Explosive behavior in a log-normal interest rate model

TL;DR

This paper analyzes a log-normal interest rate model revealing a phase transition at a critical volatility, which causes explosive behaviors and impacts derivative pricing, especially the caplet smile and skew.

Contribution

It identifies a sharp phase transition in the model's behavior at a critical volatility, highlighting limitations for high-volatility regimes.

Findings

Model exhibits two regimes: low and high volatility.

In high volatility, expectation values and convexity adjustments explode.

Low volatility regime approximates a log-normal caplet smile.

Abstract

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.