# On short-rate-models: Black-Karasinski (with constant parameters) compared to Vasicek

When modelling the term structure of interest rates, one widespread possibility is using the Black-Karasinski model, which is given by the following stochastic process

$$d\ln{r}=[\theta(t)-a(t)\ln{r}]dt+\sigma(t)dt,$$

where $\theta(t)$, $a(t)$ and $\sigma(t)$ are parameters which can be adjusted so that the model fits the current term structure. This is why it is considered a no-arbitrage model.

If we now remove the time-dependence of the parameters, we end up at an equilibrium-model with a log-normal distribution of the interest rate $r$. This is basically a logarithmic equivalent of the so-called Vasicek model, which is given by

$$dr=a(b-r)dt+\sigma dz.$$

The most striking difference between the two models is now the fact that in the logarithmic one, negative interest rates cannot occur. In literature, the absence of negative rates is presented as an advantage. However, due to negative interest rates actually appearing in derivative markets, this statement has to be reconsidered.

What I want to know is the following: are there any other advantages (if there are any) of using a log-normal equilibrium model (Black-Karasinski with constant parameters) in comparison to a normally distributed equilibrium model (Vasicek)?

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In chapter 3 (One-factor short-rate models) they have a very nice table which lists some of the properties of instantaneous short rate models. In both of your models you know the distribution of $r_t$. The huge difference between the two models is the following: For Vasicek you have both, analytical bond and analytical option prices. Instead, the Black-karasinski model does not provide an analytical solution to neither bond nor option prices. This is the main difference and of course a disadvantage of the Black-Karasinski model.