  We consider an interest rate model with log-normally distributed rates in the
terminal measure in discrete time. Such models are used in financial practice
as parametric versions of the Markov functional model, or as approximations to
the log-normal Libor market model. We show that the model has two distinct
regimes, at high and low volatilities, with different qualitative behavior. The
two regimes are separated by a sharp transition, which is similar to a phase
transition in condensed matter physics. We study the behavior of the model in
the large volatility phase, and discuss the implications of the phase
transition for the pricing of interest rate derivatives. In the large
volatility phase, certain expectation values and convexity adjustments have an
explosive behavior. For sufficiently low volatilities the caplet smile is
log-normal to a very good approximation, while in the large volatility phase
the model develops a non-trivial caplet skew. The phenomenon discussed here
imposes thus an upper limit on the volatilities for which the model behaves as
intended.
