(Hagan et al. (2002)) Consider the European option priced as $e^{-r T} \mathbb{E}^{*}\left[\left(S_{T}-ight.ight.$ $\left.K)^{+}ight]$in a local volatility model $d S_{t}=\sigma_{\text {loc }}\left(S_{t}ight) S_{t} d B_{t}$. The implied volatility $\sigma_{\text {imp }}\left(K, S_{0}ight)$, computed from the equation

$$
\operatorname{Bl}\left(S_{0}, K, T, \sigma_{\mathrm{imp}}\left(K, S_{0}ight), right)=\mathrm{e}^{-r T} \mathbb{E}^{*}\left[\left(S_{T}-Kight)^{+}ight]
$$

is known to admit the approximation

$$
\sigma_{\mathrm{imp}}\left(K, S_{0}ight) \simeq \sigma_{\mathrm{loc}}\left(\frac{K+S_{0}}{2}ight)
$$

a) Taking a local volatility of the form $\sigma_{\text {loc }}(x):=\sigma_{0}+\beta\left(x-S_{0}ight)^{2}$, estimate the implied volatility $\sigma_{\text {imp }}(K, S)$ when the underlying asset price is at the level $S$.

b) Express the Delta of the Black Scholes call option price given by

$$
\mathrm{Bl}\left(S, K, T, \sigma_{\mathrm{imp}}(K, S), right)
$$

using the standard Black-Scholes Delta and the Black-Scholes Vega.