Solution
= Solution
{c}
Differentiate the <Black-Scholes formula> for the <European put option> with respect to its strike. The <chain rule> gives
$$
\frac{\partial P}{\partial K}
=e^{-r\tau}\Phi(-d_-)
-Ke^{-r\tau}\phi(d_-)\frac{\partial d_-}{\partial K}
+S\phi(d_+)\frac{\partial d_+}{\partial K}.
$$
The final two terms cancel by both identities in part <3/a/i>. Hence the <put-price strike derivative> is
$$
\boxed{\frac{\partial P}{\partial K}=e^{-r\tau}\operatorname N(-d_-).}
$$