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

  • $\begingroup$ Now that you accepted the answer, can you please add a simple comment to explain what the issue was ? Just to close it. $\endgroup$
    – Dom
    Feb 4 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.


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