a) Compute the Black-Scholes call and put Vega by differentiation of the Black-Scholes function

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

with respect to the volatility parameter \(\sigma>0\), knowing that

\[
\begin{align*}
-\frac{1}{2}\left(d_{-}(T-t)ight)^{2} & =-\frac{1}{2}\left(\frac{\log (x / K)+\left(r-\sigma^{2} / 2ight)(T-t)}{|\sigma| \sqrt{T-t}}ight)^{2} \\
& =-\frac{1}{2}\left(d_{+}(T-t)ight)^{2}+(T-t) r+\log \frac{x}{K} \tag{6.43}
\end{align*}
\]

How do the Black-Scholes call and put prices behave when subjected an increase in the volatility parameter \(\sigma\) ?

b) Compute the Black-Scholes Rho by differentiation of the Black-Scholes function \(g_{\mathrm{c}}(t, x)\). How do the Black-Scholes call and put prices behave when subjected an increase in the interest rate parameter \(r\) ?