126 Valuation and Risk Management of Weather Derivatives: The Application of CME Rainfall Index Binary Contracts Table 1 Transition States, Marks and Counts Transition No Rain to No Rain No Rain to Rain Rain to No Rain Rain to Rain Marks 00 01 10 11 Counts n00(t) n01(t) n10(t) n11(t) Step 4: The historical transition probabilities are calculated as (2.13) and (2.14). As mentioned above, p01(t) = 1 - p00(t) and p11(t) = 1 - p10(t). 00( ) = 00( ) 00( ) + 01( ) . (2.13) 10( ) = 10( ) 10( ) + 11( ) . (2.14) The logarithm of the likelihood function is: U = U0 + U1 . (2.15) 0 = 00( ) log 00( ) + 01( ) log[1 00( )] 365 =1 . (2.16) 1 = 10( ) log 10( ) + 11( ) log[1 10( )] 365 =1 , (2.17) since 10 0 ( ) = 0 and 00 1 ( ) = 0, U can be considered the sum of U0 and U1, which are log-likelihood functions of p00(t) and p10(t). Therefore, we could maximize U0 and U1 separately, but notice that the constraints are 0 ≤ px0(t) ≤ 1. (2) The log-likelihood functions U2 for the magnitude model: We assign Φ2 to be a vector whose elements are the coefficients of the Fourier series describing α, β1, and β2. The maximum likelihood estimate Φ2 of Φ2 is found by maximizing U2, the logarithm of likelihood function L(Φ2│Mt ):
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