Consider a One-Factor HJM Model on a Binomial Model with Time

Consider a one-factor HJM model on a binomial model with time steps of one year and a probability of an up shift of $q$ . Each $j$ -th forward rate has constant volatility $\sigma ( j )$ . For each forward period of one year what is the risk-neutral drift term in the $n$ -th period equal to?

A) $\alpha ( n ) = \ln \left[ q \exp \left( \sum _ { j = 1 } ^ { n } \sigma ( j ) \right) + ( 1 - q ) \exp \left( - \sum _ { j = 1 } ^ { n } \sigma ( j ) \right) \right] - \sum _ { j = 1 } ^ { n - 1 } \alpha ( j )$
B) $\alpha ( n ) = \ln [ q \exp ( \sigma ( n ) ) + ( 1 - q ) \exp ( - \sigma ( n ) ) ]$
C) $\alpha ( n ) = \exp \left[ q \ln \left( \sum _ { j = 1 } ^ { n } \sigma ( j ) \right) + ( 1 - q ) \ln \left( - \sum _ { j = 1 } ^ { n } \sigma ( j ) \right) \right] - \sum _ { j = 1 } ^ { n - 1 } \alpha ( j )$
D) $\alpha ( n ) = \exp [ q \ln ( \sigma ( n ) ) + ( 1 - q ) \ln ( - \sigma ( n ) ) ]$

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