It Is Time to Shift Log-Normal

Abstract

The Libor market model (LMM) and other models for interest rate processes assume the instantaneous short rate is log-normal. It made sense that the rate should be bounded below by zero and that volatility in basis points be proportional to the level of the rate. But there have been some anomalous periods when the model has exhibited substantial problems, such as the 2008 financial crisis and during periods when interest rates fell into negative territory in several major economies, and the log-normal assumption was plainly violated. In this article, the authors show how the assumption that the Libor rate is log-normal can be replaced by assuming 1/(1 + Libor), that is, the price of a zero-coupon bond, is log-normal instead. Under this assumption, the forward short rate follows a shifted log-normal and the drift term in the short rate equation must be modified. With this assumption, the rate is approximately log-normal when rates are high, but it becomes closer to normal when rates are low. The performance of the modified model is illustrated for caps, swaps, and swaptions.