Using Black-Scholes with Negative Interest Rates

Summary

The document asks whether negative short-term risk-free rates require changes to the Black-Scholes option pricing formula. Its answer says the standard model does not require the rate to be positive, so a negative rate can be used directly. Under the stated constant-rate assumption, lowering the rate lowers the price of a call option.

It notes that interest rates may instead be modeled as time-varying or random, citing Hull-White and shifted CIR as examples of extensions. These approaches can represent rates more realistically, but the document does not derive them or provide comparisons or numerical evidence. It also cautions that Black-Scholes remains a simplified framework whose assumptions do not capture all real-world price dynamics, so applying it requires care.

Key ideas

• The Black-Scholes model does not require the risk-free rate to be positive.
• A negative rate can be substituted directly when the model’s constant-rate assumption is retained.
• The answer states that lower rates reduce call option prices.
• Time-dependent or stochastic rates can be handled with extensions such as Hull-White or shifted CIR.
• Black-Scholes remains a simplified model and may not represent real market dynamics fully.

Tags

# Option pricing with negative short-term interest rates


# Option pricing with negative short-term interest rates












In countries with negative short-term risk-free interest rates, do you just use a negative "r" in the Black-Scholes formula, or do adjustments need to be made?

## Answer by Kevin (score 3)

https://quant.stackexchange.com/a/47486

The Black Scholes world does not assume $r>0$. So, you can just plug in a negative number for $r$. Note that the lower $r$, the lower a call option price.

The only assumption regarding interest rates is that they are constant during the lifetime of the option.

It is possible though to generalise the model and allow for time-dependent or random interest rate (e.g. hull white model or shifted CIR model). This can provide a more accurate incorporation of interest rates.

Of course, other caveats remain and the Black Scholes setting is very simplistic and fails to capture real world price dynamics. One ought to be careful to apply it in real world.

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