The Black-Karasinski Model: Lognormal Rates with Mean Reversion

Swaption traders in the 1990s found that Gaussian short-rate models systematically misprice the swaption volatility surface: Gaussian volatility declines with the level of rates, but observed vol is proportional to rates. The Black-Karasinski (BK) model, published in 1991, fixes this by modelling the logarithm of the short rate as an Ornstein-Uhlenbeck process, delivering both guaranteed positive rates and a proportional-volatility structure that tracks the market's own convention for quoting cap and swaption volatilities in Black's formula. The price is tractability: no closed-form bond price exists, and all pricing must go through a trinomial tree. Understanding where BK sits relative to Hull-White, BDT, and the CIR model sharpens the fundamental tractability-versus-realism trade-off running through all short-rate modelling.