Analyze Newton-Raphson Convergence, Choose Sigma, and Derive Vega

You are valuing options under the Black–Scholes model and need to compute implied volatility. 1) Explain the Newton–Raphson method for root-finding in this context and detail the conditions under which it fails to converge (e.g., properties of the objective function and its derivative, step-size issues, and edge cases). 2) How would you select an initial volatility (sigma) seed before running Newton–Raphson, and what practical heuristics or data would you use to justify that choice? 3) Write the closed-form formula for Vega under Black–Scholes for a European option, define all variables and units, and explain how Vega’s magnitude influences Newton–Raphson updates.