# Risk free rate in black-scholes model

Currently reading A. Damodaran‘s book Investment Valuation. In chapter 5 in order to value an option using black-scholes model he adjusts risk free rate using the following formula: $$1-e^{-r}$$ I. E. If given risk free rate is $$3,6%$$ it becomes $$1-e^{(-0,036)} = 3,54\%$$ Why is that? I wasn’t able to find any explanation on that

• Now that you accepted the answer, can you please add a simple comment to explain what the issue was ? Just to close it.
– Dom
Feb 4, 2021 at 22:44

Black Scholes uses a continuously compounded rate $$r$$. To go from a $$T$$-year annually compounded rate $$\hat{r}$$ to a $$T$$-year continuously compounded $$r$$ you use the formula

$$e^{rT} = (1+\hat{r})^T$$

So to solve for the Black-Scholes continuously compounded rate you take logs and simplify which gives

$$r = \ln(1+\hat{r})$$.

This is what Damodoran quotes on page 132 (chapter 5, page 12) on the second edition of his book (I found a free online version).

So if $$\hat{r} = 0.036$$ then $$r=0.0353$$. It's very close to the result you got, but not identical.

If you are converting in the opposite direction in order to solve for $$\hat{r}$$ the formula is

$$\hat{r} = e^r - 1$$

which is not the same either.

I don't know where the formula you get comes from. It's almost the correct formula - but the signs of the interest rates are incorrectly reversed.