Fischer Black published a paper shortly before his death in 1995 considering interest rates as having embedded options when considering the "shadow real interest rate" and the real interest rate. From the paper:

The nominal short rate is the "shadow real interest rate" (as defined by the investment opportunity set) plus the inflation rate, or zero, whichever is greater. Thus the nominal short rate is an option.

The paper is very short, but was published without incorporating details from a referee review.

Can someone flesh out more the shadow real interest rate and how it is used (as proposed) in quantitative models of interest rates? Has this line of research been picked up on in the last twenty five years?

  • $\begingroup$ I find this paper amazing, because it defines money. We have had many many papers about money, but Fischer Black defines money in terms of something we knew about: options. The work has been continued by Wu and by Krippner. I will write more about the shadow rate on this page on StackExchange. $\endgroup$
    – user118316
    Mar 17, 2021 at 3:26

1 Answer 1


Nominal rates have been negative in Europe for a while now. So the idea that rates should be non negative (the usual argument being that one would keep his money in his wallet rather than paying to lend it) is no longer a "first principle" of mathematical finance.

  • $\begingroup$ I agree that’s the idea of the paper. Fischer Black might point out that rates in Europe are only slightly negative (-0.5% area), and there is resistance to moving deposit rates for retail investors below zero. So the idea has some merit still. $\endgroup$
    – dm63
    Dec 2, 2019 at 11:21
  • $\begingroup$ Do you think the observed negative nominal rates invalidates the idea proposed in this paper? (Asking in earnest). It seems to me the lower bound for the trees, the embedded option, could then just be translated down to the new bound, which was previously thought to be zero. $\endgroup$
    – Jared
    Dec 2, 2019 at 22:49
  • $\begingroup$ You are right and that is the comment made by @dm63. But how do you choose the option "strike" ? You have to make an assumption on what the central bank minimum deposit rate would be (since it essentially drives the interbank market), and recent past has shown that it is not zero. $\endgroup$ Dec 3, 2019 at 10:25

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