In 1995, Fischer Black submitted a paper on “Interest rates as options” when he was terminally ill with cancer. While publishing the paper (Journal of Finance, 1995, 50(5), 1371-1376), the Journal noted:
Note from the Managing Editor: Fischer Black submitted this paper on May 1, 1995. His submission letter stated: “I would like to publish this, though I may not be around to make any changes the referee may suggest. If I’m not, and if it seems roughly acceptable, could you publish it as is with a note explaining the circumstances?” Fischer received a revise and resubmit letter on May 22 with a detailed referee’s report. He worked on the paper during the Summer and had started to think about how to address the comments of the referee. He died on August 31 without completing the revision.
The paper contained an interesting idea to deal with the problem of negative interest rates – assume that the true or ‘shadow short rate’ can be negative, but the rate that we do observe is never negative because currency provides an option to earn a zero interest rate instead. Viewed this way, the interest rate can itself be viewed as an option (with a strike price of zero). What Black found attractive about this idea was that it made modelling easy: one could for example assume that the shadow rate follows a normal (Gaussian) distribution. Whenever the Gaussian distribution produces a negative interest rate, we simply replace it by zero. We do not need to assume a log normal or square root process just to avoid negative interest rates.
While interesting in theory, the model did not prove very popular in practice. But five years of zero interest rates in the US has changed this. Neither the lognormal nor the square root process can easily yield a persistent zero interest rate. Black’s shadow rate achieves this in a very easy and natural manner. More than the finance community, it the macroeconomics world that has rediscovered Black’s model. For example, Wu and Xia have a paper in which they show that macroeconomic models perform nicely even at the zero lower bound (ZLB) if the actual short rate is replaced by the shadow rate (h/t Econbrowser). The shadow rate has the same correlations with other macroeconomic variables at the ZLB as the actual rate has during normal times.
As I have mentioned previously on this blog, modelling interest rate risk at the ZLB is problematic and different clearing corporations have taken different approaches to the problem. Maybe, they should take Black’s shadow short rate more seriously.