Implied volatility

(a) Hull's call is quoted at 4.76 instead of the model value 4.759422. Without a solver, estimate the implied volatility using vega, then confirm with one Newton step. (b) Derive the Brenner–Subrahmanyam approximation for an option struck at the forward: $\sigma\approx\dfrac{c\,e^{rT}}{F}\sqrt{\dfrac{2\pi}{T}}$. State the error order. (c) Apply (b) to the Dec 24,700 call (476.90). Why is the answer 12.68% rather than the exact 12.43%? (d) Prove that $\partial^2c/\partial\sigma^2=\mathcal V d_1d_2/\sigma$, and explain why starting Newton at $\sigma^*=\sqrt{2|\ln(F/K)|/T}$ is safe while starting at a very small $\sigma$ for a deep OTM option is not.