# From BS formula how to show Euro-call values go up and Euro-put values go down with interest rates?

The BS formula gives, as quoted from Wikepdia:

{\displaystyle {\begin{aligned}C(S_{t},t)&=N(d_{1})S_{t}-N(d_{2})Ke^{-r(T-t)}\\d_{1}&={\frac {1}{\sigma {\sqrt {T-t}}}}\left[\ln \left({\frac {S_{t}}{K}}\right)+\left(r+{\frac {\sigma ^{2}}{2}}\right)(T-t)\right]\\d_{2}&=d_{1}-\sigma {\sqrt {T-t}}\\\end{aligned}}}

in which all notations are standard (I don't think any notational ambiguity exists; otherwise please refer to this article directly.)

I want to show that when everything else is fixed, $C$ will go up with $r$. Or alternatively, that $\partial C/\partial r\ge 0$. But it turns out the sign of the partial derivative is not that obvious.

To convince myself somehow, I drew a couple of $C$-$r$ plots on my computer using various sets of parameters and found out the curve always went upwards as expected. But I still want a rigorous proof in the mathematical sense. So am I missing something here? Is there anybody who can help? Thanks.

• At time $t$, $S_t$ is a known quantity, we do not treat it as a function of $t$. At time $0$, $S_t$ depends on $r$, but is also a random number. – Gordon Mar 25 '17 at 13:15
• I added some details for how to convert $\phi(d_2)$ to $\phi(d_1)$. – Gordon Mar 25 '17 at 13:48