of CME Rainfall Index Binary Contracts random variable u1(t) ~ U[0,1] and the initial value X0: 0, if u1(t) ≤ px0(t) , Xt = ⎧ ⎨ ⎩ (3.1) 1, otherwise. where px0(t) is the estimated transition probability function, and the use of either p00(t) or p10(t) is determined by Xt-1 = 0 or 1. The variable X0 equals to the initial value of X0. 3.1.2 Simulation of the Magnitude Model Recalling the PDF of the mixed exponential distribution followed by the magnitude model ( ) = ( ) 1( ) 1( ) + 1 ( ) 2( ) 2( ) , (3.2) where 0 ≤ α(t) ≤ 1 and 0 < β1(t) < β2(t) for all t. The magnitude model could be simulated with two independent uniform random variables u2(t), u3(t) ~ U[0,1], which are also independent of u1(t). The simulation of the magnitude model is Mt = Mmin - δtl n[u3(t)] , (3.3) where Mmin is the minimum magnitude of precipitation detected as rain (0.01 inch) 4 , and δt is given by β1(t), if u2(t) ≤ α(t), δt = ⎧ ⎨ ⎩ (3.4) β2(t), if u2(t) >α(t). The uniform random variable u2(t) decides which exponential distribution is chosen. If simulates enough times, the distributions will greatly mixed. Since the cumulative distribution function (CDF) of the exponential distribution is 4 Rainfall measuring less than 0.01 inch is defined as the trace precipitation.
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