Interest Rates as Options



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    • Goldman, Sachs & Co. I am grateful to Chi-Fu Huang and Wei-Tong Shu for discussions of these issues, and to Wei-Tong Shu for producing the graphs. [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.]


Since people can hold currency at a zero nominal interest rate, the nominal short rate cannot be negative. The real interest rate can be and has been negative, since low risk real investment opportunities like filling in the Mississippi delta do not guarantee positive returns. The inflation rate can be and has been negative, most recently (in the United States) during the Great Depression. 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. Longer term interest rates are always positive, since the future short rate may be positive even when the current short rate is zero. We can easily build this option element into our interest rate trees for backward induction or Monte Carlo simulation: just create a distribution that allows negative nominal rates, and then replace each negative rate with zero.