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.

Keywords: Libor Reform, RFR, OIS, Risk-Free Rates, Sonia, Fallback, Arc-Sine, Occupation Time, Median, Percentile, Fallback Spread, Libor Adjustment Spread, Quantile

JEL Classification: C61, G13, G15, G18, G21, C51