Prof. Jayanth R. Varma’s Financial Markets Blog

A blog on financial markets and their regulation

Revisiting Fischer Black’s deathbed paper

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.

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