I've implemented in the past a calibration method for Hull-White (HW) that estimates Sigma and Alpha based on market caplet vols. To do this I minimized the least-squares differences between market vols/prices and calculated values based on HW closed-form formula for pricing options on pure discount bonds (modified Black-Scholes (BS) in terms of Sigma and Alpha).I’m interested in doing something similar now for Black-Karasinski (BK), but since BK uses a lognormal rate distribution, I cannot use the same modified BS closed formula I was using for HW, which - being BS - assumes normally distributed interest rates.So far I have not been able to find a closed-form formula I can use for BK calibration. Does anyone know of one, or is a numerical approach (i.e. lattices) the only way to calibrate BK to market data?Thanks Much!-LB

As far as I know, you'll have to use lattices

Why would you use Black-Karasinski?

use trees or lattices