With the notation used in this chapter:
(a) What is $N^{\prime}(x)$ ?
(b) Show that $S N^{\prime}\left(d_1\right)=K e^{-r(T-t)} N^{\prime}\left(d_2\right)$, where $S$ is the stock price at time $t$ and
$$
d_1=\frac{\ln (S / K)+\left(r+\sigma^2 / 2\right)(T-t)}{\sigma \sqrt{ } T-t}, \quad d_2=\frac{\ln (S / K)+\left(r-\sigma^2 / 2\right)(T-t)}{\sigma \sqrt{ } T-t}
$$
(c) Calculate $\partial d_1 / \partial S$ and $\partial d_2 / \partial S$.
(d) Show that when $c=S N\left(d_1\right)-K e^{-r(T-t)} N\left(d_2\right)$, it follows that
$$
\frac{\partial c}{\partial t}=-r K e^{-r(T-t)} N\left(d_2\right)-S N^{\prime}\left(d_1\right) \frac{\sigma}{2 \sqrt{T-t}}
$$
where $c$ is the price of a call option on a non-dividend-paying stock.
(e) Show that $\partial c / \partial S=N\left(d_1\right)$.
(f) Show that $c$ satisfies the Black-Scholes-Merton differential equation.
(g) Show that $c$ satisfies the boundary condition for a European call option, i.c., that $c=\max (S-K, 0)$ as $t \rightarrow T$.