An interest rate derivative pricing model that was developed by Black and Karasinski in 1991 in an attempt to overcome a major flaw in the Ho-Lee and Hull-White models, literally the occurrence of negative short-term interest rates. The value of the short-term rate in this model, as in the Hull-White model, follows a Wiener process (a drawing from a standard normal stochastic process). However, this value at a future time is normal in the previous models, whereas it is lognormal in the Black-Karasinski (BK) model. Mathematically, the Hull-White model determines the short-term rate as follows:
dr = [Θ(t)- ar] dt + σ dz
where: dr is the change in the short-term rate over a small interval; r is the short-term interest rate; Θ(t) is a function of time which determines the average course followed by (r) (supposed to be consistent with spot zero yield curve; a is a mean reversion rate (proxy for relationship between short and long term rate volatilities); dt is a small shift in time; σ is the annual standard deviation of interest rate; dz is a Wiener process.
The BK model, however, determines the short-term rate using this formula:
d ln r = [Θ(t)- a (t) ln r] dt + σ (t) dz
where: ln r is the lognormal of interest rate.
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