Arc-Sine Law and the Libor Reform

Abstract

The fallback "Libor adjustment spread" spread to be used for calculating Libor replacement rates in the future is defined as the median (50%-th percentile) of five years of historical observations of the spread between Libor and compounded OIS rates, calculated on the future date of Libor cessation announcement. Some of the observations entering this calculation have already occurred and some are still in the future. In this note we assert that the future realized median is a non-linear function of future, yet unknown, spread observations and therefore its fair value calculation must account for spread dynamics and not just forward values. We propose a model of the future evolution of spreads and derive a very numerically-efficient algorithm for calculating the fair value of the median that incorporates both the historical observations and the future dynamics of the spread. We establish that, given our model, the market expectations of the fallback spreads are at, or somewhat above, the upper range of theoretically-justifiable values. The approximation method we develop is based in part on the Arc-Sine Law and should be of independent interest to math finance professionals.