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?


1 Answer 1


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.

  • $\begingroup$ When rates are positive, an individual investor can earn the risk-free rate by purchasing a money-market fund or short-term bank certificate of deposit. My understanding is that in the Eurozone, most indivdiual investors do not face negative interest rates, but investors with more than 1 million euros do. So I wonder if the risk-free that fits that data empirically is zero or the official ECB deposit rate. $\endgroup$
    – Fortranner
    Sep 4, 2019 at 12:39
  • $\begingroup$ I think you can verify that the risk-free rate implied in option prices is indeed negative by applying the put-call parity. For instance, yesterday's settlement prices at Eurex for Dec '19 12150 options are C=383.20 and P=426.90 with S=12126.50 and therefore put-call parity only holds when the risk-free rate is negative. $\endgroup$
    – Cornholio
    Sep 6, 2019 at 7:29

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