Statistical Methods in Finance 2016

Abstract

Interest rate modelling with negative rates

by Vineet Virmani

What happens when a stochastic process crosses an inaccessible boundary? The earliest models of interest rates precluded simple stochastic processes like the Brownian Motion because they would produce negative interest rates pretty soon. Back then, interest rates were taken for granted to be non-negative because of the existence of cash. So interest rate models were chosen to make zero an inaccessible boundary, and Feller's boundary classification theorem became the bible of interest rate modelling. Until the Global Financial Crisis of 2008 came along that is, when interest rates actually hit the supposedly inaccessible boundary of zero. And the problem then was how to model interest rates after they had hit zero. In some models, zero was an absorbing barrier, so one was doomed: there was no escape from the crisis. In other models, zero was like a reflecting barrier: even if you somehow got there, you would bounce off instantly. But the rates not only remained near-zero for years, they also frequently journeyed between zero and positive values. Today, of course, the interest rates are actually negative in many countries. So the barrier was not just hit, it was breached. In many models, that meant dealing with logarithms and square roots of negative quantities. In this paper, we discuss some solutions that have been proposed to deal with these issues. The popular choices range from standard Brownian Motion to shifted Geometric Brownian Motion to regime switching models that sort of magically transport us back and forth across the inaccessible boundary. None of these are satisfactory for risk management purposes. Likely some new and deeper mathematics is needed.

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