# Prof. Jayanth R. Varma’s Financial Markets Blog

A blog on financial markets and their regulation

## Revisiting Fischer Black’s deathbed paper

November 29, 2013

Posted by on 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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