In order to prevent the possibility of negative rates, some prefer to work with what can be described as the lognormal version of the HW model. This alternative can also be defined as a restricted version of the Black and Karasinski (BK) model. The general specification for the BK model has a total of three timedependent functions, which allows the model to be fit to the zero curve, rate volatility curve, as well as the atthemoney differential rate curve. 



where, 
ln r = the natural logarithm of the instantaneous short rate, r q(t) = a time dependent mean reversion level a(t) = a time dependent mean reversion rate s(t) = time dependent volatility dz = an increment in a standard Weiner process 

If the mean reversion rate and volatility are assumed to be constant, then the resulting simplified model is fit to the zero curve only. 

Construction of trees consistent with this process closely follows the procedures outlined in both Building HW Trinomial Trees with a Constant Timestep and Building HW Trinomial Trees with Changing Timestep. However, because the tree contains values for x = ln r as opposed to the short rate itself, the displacement factors at each timestep are now calculated as: 



to reflect the fact that the short rate at each node must be recovered from the corresponding value of x_{i, j}: 




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