a) Check that the Black-Scholes formula for European call options

\[ g_{\mathrm{c}}(t, x)=x \Phi\left(d_{+}(T-t)ight)-K \mathrm{e}^{-(T-t) r} \Phi\left(d_{-}(T-t)ight), \]

satisfies the following boundary conditions:

i) at \(x=0, g_{\mathrm{c}}(t, 0)=0\), ii) at maturity \(t=T\),

\[
g_{\mathrm{c}}(T, x)=(x-K)^{+}= \begin{cases}x-K, & x>K \\ 0, & x \leqslant K\end{cases}
\]
and \[
\lim _{t earrow T} \Phi\left(d_{+}(T-t)ight)= \begin{cases}1, & x>K \\ \frac{1}{2}, & x=K \\ 0, & x\]

\[
g_{\mathrm{c}}(T, x)=(x-K)^{+}= \begin{cases}x-K, & x>K \\ 0, & x \leqslant K\end{cases}
\]
and \[
\lim _{t earrow T} \Phi\left(d_{+}(T-t)ight)= \begin{cases}1, & x>K \\ \frac{1}{2}, & x=K \\ 0, & x\]

iv) as time to maturity tends to infinity, \[
\lim _{T ightarrow \infty} \mathrm{Bl}_{\mathrm{p}}\left(S_{t}, K, \sigma, r, T-tight)=0, \quad t \geqslant 0 \]