Shadow Real Rates and the Option in Nominal Rate Models

Summary

The document introduces Fischer Black’s idea that the nominal short rate can be viewed as an option: it is the greater of zero and the sum of a shadow real interest rate and inflation. The shadow rate is described as tied to the investment opportunity set, while the zero floor reflects the proposed lower bound in the model. This framing treats nominal rates as having an embedded option-like feature rather than as a simple additive quantity.

The discussion is brief and mainly poses questions about how to define the shadow rate and use it in quantitative interest-rate models. It notes that Black’s paper was short and published without incorporating details from a referee review. A response points out that negative nominal rates observed in Europe challenge the conventional claim that nominal rates must remain nonnegative. The excerpt gives no model implementation, empirical tests, or account of subsequent research, so it offers a conceptual starting point rather than a worked method.

Key ideas

• Black’s proposal expresses the nominal short rate as the greater of zero and the sum of a shadow real rate and inflation.
• The shadow real rate is linked to the investment opportunity set.
• The option interpretation depends on a zero lower bound for nominal rates.
• Observed negative nominal rates challenge the usual argument for a strict zero floor.
• The excerpt does not explain calibration, implementation, or empirical performance.

Tags

# Interest Rates as Options


# Interest Rates as Options












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?

## Answer by Antoine Conze (score 6)

https://quant.stackexchange.com/a/50051

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.

Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)

This summary was written by Stratmill's research agent from the original; it is not a copy of the source.